ON CERTAIN WEYL MODULES AND SIMPLE MODULES FOR ALGEBRAIC GROUPS OF TYPE B AND D
By ELIZABETH WIGGINS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2015
1 c 2015 Elizabeth Wiggins
2 To Bonnie Binford Rees
3 ACKNOWLEDGMENTS First and foremost, I want to thank my advisor, Dr. Peter Sin. None of this would be possible without his patience, guidance, and support over the last five and a half years. He is a wonderful advisor and mentor. I would also like to thank the rest of my supervisory committee for their comments and kind words: Dr. Janice Carrillo, Dr. Richard Crew, Dr. Kevin Keating,
and Dr. Alexandre Turull. Also thank you to the creators of SAGE (www.sagemath.org) for developing such a useful software. Many examples were computed using this software. The support of the Mathematics Department at UF has been incredible. Thank you to the faculty, staff, and my fellow graduate students here. This department has provided the support and coffee needed to accomplish this. Thank you to my family, for being encouraging and understanding why I’ve been in school for so long, especially my parents and my siblings. I hope that I make them proud. Also to Christine Leverenz and Will Harris, thank you for seeing potential in me and inspiring me to get where I am today. They began and nurtured my love of mathematics. I am forever indebted to them. Finally, thank you to the love of my life, Ben Russo. I would not have accomplished this without his unending support. He always believed that I could be here.
4 TABLE OF CONTENTS page ACKNOWLEDGMENTS...... 4 LIST OF TABLES...... 7 LIST OF FIGURES...... 10 ABSTRACT...... 11
CHAPTER 1 INTRODUCTION...... 13 2 LIE THEORY AND CHEVALLEY GROUPS...... 16 2.1 Lie Algebras...... 16 2.1.1 Root Systems...... 18 2.1.2 A Chevalley Basis...... 20 2.1.3 Weights...... 21 2.2 The Universal Enveloping Algebra...... 24 2.2.1 Representations of g ...... 24 2.2.2 Weyl’s Character Formula...... 27 2.2.3 Kostant Z-form...... 27 2.3 Chevalley Groups...... 29 2.3.1 Construction of the Chevalley Groups...... 29 2.3.2 Subgroups of a Chevalley Group...... 30 2.3.3 Bruhat Decomposition...... 30 2.3.4 Weyl Modules...... 32 3 ALGEBRAIC GROUPS...... 34 3.1 Algebraic Groups and Chevalley Groups...... 34 3.1.1 Weights of G with Respect to a Maximal Torus...... 35 3.1.2 Simply Connected Chevalley Groups...... 37 3.2 Weyl Modules...... 37 4 JANTZEN SUM FORMULA AND APPLICATIONS...... 41 4.1 Tools for Calculation...... 42 4.2 Weyl Modules of Simple Algebraic Groups of Type B4...... 44 4.3 Weyl Modules of Simple Algebraic Groups of Type D4 ...... 49 5 BUILDINGS...... 56 5.1 Classical Groups and Chevalley Groups...... 56 5.1.1 (B, N)-pairs...... 57 5.1.2 Twisted Groups...... 58
5 5.2 Buildings...... 60 5.3 Oppositeness in Buildings...... 62 5.3.1 Incidence Relations...... 62 5.3.2 Oppositeness...... 62 5.3.3 Chevalley Groups...... 63 5.3.4 Twisted Groups...... 64 5.4 Examples...... 65 5.4.1 Groups of Type B4 ...... 65 5.4.2 Groups of Type D4 ...... 67
APPENDIX
A TABLES FOR CALCULATIONS IN GROUPS OF TYPE B4 ...... 69
B TABLES FOR CALCULATIONS IN GROUPS OF TYPE D4 ...... 75 REFERENCES...... 81 BIOGRAPHICAL SKETCH...... 83
6 LIST OF TABLES Table page
A-1 B4, p = 2. Noncontributors for λ = ω4...... 69
A-2 B4, p = 3. Noncontributors for λ = ω4...... 69
A-3 B4, p = 3. Noncontributors for λ = 2ω4...... 70
A-4 B4, p = 3. Contributors for λ = 2ω4...... 70
A-5 B4, p = 5. Noncontributors for λ = ω4...... 70
A-6 B4, p = 5. Noncontributors for λ = 2ω4...... 70
A-7 B4, p = 5. Contributors for λ = 2ω4...... 70
A-8 B4, p = 5. Noncontributors for λ = 3ω4...... 71
A-9 B4, p = 5. Contributors for λ = 3ω4...... 71
A-10 B4, p = 5. Noncontributors for λ = ω1 + ω4...... 71
A-11 B4, p = 5. Noncontributors for λ = 4ω4...... 71
A-12 B4, p = 5. Contributors for λ = 4ω4...... 71
A-13 B4, p = 5. Noncontributors for λ = ω1...... 71
A-14 B4, p = 5. Noncontributors for λ = 2ω1...... 72
A-15 B4, p = 5. Contributors for λ = 2ω1...... 72
A-16 B4, p ≥ 7. Noncontributors for λ = (p − 5)ω4...... 72
A-17 B4, p ≥ 7. Noncontributors for λ = (p − 4)ω4...... 72
A-18 B4, p ≥ 7. Noncontributors for λ = (p − 3)ω4...... 72
A-19 B4, p ≥ 7. Contributors for λ = (p − 3)ω4...... 72
A-20 B4, p ≥ 7. Noncontributors for λ = (p − 2)ω4...... 72
A-21 B4, p ≥ 7. Contributors for λ = (p − 2)ω4...... 73
A-22 B4, p ≥ 7. Noncontributors for λ = ω3 + (p − 6)ω4...... 73
A-23 B4, p ≥ 7.Contributors for λ = ω3 + (p − 6)ω4...... 73
A-24 B4, p ≥ 7. Noncontributors for λ = ω1 + (p − 4)ω4...... 73
A-25 B4, p ≥ 7. Contributors for λ = ω1 + (p − 4)ω4...... 73
7 A-26 B4, p ≥ 7. Contributors for λ = (p − 1)ω4...... 73
A-27 B4, p ≥ 7. Contributors for λ = ω2 + (p − 7)ω4...... 73
A-28 B4, p ≥ 7. Contributors for λ = ω1 + ω3 + (p − 7)ω4...... 74
A-29 B4, p ≥ 7. Contributors for λ = 2ω1 + (p − 5)ω4...... 74
B-1 D4, p = 2. Noncontributors for λ = µ...... 75
B-2 D4, p = 2. Contributors for λ = µ...... 75
B-3 D4, p = 2. Noncontributors for λ = ω1...... 75
B-4 D4, p = 3. Noncontributors for λ = µ...... 76
B-5 D4, p = 3. Noncontributors for λ = 2µ...... 76
B-6 D4, p = 3. Contributors for λ = 2µ...... 76
B-7 D4, p = 3. Noncontributors for λ = ω2...... 76
B-8 D4, p = 3. Contributors for λ = ω2...... 76
B-9 D4, p = 3. Noncontributors for λ = ω1 + µ...... 76
B-10 D4, p = 3. Contributors for λ = ω1 + µ...... 77
B-11 D4, p = 5. Noncontributors for λ = µ...... 77
B-12 D4, p = 5. Noncontributors for λ = 2µ...... 77
B-13 D4, p = 5. Contributors for λ = 2µ...... 77
B-14 D4, p = 5. Noncontributors for λ = 2ω1...... 77
B-15 D4, p = 5. Noncontributors for λ = 3µ,...... 77
B-16 D4, p = 5. Noncontributors for λ = 4µ...... 77
B-17 D4, p = 5. Contributors for λ = 4µ...... 78
B-18 D4, p = 5. Noncontributors for λ = ω2 + 2µ...... 78
B-19 D4, p = 5. Contributors for λ = ω2 + 2µ...... 78
B-20 D4, p = 5. Noncontributors for λ = ω1 + 3µ...... 78
B-21 D4, p = 5. Contributors for λ = ω1 + 3µ...... 78 D p 7 = p−3 B-22 4, ≥ . Noncontributors for λ 2 µ...... 78 D p 7 = r p−1 r p 4 B-23 4, ≥ . Contributors for λ µ, for 2 ≤ ≤ − ...... 78
8 B-24 D4, p ≥ 7. Contributors for λ = (−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ...... 79
B-25 D4, p ≥ 7. Contributors for λ = (−p + 2r + 3)ω1 + (p − r − 3)µ...... 79
B-26 D4, p ≥ 7. Noncontributors for λ = (p − 3)µ...... 79
B-27 D4, p ≥ 7. Contributors for λ = (p − 3)µ...... 79
B-28 D4, p ≥ 7. Noncontributors for λ = (p − 3)ω1...... 79
B-29 D4, p ≥ 7. Noncontributors for λ = (p − 2)µ...... 79
B-30 D4, p ≥ 7. Noncontributors for λ = (p − 1)µ...... 79
B-31 D4, p ≥ 7. Contributors for λ = (p − 1)µ...... 80
B-32 D4, p ≥ 7. Noncontributors for λ = ω2 + (p − 3)µ...... 80
B-33 D4, p ≥ 7. Contributors for λ = ω2 + (p − 3)µ...... 80
B-34 D4, p ≥ 7. Noncontributors for λ = ω1 + (p − 2)µ...... 80
B-35 D4, p ≥ 7. Contributors for λ = ω1 + (p − 2)µ...... 80
9 LIST OF FIGURES Figure page
2-1 Dynkin diagrams of root systems of type B` and D`...... 19 5-1 Possible symmetries of Dynkin diagrams...... 59
10 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON CERTAIN WEYL MODULES AND SIMPLE MODULES FOR ALGEBRAIC GROUPS OF TYPE B AND D By Elizabeth Wiggins December 2015 Chair: Peter Sin Major: Mathematics If g is a complex Lie algebra, the irreducible finite dimensional representations of g are parameterized by the dominant weights of g and have characters given by Weyl’s Character Formula. We construct the Chevalley groups over arbitrary fields from these representation spaces, and by construction, we obtain a representation for the universal Chevalley group from each of the irreducible finite dimensional representations for g. These modules are the Weyl modules, and because Weyl’s Character Forumla does not depend on field characteristic, the formula holds for the Weyl modules of the Chevalley group. Let G be a Chevalley group over an algebraically closed field of characteristic p > 0. The Weyl modules are no longer necessarily irreducible as they were in characteristic 0, but for each dominant weight, the corresponding Weyl module has a unique simple quotient. These simple modules are the structures we study in this dissertation. One of our major interests is to develop a character formula for some of these simple modules.
In particular, for G a simply connected, semisimple algebraic group of type B4 or D4 over an algebraically closed field of characteristic p > 0, we determine the characters of certain classes simple modules for these groups by calculating the composition factors of the Weyl
modules. For groups of type B4, we find the characters of the simple modules L(rω4), for
r a positive integer. For groups of type D4, we find the characters of the simple modules
L(r(ω3 + ω4)), for r a positive integer.
11 We also study an application of the resulting character formulas in the theory of spherical buildings. We introduce the incidence relation of oppositeness, and determine what it means in terms of the Chevalley groups of type B4 and D4. The result on characters of the simple modules will provide a method to determine the rank of the incidence matrix.
12 CHAPTER 1 INTRODUCTION Groups are mathematical structures that have long been an interest of mathematicians. In particular, in order to study finite groups, we can first direct our attention to the simple finite groups, which act as building blocks of the finite groups. Because many problems in finite group theory can be reduced to problems about finite simple groups, the classification of finite simple groups has been an ongoing project in mathematics for two centuries. In this classification, one of the major classes of finite simple groups is finite groups of Lie type. The first groups of Lie type were studied in the late 19th century and are now called the classical groups, named by Hermann Weyl. The classical groups are made up of matrix groups over the real, complex, or quaternion fields, and quotients of these groups. Classical groups are known to have applications in geometry and physics, making them useful in many disciplines. When these classical groups are constructed over finite fields instead of real or complex fields, they give rise to some of the finite simple groups of Lie type. In 1955, Claude Chevalley constructed the Chevalley groups, which expanded the theory of finite groups of Lie type by constructing groups based on semisimple Lie algebras. His construction included the classical groups, and included the construction of the exceptional groups of Lie type. A major contribution from Chevalley’s work was his construction of the Chevalley basis of the Lie algebras. This special basis has integer structure constants, which allows groups of Lie type to be constructed from their corresponding complex Lie algebras over an arbitrary field. In this dissertation, we will consider Chevalley groups constructed over fields of positive characteristic. One of the methods used to study groups is by the use of representations. A representation of a group G is a group homomorphism ρ : G → GL(V ) for V a vector space over a field k. Essentially, representation theory is the study of how groups of interest act on vector spaces. By changing the study of abstract groups to actions on vector spaces, representation theory changes group theory questions into problems in linear algebra. Just as simple groups are the building blocks of groups, representations can be decomposed into irreducible
13 representations. Thus to study representation theory, one first needs to study the theory of irreducible representations. In order to study the representation theory of the Chevalley groups of interest, it is useful to consider the representation theory of Lie algebras defined over the field of complex numbers. These two theories are closely related, and representations for complex Lie algebras have well developed theory. In particular, the representations of a complex Lie algebra are completely classified by the set of weights of the Lie algebra, and there is a character formula for the irreducible representations. When we construct the Chevalley groups, we will describe the relationship between the representations in detail. Our major interest lies in the representation theory of the Chevalley groups. Although the theory is closely related, many of the desirable properties of representations of complex Lie algebras do not carry over when we consider the corresponding representations of Chevalley groups. In particular, in order to construct a Chevalley group over a field k of characteristic p > 0, we use a complex Lie algebra g and an irreducible g-representation V . The construction of the Chevalley group G results in a G-representation V (k). The character formulas of V and V (k) are equal, but the irreducibility of V does not guarantee that V (k) will be irreducible. Thus one of the advancements that can be made for the representation theory of Chevalley groups is to determine a character formula for irreducible representations. A conjecture by Lusztig was made in 1980 for such a formula [11]. In the following decades, it was proved that for a fixed root datum, there exists a lower bound m such that the conjecture is true for char k > m [1]. However, a reasonable bound has not been established for the formula, and so the problem remains largely open. In Chapter4 of this dissertation, we will focus on two particular Chevalley groups, namely Chevalley groups of type B4 and D4 constructed over a field k of characteristic p > 0. For each of these groups, our main results are the characters of the simple modules for a class of weights and arbitrary characteristic. In the groups of type
B4, we will examine the class of weights rω4 for positive integers r. For groups of type D4, we study the class of weights r(ω3 + ω4) for positive integers r.
14 After proving the character formulas for the Chevalley groups, we study an application of these results in the theory of spherical buildings in Chapter5. The theory of spherical buildings was developed by Jacques Tits in order to gain a geometric interpretation of semisimple Lie groups and in particular, to gain intuition for the exceptional Lie groups. We will consider the incidence relation of oppositeness in buildings. As with any incidence relation, the invariants of the incidence matrix are of interest. In particular, we will determine the rank of the incidence matrix for the relationship of oppositeness in the Chevalley groups of type B4 and D4 studied in Chapter4.
15 CHAPTER 2 LIE THEORY AND CHEVALLEY GROUPS This chapter introduces the structures that we will study in detail in this dissertation. We begin with a summary of the theory of Lie algebras and representations of Lie algebras. Chevalley groups are also constructed here, which are the groups of our primary focus. We show how the representation theory of Lie algebras leads to the construction of the Chevalley groups, and define some of the important representations which we will study in greater detail later. This chapter provides the necessary theory to study Chevalley groups. 2.1 Lie Algebras
Definition 2.1.1. A vector space g over a field F with an operation g × g → g denoted (x, y) 7→ [x, y] is a Lie algebra if the following axioms are satisfied: (1) [·, ·] is bilinear, (2) [x, x] = 0 for all x ∈ g, (3) [x,[y, z]] + [y,[z, x]] + [z,[x, y]] = 0 for all x, y, z ∈ g. The element [x, y] is referred to as the bracket of x and y, and Axiom3 is called the Jacobi identity. From the definition of a Lie algebra, one can easily deduce definitions of familiar notions such as subalgebras, homomorphisms, etc. We will now explicitly discuss a few of the necessary parts of the theory. Define a subspace I of g to be an ideal if for all x ∈ g and all y ∈ I , then [x, y] ∈ I . Consider the sequence of ideals I (n) defined by I (0) = g, I (1) = [g, g], and I (n) = [I (n−1), I (n−1)]. We say that a Lie algebra g is solvable if I (n) = 0 for some n. A Lie algebra is semisimple if the only solvable ideal of g is the zero ideal. We define a similar sequence of ideals I n by I 0 = g, I 1 = [g, g], and I n = [g, I n−1]. A Lie algebra g is said to be nilpotent if I n = 0 for some n.
Let g be a semisimple Lie algebra over C or an algebraically closed field of characteristic 0 with bracket [·, ·]. Define a Cartan subalgebra to be a nilpotent subalgebra of a Lie algebra that is self-normalizing. The Cartan subalgebras of a semisimple complex Lie algebra are all
16 isomorphic and in fact, they are conjugate [8]. Let h be a fixed Cartan subalgebra. The Lie algebra g has a decomposition M g = h ⊕ gα, α∈h∗ ∗ where gα = {x ∈ g | [h, x] = α(h)x for all h ∈ h}. For each α ∈ h , gα is an h-invariant 1-dimensional subspace of g [8]. This decomposition is called the root space decomposition,
∗ and if gα is nonzero, then we call α ∈ h a root. Define a Borel subalgebra of g to be a maximal solvable subalgebra. Example 2.1.2. Let V be a three-dimensional vector space. Define sl(V ) to be the space of endomorphisms of V with trace zero. sl(V ) is a Lie algebra with bracket defined by
[A, B] = AB − BA. Let eij be defined to be the 3 × 3 matrix with a 1 in the i, j position, and zeros elsewhere. sl(V ) is an 8-dimensional complex vector space with basis {eij , h1, h2 | i 6= j}, where h1 = e11 − e22 and h2 = e22 − e33. An example of a Cartan subalgebra is the subalgebra h of diagonal matrices, i.e., the subalgebra generated by h1 and h2. Each non-diagonal element of the basis eij generates a one-dimensional space. These subspaces are the gα in the root space decomposition. An example of a Borel subalgebra is the subalgebra of upper triangular matrices, which is generated by h and the eij with 1 ≤ i < j ≤ 3. The set of lower triangular matrices is another example of a Borel subalgebra. Let R be the subset of roots in h∗. The rational span of R in h∗ is of dimension ` over
= dim ∗ Q, where ` C h [8]. We say that the integer ` is the rank of g. By extending the base E = (span R) field from Q to R, we obtain an `-dimensional real vector space Q ⊗Q R, which is spanned by R. Next we construct a form on E by first considering the Killing form κ, a nondegenerate form on g. We find that κ is also nondegenerate when restricted to h, and thus there exists an
∗ element hγ ∈ h for each γ ∈ h such that κ(h, hγ) = γ(h) for all h ∈ h. We may transfer
∗ ∗ the form to h by defining (γ, δ) = κ(hγ, hδ) for γ, δ ∈ h . This form extends to E, and is a symmetric, nondegenerate, positive bilinear form. This results in constructing E into a Euclidean space [8]. Thus we can consider roots as elements of euclidean space. Roots will
17 play an important role in the classification and representation theory of Lie algebras. Next we will discuss them in more detail and develop some useful theory. 2.1.1 Root Systems
∗ ∨ = 2α w E If α ∈ h is a root, then define the coroot of α by α (α,α) . A reflection α in ∨ determined by any nonzero vector α ∈ E is defined by wα(v) = v − (v, α )α. Geometrically,
wα(v) for v ∈ E represents the reflection of v in the hyperplane orthogonal to α. Thus wα leaves the vectors orthogonal to α fixed, and sends α to −α (and −α to α). We now present an axiomatic definition of a root system. Definition 2.1.3. A root system R is a finite subset of the euclidean space E if (R1) R spans E and does not contain 0. (R2) If α ∈ R, the only multiples of α in R are ±α.
(R3) If α ∈ R, the reflection wα leaves R invariant.
(R4) If α, β ∈ R, then (β, α∨) ∈ Z. Elements of the root system are called roots. As we consider roots as elements of Euclidean space, we can use familiar properties of E to learn more about the root system. For example, consider the formula kαkkβk cos θ = (α, β), where θ is the angle between α and β, and kαk = (α, α)1/2 is the root length. We can deduce with Axiom4 that there are a finite number of possible angles between roots within a root system. In particular, the number of possible ratios kβk/kαk is limited for a fixed root system. This restricts the number of possible root lengths. In fact, for an irreducible root system, the roots will either all have the same length or have exactly two lengths [8]. Thus if R is an irreducible root system, we can define short roots and long roots in the obvious way. Definition 2.1.4. A subset S of R is a base if (a) S is a basis for E. R = P k (b) Each root β ∈ can be written as β α∈S αα with integer coefficients which are all nonnegative or nonpositive.
18 1 ` − 1 ` B` ···
` − 1
D` ··· 1 ` − 2 `
Figure 2-1. Dynkin diagrams of root systems of type B` and D`.
We call elements of a base S the simple roots. The second condition of the definition allows for a partition of R into positive roots and negative roots, denoted R+ and R− respectively. The positive roots consist of roots that have positive coefficients when written as the sum of simple roots, and negative roots are defined similarly. This leads to a partial order on E using the base S in the following way. For µ, λ ∈ E, define µ < λ when λ − µ is a sum of positive roots. Root systems arise naturally in the theory of Lie algebras as elements of h∗. They are critical in the classification of Lie algebras because for each root system, there exists a semisimple Lie algebra with an equivalent root system, and any two semisimple complex Lie algebras with equivalent root systems are isomorphic. Thus for g a semisimple Lie algebra, the pair (g, h) is determined by the root system R up to isomorphism [8].
We can classify irreducible root systems into types A`, B`, C`, D`, E6, E7, E8, F4, and
G2. For example, a Lie algebra of type A` is sl`+1, and so Example 2.1.2 is a description of a
Lie algebra of type A2. Although we do not want to completely describe each root system in detail, we will use types B` and D` as examples throughout this dissertation.
We begin our description of the root systems of type B` and D` with the Dynkin diagrams for the root systems. The Dynkin diagram of a root system of rank ` is a graph
∨ ∨ of ` nodes with the ith node joined with the jth node by (αi , αj )(αj , αi ) edges, where the set {α1, ... , α`} is a base of the root system. In the case of a double or triple edge, we add an arrow pointing to the shorter root. For example, Figure 2-1 shows the Dynkin diagrams for root systems of type B` and D`.
19 We define the Weyl group W to be group generated by the reflections wα for α ∈ R. W is a finite group, and by the definition of a root system, each element w of W will fix the root system as w acts on the vector space E. If S = {α1, ... , α`} is a set of simple roots of a w root system, then the Weyl group is generated by the simple reflections αi [8]. A set of simple roots S defines a length function ` on W . Since W is generated by simple reflections, each w W w = w w w S n ∈ can be expressed as β1 β2 ··· βn , with βi ∈ . If is minimal, then we define the length of w to be `(w) = n. Example 2.1.5. Earlier we constructed a real vector space E of dimension `. This allows us to represent the root systems as subsets of a standard `-dimensional vector space. Let
{ei | i = 1, ... , `} be the standard orthonormal basis of E. The inner product is the usual dot product.
Suppose R is a root system of type B`. We can take R to be R = {±ei ± ej , ±ek | 1 ≤ i < j ≤ `, 1 ≤ k ≤ `}. The simple roots S of R can be taken to be αi = ei − ei+1 for i = 1, ... , ` − 1 and α` = e`. The Weyl group acts by permuting the components of vectors in E and by sign changes of components.
Similarly, let R be a root system of type D`. R can be taken as R = {±ei ± ej | 1 ≤ i < j ≤ `}. The simple roots S are αi = ei − ei+1 for i = 1, ... , ` − 1 and α` = e`−1 + e`. The Weyl group acts by permuting the components of vectors in E and by an even number of sign changes of components. 2.1.2 A Chevalley Basis
Our next goal is to construct a special basis of the Lie algebra. We are currently working with a complex Lie algebra, but our eventual goal is to transfer to an arbitrary field. In order to
do so, we need integer structure constants. That is, if {v1, ... , vm} is a basis of the Lie algebra,
the structure constants are the field elements aij such that [vi , vj ] = aij vk for all i, j = 1, ... , m.
R h0 Definition 2.1.6. For each root α ∈ , define α ∈ h to be the element such that (h, h0 ) = (h) h h = 2 h0 h = h S = , ... , α α for all ∈ h, and α (α,α) α. Let i αi for {α1 α`}.
20 We use these elements hi of h in order to set up the desired basis, called a Chevalley basis. We will eventually define the Chevalley groups, originally constructed by Chevalley in 1955 [6]. The following theorem is the existence theorem of the Chevalley basis, and is proved in many references [8], [15].
Theorem 2.1.7. There exist root vectors xα for each α ∈ R and elements of hi ∈ h for each
αi ∈ S such that the set {xα, hi | α ∈ R, 1 ≤ i ≤ `} is a basis of g with the following structure properties:
(a) [hi , hj ] = 0 for 1 ≤ i, j ≤ `.
∨ (b) [hi , xα] = (α, αi )xα for 1 ≤ i ≤ ` and α ∈ R. [x , x ] = h h , ... , h (c) α −α α is a Z-linear combination of 1 `.
(d) If α, β are independent roots, β − rα, ... , β + qα the α string through β, then [xα, xβ] = 0
if q = 0, and [xα, xβ] = ±(r + 1)xα+β if α + β ∈ R.
The basis in Theorem 2.1.7 is called a Chevalley basis. One should note that the hi ’s in
the Chevalley basis are the same hi ’s in Definition 2.1.6. Also note that a Chevalley basis is not
completely unique. The hi ’s are completely determined by choice of base in the root system,
but the xα’s may vary by constants [8], [15]. 2.1.3 Weights
Let £ be the subset of the vector space E consisting of vectors λ ∈ E such that
(λ, α∨) ∈ Z for all α ∈ R. Elements of £ are called integral weights, and £ is a lattice in the real vector space E in the sense that £ is the Z-span of an R-basis for E. If S is a base of the root system R, then we say a weight λ ∈ £ is dominant integral if (λ, α∨) is nonnegative for
+ all α ∈ S. Let £ denote the set of dominant weights. Note that R ⊂ £. ZR is called the root lattice, which we can consider as a sublattice of £. Note that because E is a real vector space with basis S, the set of coroots {α∨ | α ∈ S}
∨ is also a basis of E. Let λi , i = 1, ... , ` be the dual basis to the set of coroots α , α ∈ S,
( , ∨) = ` £ defined by λi αj δij . We call {λi }i=1 the fundamental weights. is the Z-span of the
fundamental weights λi [8].
21 Example 2.1.8. Let R be a root system with S a set of simple roots.
Suppose R is a root system of type B`. If a root α ∈ S is of the form ei ± ei+1, then ( , ) = 2 ∨ = 2α = = e ( , ) = 1 ∨ = α α and so α (α,α) α. If α `, then α α and α α. The fundamental
weights λi can be taken to be λi = (ai1, ai2, ... , ai`) with 1 if j ≤ i aij = 0 if j > i
1 i 1 = 1 , 1 , ... , 1 for ≤ ≤ ` − , and λ` 2 2 2 .
Suppose R is a root system of type D`. Simple roots α are of the form ei − ei+1.
∨ Because (α, α) = 2, we have α = α. The fundamental weights λi can be defined λi =
(ai1, ai2, ... , ai ) with ` 1 if j ≤ i aij = 0 if j > i 1 i 2 = 1 , ... , 1 , 1 = 1 , 1 , ... , 1 for ≤ ≤ ` − , λ`−1 2 2 − 2 , and λ` 2 2 2 .
Consider the finite group £/ZR, which is called the fundamental group. The structure of £/ZR can be computed by reducing the Cartan matrix, which is the matrix C defined by
∨ Cij = (αi , αj ), into diagonal form to find the elementary divisors.
Example 2.1.9. In type B`, the Cartan matrix is 2 −1 −1 2 .. . CB = . 2 −1 −1 2 −2 −1 2
22 The reduced diagonal matrix is 2 1 C 0 = . B . .. 1
∼ Thus the fundamental group is £/ZR = Z/2Z.
In type D`, the Cartan matrix is 2 −1 −1 2 . .. CD = 2 1 . − −1 2 −1 −1 −1 2 0 −1 0 2
In the case of type D`, The diagonal form is different depending on whether ` is odd or even. The diagonal forms for odd and even ` are 2 4 2 1 0 0 CD, = , CD, = 1 . odd .. even . .. . 1 1
D £ R ∼ 4 £ R ∼ Thus we have that in type `, if ` is odd, then /Z = Z/ Z and if ` is even, then /Z = Z/2Z × Z/2Z.
23 2.2 The Universal Enveloping Algebra
Let K be an arbitrary field. Let g be a Lie algebra over a field K, and let U be an associative algebra with 1 over K. Suppose that R : g → U is linear and preserves the bracket on g, i.e. R([x, y]) = R(x)R(y) − R(y)R(x). The map R is called a homomorphism. Define a universal enveloping algebra of g to be a couple (U, R) such that U is an associative algebra with 1, R is a homomorphism of g into U, and for any other such pair (A, ψ), there exists a unique homomorphism θ : U → A such that θ ◦ R = ψ and θ(1) = 1. To construct such an algebra, consider the tensor algebra T (g) of g. Let I be the two sided ideal of T (g) generated by all elements x ⊗ y − y ⊗ x − [x, y] for x, y ∈ g. Now we let U be the algebra T (g)/I [8]. We would like to be able to consider g with its image in U. The following theorem allows us to do so. A proof can be found in [8]. Theorem 2.2.1 (Birkhoff-Witt Theorem). Let g be a Lie algebra over a field K and (U, R) its universal enveloping algebra. Then (a) The map R is injective.
(b) If g is identified with its image in U and if x1, ... , xr is a linear basis for g, the monomials
k1 kr x1 ··· xr form a basis for U, where each ki is a nonnegative integer. 2.2.1 Representations of g
A representation of a complex Lie algebra g is a homomorphism ϕ : g → gl(V ), for V a
vector space over C. We refer to the representation space V as a g-module, which is defined
with an operation g × V → V denoted (g, v) 7→ g v satisfying the following properties for all a, b ∈ C, g, h ∈ g, and u, v ∈ V .
1. (ag + bh) v = a(g v) + b(h v). 2. g (au + bv) = a(g u) + b(g v). 3. [g, h] u = g (h u) − h (g u). With this definition of g-modules, the familiar notions of submodules, homomorphisms, etc. come naturally. We call a nontrivial g-module irreducible if it has no proper nontrivial submodules.
24 Suppose V is a finite dimensional g-module. A Cartan subalgebra h will act diagonally on V V = X V V = v V h v = (h)v h V = 0 , and so λ, where λ { ∈ | λ for all ∈ h} [8]. If λ 6 , λ∈h∗ then we call Vλ a weight space, λ a weight of V , and v a weight vector. Let £(V ) denote the
Z-span of the set of weights of V . We will see the importance of weights of a representation throughout this section. Also the connection between weights of a module and the weights defined in Section 2.1.3 will become evident. In order to further investigate repesentations of g, the following properties are required, which establish necessary elements to the theory. Theorem 2.2.2 ([8], [15]). If g is a semisimple Lie algebra with Cartan subalgebra h, then (a) Every finite dimensional irreducible g-module V contains a nonzero vector v + such that v + x v + = 0 x R+ is a weight vector to some weight λ and α for all α ∈ gα, α ∈ . We call λ the highest weight of V , and v + the highest weight vector.
(b) If λ is the highest weight of V , then dim Vλ = 1, then each weight µ of V is of the form µ = λ − (nonnegative linear combination of positive roots). Next we wish to construct certain g-modules which will play an important role later. A g-module V is said to be a highest weight module of weight λ if there exists v + ∈ V with V = Uv +. Lemma 2.2.3 ([8]). A highest weight module V of weight λ has a unique maximal submodule. This is because for each submodule of V , V is the direct sum of its weight spaces. Thus
for any proper submodule U of V , Vλ cannot lie in U because V is generated by Vλ. Consider the sum W of all proper submodules of V ; W is a maximal proper submodule of V [8].
∗ V ( ) Theorem 2.2.4 ([8]). Let λ ∈ h . There exists an irreducible highest weight g-module λ C of highest weight λ, which is unique up to isomorphism. V ( ) The construction of λ C is described in [8], which is briefly summarized here. First D v + D = v + define a one-dimensional vector space λ generated by a vector , i.e., λ C . Let
25 M b = h⊕ gα be a Borel subalgebra of g. We make Dλ into a b-module by defining the action α>0 ! h + X x v + = h v + = (h)v +. α λ α>0
We extend the action to U(b) to make Dλ into a U(b)-module. Now let M(λ) = U(g) ⊗U(b)
+ Dλ, defining M(λ) as a left U(g)-module. Note that M(λ) is generated by 1 ⊗ v . Observe 0 x 1 v + = 1 x v + = 0 h 1 v + = 1 h v + = 1 (h)v + = (h)(1 v +) that for α > , α ⊗ ⊗ α and ⊗ ⊗ ⊗λ λ ⊗ by definition. This shows that M(λ) is a highest weight module of weight λ with highest weight vector 1 ⊗ v +, which we will denote v +. From this module M(λ), we wish to construct an irreducible module. From Lemma 2.2.3, we know that M(λ) has a unique maximal submodule Y (λ). Note that the weight space
M(λ)λ does not lie in Y (λ), and so the quotient M(λ)/Y (λ) is an irreducible highest weight V ( ) M( ) Y ( ) module of weight λ. Thus the desired module λ C is defined to be λ / λ . Theorem 2.2.4 gives a one-to-one correspondance between λ ∈ h∗ and the irreducible highest weight modules, but says nothing of finiteness of dim V , which we address next. If
∗ (h ) S λ ∈ h is any linear function with the property that λ αi ∈ Z for all αi ∈ , then λ is called (h ) an integral weight. If all λ αi are nonnegative integers, then λ is dominant integral. Theorem 2.2.5 ([8]). Let V be an irreducible g-module. V is finite dimensional if and only if the highest weight of V is a dominant integral weight. In fact, we can say that if V is a finite dimensional g-module, then any weight µ of V is
( , ∨) = (h ) an integral weight, i.e., µ αi µ i ∈ Z [8], [15]. Recall that £ is the set of elements µ in E with (µ, α∨) ∈ Z for all α ∈ R. Let µ be a weight of a finite dimensional g-module V . Let hα for α ∈ R be defined as in Definition 2.1.6. h h ∗ ( , ∨) Each α is a Z-linear combination of the i ’s, so by linearity of µ ∈ h , µ α ∈ Z for all α ∈ R. Thus µ ∈ £, which implies that if ¥(V ) is the set of weights of V , we can say that
¥(V ) ⊂ £, and for £(V ) the Z-span of ¥(V ), £(V ) ⊂ £ [15].
26 2.2.2 Weyl’s Character Formula £+ V = V ( ) Let λ ∈ be a dominant weight, and let λ C. We now begin to discuss the significance of these modules. First we present Weyl’s formulas, which demonstrate that these modules have well developed theory. Let ρ be half the sum of the positive roots. Define the
weight multiplicity mλ(µ) of a weight µ in V by mλ(µ) = dim Vµ. Note that mλ(µ) = 0 if µ V V Ch V ( ) is not a weight of . For a dominant weight λ, denote the formal character of by λ C or Chλ, which is defined by X Chλ = mλ(µ)e(µ), µ∈£ where the e(µ) form a basis for the group ring Z(£) for all µ ∈ £. Weyl’s character formula V ( ) gives us the character of λ C. Proofs can be found in many references [8], [9]. Theorem 2.2.6 (Weyl’s character formula). Let λ ∈ £+. Then
P (−1)`(w)e(w(λ + ρ)) Ch = w∈W . λ P ( 1)`(w)e(w( )) w∈W − ρ
From the character formula, we can derive a dimension formula as a corollary. Corollary 2.2.7 ([8]). Let λ ∈ £+.
Q (λ + ρ, α) dim V (λ) = α>0 . C Q ( , ) α>0 ρ α V ( ) £+ The modules λ C are crucial to the representation theory of Lie algebras; for λ ∈ , they are irreducible, finite dimensional, and parameterized by £+ (by Theorem 2.2.4). Because of Weyl’s character formula, their characters are known and calculable.
2.2.3 Kostant Z-form R S = , ... , Suppose is a root system with base {α1 α`}. Let UZ be the Z-algebra x m m! m + R generated by all α / , ∈ Z , α ∈ . The algebra UZ is called the Kostant Z-form of the universal enveloping algebra U.
Definition 2.2.8. If V is a vector space over C and M is a finitely generated subgroup of V which has a Z-basis which is a C-basis for V , we say M is a lattice in V .
27 + Fix an ordering of R , α1, ... , αm. Denote m-tuples and `-tuples of nonnegative integers
A = (a1, ... , am), B = (b1, ... , b`), and C = (c1, ... , cm).
Then define elements fA, hB ,and eC of U, where
x a1 x am h h x c1 x cm −α1 −αm 1 ` α1 αm fA = ··· , hB = ··· , and eC = ··· . a1! am! b1 b` c1! cm!
Kostant proved the following result in 1966 [10]. Proofs may be found in many references [8], [15].
Theorem 2.2.9 (Kostant’s Theorem). Let B be the lattice in U with Z-basis consisting of all f h e = A B C . We have B UZ.
Theorem 2.2.9 implies that any product of fA’s, hB ’s, and eC ’s can be rearranged to be in
the form of sums of fA0 hB0 eC 0 . This easily implies Corollary 2.2.10. We will use these properties to construct a particular lattice in a finite dimensional g-module V . Corollary 2.2.10 ([8], [15]). U = U − U 0 U +, where U −, U 0, and U + are the -algebras Z Z Z Z Z Z Z Z generated by x m/m! for α < 0, hi , and x m/m! for α > 0 respectively. α ni α Corollary 2.2.11 ([8], [15]). Every finite dimensional g-module V contains a lattice M x m m! M invariant under α / , i.e., is invariant under UZ. V V A lattice of invariant under UZ is called an admissible lattice of . Note that we can construct such a lattice by considering M = U v +. Note that U v + = U − v + because Z Z Z xm h x v + = 0 and thus α v + = 0, and i acts as scalar multiplication on v + [8], [15]. This α m! bi V lattice is invariant under UZ as desired, and has the property that every admissible lattice of contains M [8]. Thus we refer to U − v + as the minimal admissible lattice of V . Z V Corollary 2.2.12 ([8], [15]). Every UZ-invariant lattice of is the direct sum of its weight components.
For M an admissible lattice of V , let Mµ = M ∩ Vµ, the intersection of the lattice M with
the weight space Vµ. Corollary 2.2.12 allows M to be written as the direct sum of these weight
28 M components, that is, M = Mµ, where ¥(V ) is the set of weights of V . Our next goal is µ∈¥(V ) to construct a Chevalley group, and we require an admissible lattice to do so. 2.3 Chevalley Groups
2.3.1 Construction of the Chevalley Groups
The following construction of a Chevalley group is given in [8]. Let Fp be the prime field
of characteristic p with k an extension field of Fp. Let V be a faithful g-module. Denote the
Z-span of the weights of V by £(V ). Let M be an admissible lattice in V . Define gV to be the stabilizer of M in g.
Proposition 2.3.1 ([8], [15]). gV is an admissible lattice in the g-module g and gV = + P x = h (h) £(V ) hV α∈R Z α for hV { ∈ h | λ ∈ Z for all λ ∈ }. In particular, gV depends only on V . M V (k) = M k (k) = k Consider the lattices and gV . Let ⊗Z and gV gV ⊗Z . Because gV is isomorphic to a subgroup of End(M), gV (k) may be identified with a Lie subalgebra of End(V (k)). x , h ( ) x h Let { α i } be a Chevalley basis for g and define g Z to be the Z-span of the α and i , R i = 1, ... , M ( ) for α ∈ and `. Note that because is a UZ-invariant lattice, g Z is a subset of
gV . Use the inclusion map g(Z) → gV to induce a Lie algebra homomorphism g(k) → gV (k) (k) = ( ) k V V (k) (k) for g g Z ⊗Z . For each faithful g-module , is an g -module through the
homomorphism g(k) → gV (k). The module is not necessarily faithful and depends on choice of admissible lattice.
Note that for each t ∈ k, there exists a unique ring homomorphism Z[T ] → k such that
T 7→ t. Let xα(t) ∈ Aut(V (k)) be defined by considering
∞ n n X T x α ∈ Aut(M ⊗ [T ]). n! Z Z n=0 x n M [T ] n Note that α acts as zero on ⊗Z Z for large enough, so we may then specialize the
indeterminate T to t ∈ k via the above map to define xα(t) ∈ Aut(V (k)).
29 Define GV (k) = hxα(t) | t ∈ k, α ∈ Ri. Up to isomorphism, GV (k) only depends on
£(V ). We call GV (k) the Chevalley group of type £(V ) over k. 2.3.2 Subgroups of a Chevalley Group
Let G be a Chevalley group of type £(V ). We can define groups Xα for each α ∈ R to be the groups {xα(t) | t ∈ k}. The subgroups Xα are called the root subgroups of G, and we can also consider the group G to be the group generated by the root subgroups Xα for α ∈ R.
−1 −1 Define elements wα(t) = xα(t)x−α(−t )xα(t) and hα(t) = wα(t)wα(1) in G. Let T be the subgroup of G generated by all hα(t) and let N be the subgroup generated by all wα(t)
− with α ∈ R and t ∈ k. Let U be the subgroup generated by all Xα for α > 0, and similarly U the subgroup generated by Xα for α < 0. Finally, let B be the subgroup generated by U and T . Let W be the Weyl group. Consider the following technical lemma. A proof can be found in [15]. Lemma 2.3.2. 1. T is a normal subgroup of N.
2. There exists an isomorphism ϕ : W → N/T such that ϕ(wα) = wα(t)T for all α ∈ R. 2.3.3 Bruhat Decomposition
Because W is isomorphic to the quotient N/T by Lemma 2.3.2, then we adopt the convention that if n ∈ N is a representative for w ∈ W under the canonical map, then we write wB or Bw in place of nB or Bn. Consider the following technical lemmas on double cosets of B. Lemma 2.3.3 ([15]). If w ∈ W and α is a simple root, then w 0 BwB Bw B Bww B (1) If α > , then α ⊆ α . BwB Bw B Bww B BwB (2) α ⊆ α ∪ .
Lemma 2.3.4 ([15]). Let G be a Chevalley group and thus generated by root subgroups Xα for α ∈ R for R the root system. G can be generated by {Xα, ωα = wα(1) | α ∈ S} for S the set of simple roots. [ Consider the set BwB. Recall that B is the subgroup of G generated by the positive w∈W root subgroups Xα and H. Thus Xα ⊂ B for each simple root α. Also note that ωα is in
30 [ the coset BwαB, so that the generators of G in Lemma 2.3.4 are all contained in BwB. w∈W Then by Lemma 2.3.3, we see that the set is closed under multiplication of the generators, and [ so BwB is a group which is equal to G, stated in Theorem 2.3.5. This decomposition, w∈W known as the Bruhat decomposition was originally proved by Bruhat in 1956 [3]. The complete construction can be found in many references [4], [15]. S BwB = G Theorem 2.3.5 (Bruhat Decomposition). 1. w∈W . 2. If BwB = Bw 0B, then w = w 0. Theorem 2.3.5 gives a decomposition of G into a disjoint union of double cosets of G. It points out that a system of representatives for N/T is a system of representatives of B \ G/B. We now present an example from [15], which gives a useful geometric interpretation of the Bruhat decomposition.
Example 2.3.6. ([15]) Let g = sln(C). Let the Chevalley group constructed from g over a
field k be G = SLn(k). The subgroups B, T , and N are the subgroups of upper triangular matrices, diagonal matrices, and monomial matrices respectively. Thus the Weyl group W , which is isomorphic to N/T , is isomorphic to the group of permutation matrices. That is, we consider W as the group that permutes the coordinates of the underlying vector space. The Bruhat decomposition of G implies that the permutation matrices in W form a system of representatives of the double cosets in B \ G/B. We will describe a geometric interpretation of the Bruhat decomposition. Let V be the underlying n-dimensional vector
space. Define a flag in V to be a chain V1 ⊂ · · · ⊂ Vn of subspaces of V such that
dim Vi = i. Let {v1, ... , vn} be the standard basis for V . The basis has a standard flag
associated with it. The standard flag F1 ⊂ · · · ⊂ Fn is defined by Fi = hv1, ... vi i. Because G acts on V , we consider the action of G on the flags of V . The subgroup B of upper triangular matrices is the stabilizer of the standard flag of V , so we can identify the group B \ G/B with the G-orbits of pairs of flags.
Define a simplex to be a set of points {p1, ... , pn} in V which are linearly independent. A
flag V1 ⊂ · · · ⊂ Vn is said to be incident with the simplex if Vi = hpπ(1), ... pπ(i)i for some
31 permutation π ∈ Sn. Using linear algebra and induction on n, one can show that given any two flags of V , there exists a simplex incident with both flags. That is, there exists an element of
Sn that transforms one simplex into the other. This provides a correspondence between Sn and B \ G/B, and thus this is a geometric interpretation of the Bruhat decomposition. 2.3.4 Weyl Modules
Note that by construction of G in Section 2.3.1, we can consider V (k) as a GV (k)-module. We also have the following proposition, which allows us to consider the module V (k) as a sum of weight components. V (k) = M k V (k) = M V (k) Proposition 2.3.7 ([15]). Let µ µ ⊗Z . Then µ. µ∈¥(V ) Lemma 2.3.8 ([8]). Let V be a faithful representation and let £(V ) be the additive group generated by weights of V . Then £(V ) is an admissible lattice, and ZR ⊂ £(V ) ⊂ £. All lattices between ZR and £ can be realized as in the Lemma 2.3.8 for some V [15]. In particular, if £(V ) = ZR, then V is the representation space of the adjoint representation, and if £(V ) = £, then V corresponds to the sum of representations with fundamental weights as highest weights [15]. We call the Chevalley groups constructed from lattices ZR and £ the adjoint group and the universal group respectively. Proposition 2.3.9 ([15]). Let g be a Lie algebra, k a field. Suppose we construct Chevalley groups G and G 0 with g-modules V and V 0 respectively. If £(V 0) ⊇ £(V ), then there exists
: G 0 G (x 0 (t)) = x (t) R t k × a homomorphism ϕ → such that ϕ α α for all α ∈ , ∈ , and ker ϕ ⊆ Z(G 0). If £(V 0) = £(V ), then ϕ is an isomorphism. Let G be the universal Chevalley group constructed from the weight lattice £ over a field k of characteristic p. If G 0 is a Chevalley group constructed from a lattice £(V ), then Proposition 2.3.9 constructs a homomorphism which allows us to consider the modules V (k) as G-modules through the action of G 0 on V (k) and the homomorphism ϕ : G → G 0. V ( ) Recall the existence of an irreducible highest weight g-module λ C of highest weight V = V ( ) M = .v + λ in Theorem 2.2.4. Let λ C, and let UZ be the minimal admissible lattice. When we construct V (k) as a G-module as above, V (k) is generated by v + ⊗ 1, where
32 + v ∈ M ∩ Vλ. This makes V (k) a highest weight module of weight λ with maximal vector v + ⊗ 1. When the field k is fixed, we denote the module V (k) by V (λ). Furthermore, note that V (k) has a unique maximal submodule by Lemma 2.2.3 and thus a unique irreducible quotient which is a highest weight module of weight λ, which we denote by L(λ). The G-modules V (λ) are called the Weyl modules, and they are constructed with the universal property for finite dimensional G-modules. That is, if V is any finite dimensional G-module generated by a one dimensional weight space of weight λ, then V is isomorphic to a quotient of V (λ). Also note that because of Proposition 2.3.7, V (λ) is the direct sum of its weight spaces and thus Weyl’s character formula gives the character for V (λ) over any field.
33 CHAPTER 3 ALGEBRAIC GROUPS In this chapter we will draw connections between the theory of Lie algebras and the theory of algebraic groups. In particular, we see that Chevalley groups constructed over an algebraically closed field are algebraic groups, which allows us to use the theory developed for algebraic groups to study the Chevalley groups. We will also revisit the Weyl modules, which will be the representations studied in detail in the next chapter. 3.1 Algebraic Groups and Chevalley Groups
We begin with some definitions for algebraic groups. Let k be an algebraically closed
n field. A subset X of k is called an algebraic subset if there exists a subset P of k[x1, ... , xn] such that X = {x ∈ k n | p(x) = 0 for all x ∈ X }. Algebraic sets define the closed sets of
n n the Zariski topology on k . For V a subset of k , let Ik (V ) = {p ∈ k[x1, ... , xn] | p(x) = 0 for all v ∈ V }.
Example 3.1.1. Let P(x) be the polynomial in k[x0; xij ]i=1,...,n; that is defined by j=1,...,n
P(x) = 1 − x0 det xij .
n2+1 n2+1 GLn(k) = {v ∈ k | P(x) = 0} is an algebraic subset of k .
Define a matrix algebraic group G to be a closed subset of GLn(k) for some n such that
n2+1 k is an algebraically closed field and G is an algebraic subset of k . If k0 is a subfield of k, then we can consider G over k0 if Ik (G) has a basis of polynomials with coefficients in k0.A homomorphism ϕ : G → H of algebraic groups G and H is a group homomorphism such that each map of matrix coefficients gij 7→ ϕ(g)ij is a rational function. We call an algebraic group G semisimple if the radical of G is trivial and G is connected. Proposition 3.1.2 ([15]). (a) The image of an algebraic group under a rational homomorphism is algebraic. (b) A group generated by connected algebraic subgroups is algebraic and connected. It is
defined over k0 if each of the subgroups is.
34 Let G be a Chevalley group over an algebraically closed field k with lattice M. Let B, T , and N be defined as they were in Section 2.3.2. Recall that G is generated by root subgroups
Xα for α ∈ R. We can define a homomorphism xα : ka → Xα, where ka is the additive group of the field elements, defined by xα : t 7→ xα(t), where the xα(t) are the elements defined in Section 2.3. This map is a rational isomorphism of groups. Note that in the Zariski topology,
ka is a connected group, and so by Proposition 3.1.2a, we find that each Xα is a connected algebraic groups. Considering Proposition 3.1.2b, we conclude that G itself is a connected algebraic group. Since the radical of G is the maximal connected solvable normal subgroup of G, the only possibility is that rad G is a finite group and thus trivial by connectedness. This argument and similar arguments show the following theorem. A complete proof can be found in [15]. Theorem 3.1.3. (a) G is a semisimple algebraic group relative to M. (b) B is a maximal connected solvable subgroup. We call B the Borel sugroup. (c) T is a maximal connected diagonalizable subgroup. We call T a maximal torus. (d) N is the normalizer of T and N/T =∼ W .
(e) G, B, T , and N are all defined over k0 relative to M. We will study Chevalley groups viewed as algebraic groups. Because of Theorem 3.1.3, we may use the language of algebraic groups to discuss the subgroups of G. 3.1.1 Weights of G with Respect to a Maximal Torus
Eventually, we want to parameterize simple modules of the Chevalley groups as we did the irreducible representations of the Lie algebras. First, we generalize the definitions of weights and roots of a Lie algebra to define global characters and global roots. The term “global” typically indicates that we refer to the properties of the group instead of the Lie algebra. Let G be a Chevalley group constructed from a representation V with lattice M and weight lattice L; let T be a maximal torus of G. Let µ be an element of the weight lattice. Define a global character µ^ : T → k ∗ by
35 Y Y µ(hi ) µ^ hi (ti ) = ti , where the hi (ti )’s are defined as they were in Section 2.3.2 and
the hi ’s are defined as they were in Definition 2.1.6. µ^ is in fact defined over the prime field ^ k0 [15]. Let L be the lattice generated by µ^ for µ ∈ L. We define the global roots in a similar
∗ Q Y α(hi ) ^ way. Define α^ : T → k by α^ ( hi (ti )) = ti , and let £r be the lattice generated by all α^ for α ∈ R. This establishes a one-to-one correspondence between the weights of a lattice
£(V ) used to define a Chevalley group GV (k) and the global characters, which we will denote X (T ). Lemma 3.1.4 ([15]). Let T be a maximal torus of an algebraic group G. The rational characters of T are the elements of the lattice generated by the global characters of the representation defining G.
Theorem 3.1.5 ([15]). Given a root system R, a lattice L with ZR ⊂ L ⊂ £, and an algebraically closed field k, there exists a semisimple algebraic group G defined over k such
that the lattices ZR and L are realized as the lattices of global roots and characters relative to a maximal torus. The following Theorem contains analogue statements for modules for algebraic groups of Theorems 2.2.2 and 2.2.4 on modules for Lie algebras. This demonstrates the parallels in the representation theory of Lie algebras and Chevalley groups. Theorem 3.1.6 ([15]). Let G be a Chevalley group over an algebraically closed field with maximal torus T . (a) Every nonzero rational G-module V contains a nonzero vector v + such that v + is a weight vector for λ ∈ X (T ) and v + is U-invariant.
+ + − + (b) Assume V = kGv with v a vector as in (a). Then V = kU v , the dimension of Vλ is P P 1, every weight of V has the form λ − α for positive roots α, and V = Vµ.
+ (c) In (a), (λ, α∨) ∈ Z for all positive roots α. (d) If V is irreducible, then the highest weight λ and the line kv + of (a) are uniquely determined.
36 (e) Given any character λ : T → k ∗ satisfying (c), there exists a unique irreducible rational G-module V in which λ is realized as in (a). Let G be a semisimple, algebraic group. Let T be a maximal torus in G. Recall X (T ) = Hom(T , k ×). Let Y (T ) = Hom(k ×, T ). There is a pairing h·, ·i : X (T ) × Y (T ) → k × defined in the following way. For λ ∈ X (T ), µ ∈ Y (T ), let hλ, µi be the unique integer such that λ ◦ µ : a 7→ ahλ,µi for all a ∈ k ×. The set of roots R is a subset of X (T ). Define the
∨ ∨ × coroots α as elements of Y (T ) by α (a) = hα(a) for a ∈ k , where hα(a) are the elements of T defined in Section 2.3.2. Let R∨ denote the set of coroots in Y (T ). E = X (T ) R , Let ⊗Z R be a real vector space which contains . An inner product h· ·i can be defined on E. Then the root system R becomes a root system as in Definition 2.1.3. Y (T )
E ∨ Y (T ) 2α E can be identified with a sublattice of by associating α ∈ with (α,α) ∈ . 3.1.2 Simply Connected Chevalley Groups
Let G be an algebraic group with maximal torus H. G is said to be simply connected if the set of weights X (T ) contains all fundamental weights λi . Due to Lemma 3.1.4, this is precisely the definition of the universal Chevalley group, and thus the universal Chevalley group is a simply connected group. We will assume that G is simply connected whenever possible because the properties for the representation theory are much easier to work with.
For example, G is simply connected if and only if Y (T ) = ZR∨ [9]. Notice that due to Proposition 2.3.9, which shows that any Chevalley group is isomorphic to a quotient of the simply connected one constructed from the same Lie algebra, each irreducible module can be considered as a module for the simply connected group. Thus we assume that G is simply connected when studying irreducible representations without loss of generality. 3.2 Weyl Modules
We want to discuss Weyl modules as representations of algebraic groups, and so we will first discuss a second construction of the Weyl modules from [9]. We begin with some more general representation theory.
37 Let G be an algebraic group over a field k. Let H be a subgroup of G and M a kH-module. We can define a G-module, the induced module in the following way. Let
G −1 indH M = {morphisms f : G → M | f (gh) = h f (g) for all g ∈ G, h ∈ H}.
−1 G This is a G-module with action defined by (xf )(g) = f (x g) for x ∈ G, f ∈ indH M, and g ∈ G. Let B be the Borel subgroup of G generated by the positive root subgroups. Let λ be a weight of G so that λ : T → k ×. Because B = UT , we can extend λ to be a character of B defined by λ((u, t)) = λ(t) for u ∈ U and t ∈ T . Let kλ be the corresponding representation space to the character λ. kλ is a one-dimensional B-module, and thus isomorphic to a copy of k. Consider the induced module
G −1 indB (kλ) = {morphisms f : G → k | f (gb) = λ(b )f (g) for all b ∈ B, g ∈ G}.
0 0 G Define the induced module H (λ) by H (λ) = indB (kλ).
+ Proposition 3.2.1 ([9]). Let λ ∈ X (T ) with H0(λ) 6= 0. dim H0(λ)U = 1 and
0 U+ 0 H (λ) = H (λ)λ. Proposition 3.2.1 shows that the λ-weight space of H0(λ) is a one-dimensional U+-invariant subspace.
0 0 Corollary 3.2.2 ([9]). If H (λ) 6= 0, then socG H (λ) is simple. The socle of the induced module is in fact the simple module that we wish to study. Set
0 0 L(λ) = socG H (λ) for any λ ∈ X (T ) such that H (λ) 6= 0. In fact, λ is a dominant weight if and only if H0(λ) 6= 0 [9]. Proposition 3.2.3 ([9]).
(a) The L(λ) with λ ∈ X (T )+ are a system of representatives for the isomorphism classes of simple G-modules.
38 0 U+ U+ (b) Let λ ∈ X (T ) with H (λ) 6= 0. Then L(λ) = L(λ)λ and dim L(λ) = 1. Any weight
µ of L(λ) satisfies w0λ ≤ µ ≤ λ. The multiplicity of L(λ) as a composition factor of H0(λ) is equal to 1.
This allows us to use the dominant weights X (T )+ to parameterize all simple modules for G. We call L(λ) the simple G-module of highest weight λ, and λ is the highest weight of L(λ). We can deduce that the formal character of L(λ) is
X Ch L(λ) = e(λ) + dim(L(λ)µ)e(µ) [9]. µ<λ
From this, we find that for λ ∈ X (T )+, the formal characters Ch L(λ) are linearly independent. This means that any two finite dimensional G-modules have the same character if and only if they have the same simple composition factors with the same multiplicities [9].
Let w0 be the longest element of the Weyl group. For each dominant weight λ ∈ X (T )+, H0(λ) is a finite dimensional module [9]. Define the Weyl module of weight λ to be V (λ) =
0 ∗ 0 H (−w0λ) . We have the property that Ch V (λ) = Ch H (λ) [9], and thus L(λ) appears as a composition factor of V (λ) exactly once. To be more precise, we can dualize Proposition 3.2.3 to get the property
V (λ)/ radG V (λ) =∼ L(λ).
Furthermore, by the following lemma, we find that the V (λ) described here is the same as the Weyl modules constructed in Section 2.3.1. Lemma 3.2.4. The G-module V (λ) is generated by a B-stable line of weight λ. Any G module generated by a B-stable line of weight λ is isomomorphic to an image of V (λ). With this lemma, we get a universal property of the modules V (λ). Because the Weyl modules defined in Section 2.3.1 have the universal property, they must be isomorphic. Thus we let V (λ) be the Weyl module of weight λ, which has a unique irreducible quotient L(λ). Consider the case where a Chevalley group is defined over a field which is not algebraically
t closed. For example, consider the Chevalley group G = GV (Fq), where q = p for a
39 prime p. Let G(k) be the algebraic group constructed over the algebraically closed field k of characteristic p so that G is contained in G(k). The simple modules for G are restrictions of some of the simple rational modules for G(k). Recall that the simple rational modules for
G(k) are simply the L(λ), with λ parameterized by X+. Let
∨ t Xt = {λ ∈ X+ | 0 ≤ hλ, α i ≤ p − 1 for all α ∈ S}.
The simple modules L(λ) with λ ∈ Xt remain simple when restricted to be considered as modules for G. The set {L(λ) | λ ∈ Xt } forms a complete set of nonisomorphic simple modules for G. For a complete theory and proofs, see the original source [14].
40 CHAPTER 4 JANTZEN SUM FORMULA AND APPLICATIONS Let G be a semisimple, simply connected algebraic group over an algebraically closed field k of characteristic p > 0. In this chapter, we will further investigate the structure of Weyl modules and calculate some characters of simple modules. As we know from earlier chapters, the Weyl modules have characters given by Weyl’s character formula (Theorem 2.2.6). However, the problem of computing characters for the simple modules L(λ) is not yet complete, although many results on the subject exist. In particular, Lusztig offered a conjecture in 1980 for the characters of simple modules for p ≥ h, for h the Coxeter number [11]. Lusztig’s conjecture has been proved for p sufficiently large by Andersen, Jantzen, and Soergel [1], but a reasonable bound on p has yet to be estabilished. We would like to use the structure of Weyl modules to study the characters of simple
modules for some particular weights. Let G be of type B4 or type D4 with roots labeled as
in Example 2.1.5. If the fundamental weights are ωi , i = 1, 2, 3, 4, as they were defined in Example 2.1.8, then we compute the characters of the simple modules for the following highest weights.
1. For G of type B4, we consider the weights λ = rω4 for r ≥ 0.
2. For G of type D4, we consider the weights λ = r(ω3 + ω4) for r ≥ 0. For each weight λ listed above, we will calculate the composition factors of the Weyl modules V (λ) to provide a complete description of the strucure of V (λ). We are interested in the simple module L(λ), which appears exactly once as a composition factor of V (λ). In particular, L(λ) =∼ V (λ)/ radG V (λ). There have been several results that compute characters for the simple modules of interest. In characteristic 2, the characters of simple modules for groups of type B4 and D4 for all weights were computed [7], and in characteristic 3, the characters for all simple modules of type D4 are known [17]. For a general p > 0, there are results for some particular weights [2].
41 Our purpose in this chapter is to find the characters of the simple modules for weights listed above for all r where p is arbitrary. 4.1 Tools for Calculation
In order to find characters for L(λ) for λ ∈ X (T )+, we can reduce our calculations by considering Steinberg’s Tensor Product Theorem [14], [9]. Let X1(T ) = {λ ∈ X (T ) | 0 ≤ hλ, α∨i ≤ p − 1 for all α ∈ S} be the restricted weights. Pm i Theorem 4.1.1 (Steinberg’s Tensor Product Theorem). If λ = i=1 p λi with λi ∈ X1(T ), then
[1] [m] L(λ) =∼ L(λ0) ⊗ L(λ1) ⊗ · · · ⊗ L(λm) .
As a result of this theorem, the character of L(λ) can be determined by calculating the character of the L(λi ) modules. Thus we restrict our study to the characters of L(λ) with
λ ∈ X1(T ).
We wish to study the weights λ = rω4 in groups of type B4 and λ = r(ω3 + ω4) in groups of type D4. In both of these situations, λ is a restricted weight when 0 ≤ r ≤ p − 1. Our primarily tool for calculation is Jantzen’s sum formula [9], which gives a partial description of V (λ). For each dominant weight λ, there exists a filtration of the G-module V (λ)
V (λ)0 ⊃ V (λ)1 ⊃ V (λ)2 ⊃ · · · such that V (λ)/V (λ)1 =∼ L(λ).
The Jantzen sum is defined to be
X i X X J(λ) = Ch(V (λ) ) = − vp(mp)χ(λ − mpα). i>0 α>0 m 0 In this sum, vp(n) is the p-adic valuation of n and χ is the Weyl character, which has the following description. To calculate χ(µ), we consider the weight µ0 = w(µ + ρ) − ρ, for 42 w the element of the Weyl group such that w(µ + ρ) is dominant. If µ0 is dominant, then χ(µ) = sign(w) Ch(V (µ0)), which is given by Weyl’s character formula. Otherwise, χ(µ) = 0. Note that if µ is a dominant weight, then χ(µ) = Ch(V (µ)). We say that a root pair (α, m) is a relevant root pair if α > 0 and 0 < mp < (λ + ρ, α∨). A relevant root pair is a contributor to the Jantzen sum when χ(λ − mpα) is nonzero. For this to occur, we need to calculate w((λ − mpα) + ρ) − ρ, with w((λ − mpα) + ρ) the dominant conjugate of (λ − mpα) + ρ. Because the sum formula gives the sum of characters of the filtration, some of the composition factors are counted more than once. Thus using the formula does not always give all of the information about the structure of the module. However, if V (λ) is a simple module, then its corresponding Jantzen sum will be zero. In the case where J(λ) is nonzero and depends on other weights, we use the Jantzen sum formula again to determine the structure of the Weyl modules of the related weights. To ease our calculations, consider the following standard lemma. Lemma 4.1.2. If λ + ρ − mpα is orthogonal to some root, then the pair (α, m) contributes nothing to the Jantzen sum. Proof. If λ + ρ − mpα is orthogonal to a root β, then (λ + ρ − mpα, β∨) = 0. There exists an element w of the Weyl group such that µ = w(λ + ρ − mpα) is dominant, and (µ, w(β∨)) = 0. Let γ be the positive root such that γ = ±w(β∨). Because there exists some γ ∈ ¨+ with (µ, γ∨) = 0, then (µ − ρ, γ∨) < 0, which means that µ − ρ is not dominant, and thus χ(λ − mpα) = 0. Note that in groups of type B4 and D4, the set {±ei ± ej | 1 ≤ i < j ≤ 4} is a subset of R. A vector µ = (c1, c2, c3, c4) is orthogonal to a root of the form β = ±ei ± ej when ∨ 0 = (µ, β ) = ±ci ± cj , i.e. the absolute value of two of the coordinates of µ are the same. We will use this fact to apply the lemma. In the following sections, we see how the Jantzen sum formula can be used to calculate characters of simple modules. The proof of each of the following theorems is formatted in the 43 following way. Each proof is based on the computations for the weight. Each time we use the Jantzen sum formula for a weight λ, we must do the following. (a) Determine the relevant root pairs (α, m). (b) For each root pair, calculate λ + ρ − mpα. (c) Determine whether λ + ρ − mpα is orthogonal to a root β, i.e., the root pair (α, m) is a noncontributor. (d) If the root pair is a contributor, find the Weyl group element w. (e) Calculate w(λ + ρ − mpα) − ρ. The computations for each proof are given in the tables in AppendicesA andB. The calculations are separated by contributors and noncontributors. For a noncontributor, the information listed includes the root pair (α, m) written as the vector mα, λ + ρ − mpα written as a vector, and an orthogonal root β written as a vector. The information listed for contributors includes mα, λ+ρ−mpα, and w(λ+ρ−mpα) written as vectors, the sign of the Weyl element w, and w(λ + ρ − mpα) − ρ written as a tuple of coefficients in its expression as a sum of fundamental weights. For the sake of completeness, we include the results which were previously computed: results in characteristic 2 for B4 and D4 [7] and in characteristic 3 for D4 [17]. Remark 4.1.3. We were led to the discovery of Theorems 4.2.1 and 4.3.1 with the help of computations of several examples in Sage [13]. 4.2 Weyl Modules of Simple Algebraic Groups of Type B4 Theorem 4.2.1. Let G be a simply connected, semisimple algebraic group of type B4 over an algebraically closed field of characteristic p. Consider the Weyl modules of the form V (rω4) for 0 ≤ r ≤ p − 1. (a) If p = 2, then V (ω4) is simple. (b) If p = 3, then V (ω4) and V (2ω4) are simple. (c) If p = 5, then the following hold. (i) V (ω4) and V (2ω4) are simple. 44 (ii) V (3ω4) has a simple radical isomorphic to V (ω1 + ω4), resulting in the exact sequence 0 → V (ω1 + ω4) → V (3ω4) → L(3ω4) → 0. (iii) The radical of V (4ω4) is isomorphic to the direct sum of V (ω1) and V (2ω1), each of which is simple. This results in the exact sequence 0 → V (ω1) ⊕ V (2ω1) → V (4ω4) → L(4ω4) → 0. (d) If p ≥ 7, then the following hold. (i) If 0 ≤ r ≤ p − 3, then V (rω4) is simple. (ii) rad(V ((p−2)ω4)) =∼ L(ω1 +(p−4)ω4), rad(V (ω1 +(p−4)ω4)) =∼ L(ω3 +(p−6)ω4), and rad(V (ω3 + (p − 6)ω4)) =∼ V ((p − 6)ω4), which is a simple module. This results in the exact sequence 0 → V ((p − 6)ω4)) → V (ω3 + (p − 6)ω4) → V (ω1 + (p − 4)ω4) → V ((p − 2)ω4) → L((p − 2)ω4) → 0. (iii) rad(V (p − 1)ω4)) =∼ L(2ω1 + (p − 5)ω4), rad(V (2ω1 + (p − 5)ω4)) =∼ L(ω1 + ω3 + (p − 7)ω4), and rad V (ω1 + ω3 + (p − 7)ω4) =∼ V (ω2 + (p − 7)ω4), which is a simple module. This results in the exact sequence 0 → V (ω2 + (p − 7)ω4) → V (ω1 + ω3 + (p − 7)ω4) → V (2ω1 + (p − 5)ω4) → V ((p − 1)ω4) → L((p − 1)ω4) → 0. Proof of Theorem 4.2.1. We refer to AppendixA for all relevant tables for this proof. (a) Let p = 2. Each relevant root pair (α, m) is a noncontributor, with calculations found in Table A-1. Thus J(ω4) = 0 and V (ω4) is simple. (b) Let p = 3. If r = 1, each pair (α, m) is a noncontributing pair, as we see in Table A-2. The Jantzen sum is zero, and thus V (ω4) is simple. 45 If λ = 2ω4, then we find noncontributors and contributors given in Tables A-3 and A-4, respectively. Table A-4 indicates that the Jantzen sum is J(2ω4) = Ch(V (ω1)) − Ch(V (ω1)) + Ch(V (ω1 + ω2)) − Ch(V (ω1 + ω2)) = 0, and thus V (2ω4) is a simple module. (c) Let p = 5. (i) In the case where λ = ω4, we get Table A-5 of noncontributors. Because every pair (α, m) satisfying 0 < mp < (λ + ρ, α∨) is a noncontributor, the Jantzen sum is zero, and V (ω4) is simple. In the case where λ = 2ω4, we get Tables A-6 and A-7 for noncontributors and contributors, respectively. From Table A-7, we find the Jantzen sum is J(2ω4) = Ch(V (0)) − Ch(V (0)) = 0, which gives us that V (2ω4) is a simple module. (ii) Let λ = 3ω4. The calculations for 3ω4 and the related weights are listed in Tables A-8, A-9, and A-10. First we observe in Table A-9 that J(3ω4) = Ch(V (ω1 + ω4)). We see that all of the relevant roots for J(ω1 + ω4) are noncontributors in Table A-10, which indicates that J(ω1 + ω4) = 0 and that V (ω1 + ω4) is a simple module. Thus rad(V (3ω4)) =∼ V (ω1 + ω4). (iii) Let λ = 4ω4. The calculations for 4ω4 and related weights are listed in Tables A-11, A-12, A-13, A-14, and A-15. First we see that after simplifying the information from Table A-12, J(4ω4) = Ch(V (ω1)) + Ch(V (2ω1)). 46 Table A-13 shows that J(ω1) = 0 and Table A-15 shows that J(2ω1) = − Ch(V (ω1)) + Ch(V (ω1)) = 0, so both V (ω1) and V (2ω1) are simple modules. Finally we determine the structure of V (4ω4). Because V (ω1) and V (2ω1) are both simple, note that V (ω1) = 0 0 L(ω1) = H (ω1) and V (2ω1) = L(2ω1) = H (2ω1). Thus we apply Proposition II.4.13 from Jantzen [9], which gives us that for λ, µ ∈ X (T )+ with λ 6= µ, 1 0 ExtG (V (λ), H (µ)) = 0. 1 1 ∼ Thus we find ExtG (L(ω1), L(2ω1)) = ExtG (L(2ω1), L(ω1)) = 0, so rad(V (4ω4)) = V (ω1) ⊕ V (2ω1). (d) Let p ≥ 7. (i) Let λ = rω4, where 0 ≤ r ≤ p − 7. For these values of r, there are no pairs (α, m) satisfying 0 < mp < (λ + ρ, α∨), and thus the Jantzen sum J(λ) is zero, and the Weyl module V (rω4) is simple. Let λ = (p − 6)ω4. There is one pair (α, m) which we consider, namely m = 1, = (1, 0, 0, 0) + mp = 1 p + 1 , 1 p 1 , 1 p 3 , 1 p 5 α . We find that λ ρ − α − 2 2 2 − 2 2 − 2 2 − 2 , which is orthogonal to the root β = (1, 1, 0, 0). Thus (α, m) is a noncontributor and the Jantzen sum is zero, and V ((p − 6)ω4) is simple. ∨ Let λ = (p − 5)ω4. There are two pairs (α, m) satisfying 0 < mp < (λ + ρ, α ), and both are noncontributors, summarized in Table A-16. Thus the Jantzen sum is zero, and V ((p − 5)ω4) is simple. ∨ Let λ = (p − 4)ω4. Each pair (α, m) satisfying 0 < mp < (λ + ρ, α ) is a noncontributor, summarized in Table A-17. Thus the Jantzen sum is zero and V ((p − 4)ω4) is simple. 47 Let λ = (p − 3)ω4. The calculations for noncontributors and contributors are in Tables A-18 and A-19, respectively. The Jantzen sum is J((p − 3)ω4) = Ch(V ((p − 5)ω4)) − Ch(V ((p − 5)ω4)) = 0. Thus we have V ((p − 3)ω4) is simple. (ii) To find the structure of V ((p − 2)ω4), consider the information given in Tables A-20, A-21, A-22, A-23, A-24, and A-25. We find that the Jantzen sum J((p − 2)ω4) has contributing weights (p − 6)ω4, ω3 + (p − 6)ω4 and ω1 + (p − 4)ω4 from Table A-21. We have already shown that V ((p − 6)ω4) is simple. We consider the Jantzen sums of the other two relevant weights. We first examine J(ω3 + (p − 6)ω4), which has relevant information in Tables A-22 and A-23. We find that the only contribution is Ch(V ((p − 6)ω4)) (Table A-23), indicating that rad(V (ω3 + (p − 6)ω4)) =∼ V ((p − 6)ω4). Next we consider ω1 + (p − 4)ω4, which is described in Tables A-24 and A-25. Thus we calculate from Table A-25 J(ω1 + (p − 4)ω4) = − Ch(V ((p − 6)ω4)) + Ch(V (ω3 + (p − 6)ω4)) = − Ch(rad(V (ω3 + (p − 6)ω4))) + Ch(V (ω3 + (p − 6)ω4)). So rad(V (ω1 + (p − 4)ω4)) is isomorphic to the simple module L(ω3 + (p − 6)ω4). Finally we use this information and Table A-21 to find the radical of V ((p − 2)ω4). J((p − 2)ω4) = − Ch(V (ω3 + (p − 6)ω4)) + Ch(V ((p − 6)ω4)))+ Ch(V (ω1 + (p − 4)ω4)) = − Ch(rad(V (ω1 + (p − 4)ω4))) + Ch(V (ω1 + (p − 4)ω4))). Thus the radical of V ((p − 2)ω4) is isomorphic to the simple module L(ω1 + (p − 4)ω4). (iii) Let λ = (p − 1)ω4. The calculations for the Jantzen sum are given in Tables A-26, A-27, A-28, and A-29. From the information in Table A-26, after simplification, 48 J((p − 1)ω4) relies on the weights ω2 + (p − 7)ω4, ω1 + ω3 + (p − 7)ω4, and 2ω1 + (p − 5)ω4. We begin by considering the weight ω2 + (p − 7)ω4. Calculations from Table A-27 show that V (ω2 + (p − 7)ω4) is a simple module. Next we consider the weight ω1 + ω3 + (p − 7)ω4. Calculations from Table A-28 show that after simplifying, J(ω1 + ω3 + (p − 7)ω4) = Ch(V (ω2 + (p − 7)ω4)). Thus rad(V (ω1 + ω3 + (p − 7)ω4)) =∼ V (ω2 + (p − 7)ω4). Finally, we consider the Jantzen sum J(2ω1 + (p − 5)ω4), summarized in Table A-29. J(2ω1 + (p − 5)ω4) = − Ch(V (ω2 + (p − 7)ω4)) + Ch(V (ω1 + ω3 + (p − 7)ω4)) = − Ch(rad(V (ω1 + ω3 + (p − 7)ω4))) + Ch(V (ω1 + ω3 + (p − 7)ω4)), which shows that rad(V (2ω1 + (p − 5)ω4)) is isomorphic to L(ω1 + ω3 + (p − 7)ω4). We use all this information to find the structure of V ((p − 1)ω4) using its Jantzen sum. J((p − 1)ω4) = Ch(V (ω2 + (p − 7)ω4)) − Ch(V (ω1 + ω3 + (p − 7)ω4) + Ch(V (2ω1 + (p − 5)ω4)) = − Ch(rad(V (2ω1 + (p − 5)ω4))) + Ch(V (2ω1 + (p − 5)ω4)). So we find that rad(V ((p − 1)ω4)) is isomorphic to L(2ω1 + (p − 5)ω4). 4.3 Weyl Modules of Simple Algebraic Groups of Type D4 Theorem 4.3.1. Let G be a simply connected, semisimple algebraic group of type D4 over an algebraically closed field of characteristic p. Let µ = ω3 + ω4. Consider the Weyl modules of the form V (rµ) for 0 ≤ r ≤ p − 1. 49 (a) If p = 2, V (µ) has a simple radical isomorphic to V (ω1), resulting in the exact sequence 0 → V (ω1) → V (µ) → L(µ) → 0. (b) If p = 3, then V (µ) is simple. rad(V (2µ)) =∼ L(ω1 + µ), and rad(V (ω1 + µ)) =∼ V (ω2), which is a simple module. This results in the exact sequence 0 → V (ω2) → V (ω1 + µ) → V (2µ) → L(2µ) → 0. (c) Let p = 5. (i) V (µ) is simple. (ii) The radical of V (2µ) is isomorphic to the simple module V (2ω1), resulting in the exact sequence 0 → V (2ω1) → V (2µ) → L(2µ) → 0. (iii) V (3µ) is simple. (iv) rad(V (4µ)) =∼ L(ω1 + 3µ), rad(V (ω1 + 3µ)) =∼ L(ω2 + 2µ), and rad(V (ω2 + 2µ)) =∼ L(2µ). This results in the exact sequence 0 → V (2ω1) → V (2µ) → V (ω2 + 2µ) → V (ω1 + 3µ) → V (4µ) → L(4µ) → 0. (d) Let p ≥ 7. 1 r p−3 V (r ) (i) For ≤ ≤ 2 , µ is simple. p−1 r p 4 rad(V (r )) ∼ L(( p + 2r + 3) + (p r 3) ) (ii) If 2 ≤ ≤ − , then µ = − ω1 − − µ , rad(V ((−p+2r +3)ω1 +(p−r −3)µ)) =∼ L((−p+2r +2)ω1 +ω2 +(p−r −4)µ), and rad(V ((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ)) =∼ V ((−p + 2r + 2)ω1 + (p − r − 4)µ), which is a simple module. This results in the exact sequence 0 →V ((−p + 2r + 2)ω1 + (p − r − 4)µ) → V ((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ) → V ((−p + 2r + 3)ω1 + (p − r − 3)µ) → V (rµ) → L(rµ) → 0. 50 (iii) The radical of V ((p − 3)µ) is isomorphic to the simple module V ((p − 3)ω1), resulting in the exact sequence 0 → V ((p − 3)ω1) → V ((p − 3)µ) → L((p − 3)µ) → 0. (iv) V ((p − 2)µ) is simple. (v) rad(V ((p − 1)µ)) =∼ L(ω1 + (p − 2)µ), rad(V (ω1 + (p − 2)µ)) =∼ L(ω2 + (p − 3)µ), and rad(V (ω2 + (p − 3)µ)) =∼ L((p − 3)µ). This results in the exact sequence 0 →V ((p − 3)ω1) → V ((p − 3)µ) → V (ω2 + (p − 3)µ) → V (ω1 + (p − 2)µ) → V ((p − 1)µ) → L((p − 1)µ) → 0. Proof of Theorem 4.3.1. Let µ = ω3 + ω4. We refer to AppendixB for all relevant tables for this proof. (a) Let p = 2; let λ = µ. The calculations for V (µ) are found Tables B-1, B-2, and B-3. We see that J(µ) = Ch(V (ω1)). Table B-3 shows that V (ω1) is simple and thus rad(V (µ)) =∼ V (ω1). (b) Let p = 3. For the case λ = µ, we consider the relevant roots listed in Table B-4. We see that each root is a noncontributor, and thus the Jantzen sum is zero and J(λ) is simple. When λ = 2µ, we find the relevant roots described in Tables B-5, B-6, B-7, B-8, B-9, and B-10. We find that the Jantzen sum is (Table B-6) J(2µ) = Ch(V (0)) − Ch(V (0)) − Ch(V (ω2)) + Ch(V (ω1 + µ)). We see from Table B-8 that J(ω2) = Ch(V (0)) − Ch(V (0)) = 0, revealing that V (ω2) is a simple module. The calculations for contributors to J(ω1 + µ) are listed in Table B-10. We calculate the Jantzen sum to be J(ω1 + µ) = Ch(V (0)) − Ch(V (0)) + Ch(V (ω2)) = Ch(V (ω2)). 51 Thus we find that rad(V (ω1 + µ)) =∼ V (ω2), and then J(2µ) = − Ch(rad(V (ω1 + µ))) + Ch(V (ω1 + µ)), so that rad(V (2µ)) =∼ L(ω1 + µ), and the theorem holds. (c) p = 5 (i) In the case where λ = µ, we find that all of the relevant roots are noncontributors, shown in Table B-11. Thus the Jantzen sum is zero and V (µ) is a simple module. (ii) Let λ = 2µ. Consider Tables B-12, B-13, and B-14. The calculations for the Jantzen sum in Table B-13 show that J(2µ) = Ch(V (2ω1)). Table B-14 shows that J(2ω1) is zero, so V (2ω1) is simple, and thus rad(V (2µ)) =∼ V (2ω1). (iii) Let λ = 3µ. Each relevant pair (α, m) is listed in Table B-15 and is a noncontributor. Thus the Jantzen sum is zero and V (2µ) is simple. (iv) All of the relevant weights to 4µ are listed in Tables B-16, B-17, B-18, B-19, B-20, and B-21. First, we see from Table B-17 that J(4µ) = − Ch(V (2ω1)) + Ch(V (2µ)) − Ch(V (ω2 + 2µ)) + Ch(V (ω1 + 3µ)). In part (cii), we observed that V (2ω1) is simple and rad(V (2µ)) =∼ V (2ω1). The two other relevant weights to V (4µ) are ω2 + 2µ, and ω1 + 3µ. First consider λ = ω2 + 2µ. From Table B-19, we find that there are two contributing pairs to the Jantzen sum. Calculations show J(ω2 + 2µ) = Ch(V (2µ)) − Ch(V (2ω1)) = Ch(V (2µ)) − Ch(rad(V (2µ))). This implies that rad(V (ω2 + 2µ)) =∼ L(2µ). Finally, consider the calculation for J(ω1 + 3µ). We find the following in Table B-21. J(ω1 + 3µ) = − Ch(V (2µ)) + Ch(V (2ω1)) + Ch(V (ω2 + 2µ)) = Ch(V (ω2 + 2µ)) − Ch(rad(V (ω2 + 2µ))). 52 Then we have that rad(V (ω1 + 3µ)) =∼ L(ω2 + 2µ). We return to the calculation J(4µ), to find that J(4µ) = Ch(V (ω1 + 3µ)) − Ch(rad(V (ω1 + 3µ))). Thus we have that rad(V (4µ)) =∼ L(ω1 + 3µ). (d) Let p ≥ 7. 0 r p−5 ( , m) (i) If ≤ ≤ 2 , then there are no relevant pairs α towards the Jantzen sum, and V (rλ) is a simple module. r = p−3 If 2 , then the relevant information can be found in Table B-22. Since both J p−3 = 0 V p−3 relevant roots are noncontributors, 2 µ and so 2 µ is simple. p−1 r p 4 (ii) Let 2 ≤ ≤ − . All information about related weights is listed in Tables B-23, B-24, and B-25. First we see from Table B-23 that the Jantzen sum J(rµ) is J(rµ) = Ch(V ((−p + 2r + 2)ω1 + (p − r − 4)µ)) + Ch(V ((−p + 2r + 3)ω1 + (p − r − 3)µ)) − Ch(V ((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ)). We examine J((−p + 2r + 2)ω1 + (p − r − 4)µ) and find that there are no pairs ∨ (α, m) satisfying 1 < mp < ((−p + 2r + 2)ω1 + (p − r − 4)µ + ρ, α ). So J((−p + 2r + 2)ω1 + (p − r − 4)µ) = 0 and V ((−p + 2r + 2)ω1 + (p − r − 4)µ) is simple. Next we consider λ = (−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ. There is only one relevant pair (α, m), namely m = 1, α = (1, 1, 0, 0). The calculations for this root are listed in Table B-24. We see that J((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ) = Ch(V ((−p + 2r + 2)ω1 + (p − r − 4)µ)), 53 and so rad(V ((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ)) =∼ V ((−p + 2r + 2)ω1 + (p − r − 4)µ). Finally consider λ = (−p + 2r + 3)ω1 + (p − r − 3)µ. Our calculations in Table B-25 show that J(λ) = Ch(V ((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ)) | {z } µ1 − Ch(V ((−p + 2r + 2)ω1 + (p − r − 4)µ)) = Ch(V (µ1)) − Ch(rad(V ((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ)) = Ch(L(µ1)). Thus we find that rad(V ((−p + 2r + 3)ω1 + (p − r − 3)µ)) =∼ L((−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ). We return to J(rµ) to find that J(rµ) = Ch(V ((−p + 2r + 3)ω1 + (p − r − 3)µ)) − Ch(rad(V ((−p + 2r + 3)ω1 + (p − r − 3)µ))) = Ch(L((−p + 2r + 3)ω1 + (p − r − 3)µ)). Thus rad(V (rµ)) =∼ L((−p + 2r + 3)ω1 + (p − r − 3)µ). (iii) Consider (p − 3)µ. All calculations for weights relevant for (p − 3)µ are in Tables B-26, B-27, and B-28. We find from Table B-27 that J((p − 3)µ) = Ch(V ((p − 3)ω1)). Then we find that both relevant root pairs for (p − 3)ω1 have zero contribution for J((p − 3)ω1) (Table B-28). Thus the Jantzen sum is zero, V ((p − 3)ω1) is simple, and rad(V ((p − 3)µ)) =∼ V ((p − 3)ω1). 54 (iv) Let λ = (p − 2)µ. Each relevant pair (α, m) is a noncontributor, with calculations summarized in Table B-29. Thus J((p − 2)µ) is zero and V ((p − 2)µ) is a simple module. (v) Let λ = (p − 1)µ. Information for relevant weights for (p − 1)µ is listed in Tables B-30, B-31, B-32, B-33, B-34, and B-35. First observe from Table B-31 that J((p − 1)µ) = − Ch(V ((p − 3)ω1)) + Ch(V ((p − 3)µ)) − Ch(V (ω2 + (p − 3)µ)) + Ch(V (ω1 + (p − 2)µ)). From (diii), we recall that rad(V ((p − 3)µ)) =∼ V ((p − 3)ω1), which is a simple module. Thus it remains to consider the related weights ω2 + (p − 3)µ and ω1 + (p − 2)µ. Our calculations in Table B-33 show that J(ω2 + (p − 3)µ) = Ch(V ((p − 3)µ)) − Ch(V ((p − 3)ω1)) = Ch(L((p − 3)µ)). Thus rad(V (ω2 +(p−3)µ)) =∼ L((p−3)µ). Finally, consider the weight ω1 +(p−2)µ. Our calculations from Table B-35 simplify as follows. J(ω1 + (p − 2)µ) = Ch(V ((p − 3)ω1)) − Ch(V ((p − 3)µ) + Ch(V (ω2 + (p − 3)µ)) = − Ch(rad(V (ω2 + (p − 3)µ)) + Ch(V (ω2 + (p − 3)µ)) = Ch(L(ω2 + (p − 3)µ)). Thus rad(V (ω1 + (p − 2)µ) =∼ L(ω2 + (p − 3)µ). We apply this to J((p − 1)µ) to find that rad(V ((p − 1)µ)) =∼ L(ω1 + (p − 2)µ), which completes the proof. J((p − 1)µ) = − Ch(rad(V (ω1 + (p − 2)µ))) + Ch(V (ω1 + (p − 2)µ)) = Ch(L(ω1 + (p − 2)µ)). 55 CHAPTER 5 BUILDINGS The classification and study of finite simple groups, such as the finite simple groups of Lie type, has long been an interest for mathematicians. The first groups of Lie type to be studied were the classical groups, which include the general linear groups, certain subgroups of the general linear groups, and their quotient groups. In particular, the special linear groups, orthogonal groups, symplectic groups, and unitary groups are examples of the classical groups. This chapter focuses on the broader subject of classical groups and an associated geometric structure, namely the building. Then we will consider how these subjects relate to the Chevalley groups. First we will define some examples of classical groups and identify them with the appropriate Chevalley groups. In particular, we will examine the orthogonal groups. 5.1 Classical Groups and Chevalley Groups Let V be an n-dimensional vector space over a field k of characteristic not equal to 2 that has a nonsingular symmetric bilinear form B : V × V → V . It is well known that B determines a quadratic form f on V defined by f (x) = B(x, x). Let On(k, f ) be the matrix group of nonsingular linear isometries of V , which is called the orthogonal group with quadratic form f . That is, On(k, f ) = {x ∈ GLn(k) | B(x.u, x.v) = B(u, v) for all u, v ∈ V }. The determinant of elements of On(k, f ) is either +1 or −1, which leads us to the natural definition of SOn(k, f ), the subgroup of the orthogonal group consisting of transformations with determinant +1. Consider the commutator subgroup of On(k, f ), sometimes called the derived subgroup, which we denote n(k, f ). In fact, n(k, f ) is a subgroup of SOn(k, f ). We can also consider the projective group of the derived subgroup, defined P n(k, f ) = n(k, f )/(Z(On(k, f )) ∩ n(k, f )), where Z(G) denotes the center of a group G. Example 5.1.1. The following identifications of classical groups with Chevalley groups are proved in [4]. Certain orthogonal groups can be identified with some of the Chevalley groups. 56 Recall from Theorem 3.1.5 that each lattice L such that ZR ⊆ L ⊆ £ has a corresponding group G constructed from L. Thus the fundamental group £/ZR of R provides information on the possible lattices and Chevalley groups for a fixed Lie algebra. Let V be the underlying n-dimensional vector space over a field k. First consider the case where n = 2` + 1 is odd. Let e0, e1, ... , e`, e−`, ... , e−2, e−1 be the standard basis of V . Consider the special orthogonal Lie algebra of type B`, so(V ). Recall that the fundamental B 2 group of a root system of type ` is isomorphic to Z/ Z. When we construct a Chevalley group from the representation V , we find that GV (k) =∼ (V ). However, £(V ) is properly contained in £, and so GV (k) is not the universal group. In fact, because the dimension of V is odd, −1 is not an element of SO(V ) and thus not in (V ). In this case, P (V ) = (V ), and the adjoint Chevalley group is P 2`+1(k, fB ), for fB the quadratic form defined by 2 fB (x) = x0 + x1x−1 + x2x−2 + ··· + x`x−` P for x = i xi ei ∈ V . The universal group of type B` is isomorphic to the spin group, denoted Spin(V ). In the case where n = 2`, and V has a basis e1, ... , e`, e−`, ... , e−1. The vector space V has a quadratic form fD defined by fD (x) = x1x−1 + x2x−2 + ··· + x`x−` x = P x e D 4 for i i i . Recall that in type `, the fundamental group is Z/ Z when ` is odd, and 2 2 D Z/ Z × Z/ Z when ` is even. In both cases, the universal Chevalley group of type ` is isomorphic to Spin(V ), and the adjoint Chevalley group is P 2`(k, fD ). 5.1.1 (B, N)-pairs The theory of groups with a (B, N)-pair has been useful in deriving further properties of Chevalley groups and defining the geometric structure that we will discuss in Section 5.2. Definition 5.1.2. A (B, N)-pair in a group G is a system (B, N) consisting of two subgroups such that 57 (BN1) B and N generate G. (BN2) Let B ∩ N = H. The H is a normal subgroup of N. (BN3) The group W = N/H has a generating set of involutions {wi }i∈I . (BN4) If ni ∈ N is a representative of wi ∈ W under the natural homomorphism of N onto W , then for any n ∈ N, Bni B BnB ⊆ Bni nB ∪ BnB. (BN5) If ni is as in (BN4), then ni Bni 6= B. Note that if G is a Chevalley group, then many of these properties have already been shown. Let B and N be defined as in Section 2.3.2. Part (BN1) of Definition 5.1.2 can be deduced from Lemma 2.3.4, because B is generated by root subgroups, and the ωα ∈ N for α the simple roots. Parts (BN2) and (BN3) are immediate from Lemma 2.3.2 so that W is isomorphic to the Weyl group and the involutions wi are the simple reflections wα with α the simple roots. Part (BN4) comes from Lemma 2.3.3. A complete proof of the following summarizing proposition can be found in [4]. Proposition 5.1.3. Let G be a Chevalley group with subgroups B, N, and H defined as in Section 2.3.2. Let W be the Weyl group. G has a (B, N)-pair. We will discuss Chevalley groups as groups with (B, N)-pairs and use the properties derived from (B, N)-pairs in a later section. 5.1.2 Twisted Groups Certain classical groups or their derived subgroups are isomorphic to Chevalley groups of type A`, B`, C`, and D`, but some of the classical groups are not [4]. In particular, the unitary groups and certain orthogonal groups of even dimension are not Chevalley groups. However, these groups can be constructed by twisting particular Chevalley groups. Here we present an overview of this construction. For a complete account, see [4]. Twisted groups are subgroups of fixed points of automorphisms, and so first we must define certain types of automorphisms of Chevalley groups. Let G be a Chevalley group defined over a field k. An automorphism of G will be uniquely determined by its effect on the 58 generators of the group, i.e., the xα(t). We begin by defining a field automorphism of G. Let f be an automorphism of the field k. Consider the map defined by xα(t) 7→ xα(f (t)). We can show by checking this map on the defining relations of G that this map can be extended to an automorphism of the group G. Automorphisms of G induced by an automorphism of the field k are called field automorphisms. The other automorphisms we consider are the graph automorphisms of G. However, in order to define these automorphisms, we first define a symmetry ρ of the Dynkin diagram of a Lie algebra g. A symmetry ρ is a permutation of nodes of the Dynkin diagram which preserves the number of edges between nodes. A proposition proved in [4] shows that all possible symmetries are pictured in Figure 5-1. Each nontrivial symmetry of the Dynkin diagram induces an isometry τ on the real vector space E containing R, which acts on the simple roots as ρ acts on the corresponding nodes of the Dynkin diagram [4]. A` D` D4 E6 B2 < F 4 > G2 > Figure 5-1. Possible symmetries of Dynkin diagrams Suppose that all roots in R are the same length. In this case, the isometry τ induces a map xα(t) 7→ xτ(α)(γαt), where γα = ±1 for α ∈ R (γα can be +1 if α ∈ S or −α ∈ S)[4]. This map extends to an automorphism of G. Automorphisms of this type are called graph 59 automorphisms. For the symmetries in Figure 5-1 for the Dynkin diagrams of type B2, F4, and G2, graph automorphisms do not arise in general. In particular, they only exist over certain fields [4]. The construction of such automorphisms is more complicated. Thus for our purposes, we only consider the first case. To construct a twisted subgroup of a Chevalley group, let G ∗ be a Chevalley group such that the root system R has all roots the same length, and there exists a nontrivial symmetry ρ of the Dynkin diagram of order n as in Figure 5-1. Let τ be the isometry of E induced by ρ. ∗ Write α = τ(α). Let g be the graph automorphism of G such that g(Xα) = Xα for α ∈ S. Define an automorphism σ of G ∗ such that σ = gf for f a nontrivial field automorphism of G ∗ such that σn = 1, where n is the order of ρ. In the case that all roots have the same length, this occurs when f n = 1. Then the twisted group G is a group of fixed points of the ∗ ∗ automorphism σ of G . Note that if G is defined over a finite field Fq, then q must be of the n form q = q0 for q0 a prime power. This is needed for the existence of a field automorphism f such that σ = gf , which is necessary in the construction. 5.2 Buildings Buildings are geometric structures introduced by Tits in [16]. We will give an abstract definition of a building, construct a building from a group with a (B, N)-pair, and finally consider the building that arises from a Chevalley group. Definition 5.2.1. An abstract simplicial complex X is a building, and elements of a set of subcomplexes A are called apartments if the pair (X , A) satisfy the following conditions. (B1) Every element of codimension 1 of X is contained in at least three chambers (maximal elements) of X . (B2) Each element of A is contained in exactly two chambers of A. (B3) Any two elements of X belong to a common apartment. (B4) If two apartments A and A0 contain elements σ, σ0 ∈ X , there exists an isomorphism of A onto A0 which leaves σ, σ0, and all their faces invariant. 60 If G is a group with a (B, N)-pair, then there exists a construction of a building from G. For a subset J of the indexing set I of Definition 5.1.2 (BN3), we can define the subgroup WJ of W to be the group generated by the involutions wj for j ∈ J. Define NJ to be the subgroup of N that maps to WJ under the homomorphism N → N/H =∼ W . Let PJ = BNJ B. The subgroups of G that contain a conjugate gBg−1 of B are called parabolic subgroups. The subgroups PJ for J ⊆ I are examples of parabolic subgroups and in fact, the conjugates of the PJ are exactly the parabolic subgroups of G [4]. Now we define the elements of the building, and use them to construct it. Define a type to be a nonempty subset J of I and an object of type I \ J (or cotype J) to be a left coset of PJ in G. Let be the set of cosets gPJ where g ranges over G and J ranges over all subsets of I . Define a partial order on to be the reverse of set inclusion. The associated building is the simplicial complex such that simplices are the objects, i.e., the cosets gPJ , and the face relation is given by the partial order. From this construction, any group with a (B, N)-pair results in a corresponding building, which is shown in [4]. Example 5.2.2. Here we demonstrate a geometric interpretation to the building from the (B, N)-pair given by Tits in [16]. Note that this interpretation is similar to the geometric interpretation of the Bruhat decomposition given in Example 2.3.6. Recall from Proposition 5.1.3 that each Chevalley group has a (B, N)-pair. Let G = SL(V ) be of type A`, where V is an (` + 1)-dimensional vector space. A k-simplex can be represented by a flag of k + 1 nontrivial, proper subspaces of V . A flag of type J ⊂ I is a chain of subspaces of dimensions j ∈ J. To be more precise, suppose V has basis {e1, ... , e`+1}. If J = I , the standard flag of type J is he1i ⊂ he1, e2i, · · · ⊂ he1, e2, ... , e`i. The standard flag of type {1} is the subspace he1i. A chambers of the building is a maximal flag. To interpret an apartment A for this building, first fix a basis for V . Elements of A are flags with respect to this basis. 61 This interpretation of buildings is the standard way to consider buildings constructed from Chevalley groups. This is the way that we will discuss buildings throughout this chapter. 5.3 Oppositeness in Buildings 5.3.1 Incidence Relations A relation R between the objects of two cotypes J and K can be encoded by an incidence matrix or an incidence map of permutation modules. We will define these permutation modules and study some particular incidence maps between them. Let k be a commutative ring with 1. Define FJ to be the space of functions f : G/PJ → k. We can make FJ into a left kG-module −1 by defining the action (xf )(gPJ ) = f (x gPJ ) for x, g ∈ G and f ∈ FJ . Given a G-invariant X relation R, we can define a kG-homomorphism η : FJ → FK by η(f )(hPK ) = f (gPJ ), gPJ ∼hPK where ∼ represents that gPJ and hPK are related under R. We will study this homomorphism in order to determine the rank of the corresponding incidence matrix. 5.3.2 Oppositeness We would like to examine a specific relation, namely oppositeness. In this section, we follow the construction of oppositeness given in [12]. First, we define what it means for two objects to be opposite. Let w0 be the unique element of maximum length in W , and let l(w) denote the length of w ∈ W . Definition 5.3.1. Define two types J and K to be opposite if {−w0(αj ) | j ∈ J} = {αk | k ∈ K}. Note that if w0 = −1, then each type is opposite to itself, i.e., all connected root systems except A`, D2`+1, E6. Definition 5.3.2. Let J, K be fixed cotypes. An object gPJ of cotype J and an object hPK of cotype K are opposite each other if −1 PK h gPJ = PK w0PJ 62 An equivalent conditions for gPJ and hPK to be opposite is gPJ ⊆ hPK w0PJ . This equivalent definition will be more useful to define the opposite homomorphism η described below. Let k be an algebraically closed field of characteristic p. Let J and K be opposite cotypes. Consider the kG-modules FJ and FK . The function η : FJ → FK induced by oppositeness is defined by X η(f )(hPK ) = f (gPJ ). gPJ ⊆hPK w0PJ The following theorem gives the needed information about the function η to analyze oppositeness. The theorem comes from [12], proved as a corollary of the results in [5]. Theorem 5.3.3. The image of η is a simple module, uniquely characterized by the property that its one-dimensional U-invariant subspace has full stabilizer equal to PJ , which acts trivially on it. The dimension of the image of the incidence homomorphism η is equal to the rank of the incidence matrix. Thus we will use Theorem 5.3.3 to determine the rank of the incidence matrix of the oppositeness relation. We shift our focus to Chevalley groups, and discuss the subject of oppositeness for these groups. Our goal in the next section is to determine the simple module in Theorem 5.3.3. 5.3.3 Chevalley Groups Recall from Chapter3 that if G(K) is the algebraic group constructed as a Chevalley group over the algebraically closed field K, every simple G(K)-module is isomorphic to a module L(λ) for λ the highest weight of the module. If G is a Chevalley group defined over t k = Fq where q = p for p a prime, then G is contained in an algebraic group G(K) for K an algebraically closed field. The simple G-modules are restrictions of simple G(K)-modules L(λ) for λ a q-restricted weight. That is, if q = pt , then ∨ t λ ∈ Xt (T ) = {λ ∈ X (T ) | 0 ≤ hλ, α i ≤ p − 1}. 63 Therefore, because the image of η in Theorem 5.3.3 is a simple G-module, it must be isomorphic to L(λ) for some λ ∈ Xt (T ). Let λopp denote the q-restricted highest weight of L(λopp), the simple G-module in Theorem 5.3.3. If we determine λopp as it is shown in [12], then we can determine the dimension of the module to find the rank of the incidence matrix. So we now consider the two characterizing properties of the image of η in 5.3.3. First, consider the property that the one-dimensional U-invariant subspace of L(λopp) has full stabilizer equal to PJ . L( ) L The one-dimensional subspace of λopp is the high weight space, which we denote λopp , v + B L containing the high weight vector . Because the Borel subgroup stabilizes λopp , the stabilizer, which is a subgroup of G, must be a subgroup containing B. These are precisely the ∨ parabolic subgroups PJ for J ⊂ I . Furthermore, j ∈ J if and only if hλopp, αj i = 0. Thus X λopp = ci ωi , where {ωi | i ∈ I } is the set of fundamental weights. Next, interpret the i∈I \J P L T P condition of J acting trivially on λopp . Because is contained in J , λopp restricted to the maximal torus T is a trivial character. This is equivalent to the statement that when λopp is written as a linear combination of fundamental weights, the coefficients of the ωi must be either 0 or q − 1. Thus X λopp = (q − 1)ωi . (5–1) i∈I \J Using Equation 5–1 for λopp, we can look at some specific examples. Our goals here are to look at some specific types for particular groups, first interpreting the concept of oppositeness, and then calculating λopp to determine the rank of the incidence matrix associated with oppositeness. 5.3.4 Twisted Groups G ∗ k = q = qn I ∗ Let be a Chevalley group as in Section 5.1.2 over Fq, where 0 . Let be the indexing set of the simple roots of G ∗. Set I to be the indexing set of the ρ-orbits of I ∗. Then for any subset J of I , we can consider J∗ to be the union of the ρ-orbits of each j ∈ J, so that J∗ ⊂ I ∗. 64 Again, the simple modules for the twisted group G are the restrictions of the G(k)-modules t L(λ) with λ ∈ Xt (T ), for q = p with p a prime [14]. Thus in this case, the module in X Theorem 5.3.3 is still of the form L(λopp) for λopp ∈ Xt (T ). In fact, λopp = (q0 − 1)ωi i∈I ∗\J∗ [12]. 5.4 Examples 5.4.1 Groups of Type B4 Let G be a Chevalley group of type B4, for example, the adjoint group G = 9(Fq, fB ). The 9-dimensional vector space V is generated by vectors {e0, e1, e2, e3, e4, e−4, e−3, e−2, e−1}. V has a nonsingular quadratic form f that defines a nonsingular symmetric bilinear form B in B(u, v) = 1 f (u + v) f (u) f (v) the standard way (i.e. 2 ( − − )). The basis vectors have the following properties: for all i, j = 1, 2, 3, 4, B(ei , e−j ) = δij and B(ei , ej ) = B(e−i , e−j ) = B(ei , e0) = B(e−i , e0) = 0. Let I = {1, 2, 3, 4}, J = {4}. Now objects of cotype J correspond to 4 dimensional (maximal) isotropic subspaces of V . Type J is opposite to itself because w0 = −1 in B4. Recall the geometric interpretation of buildings in Example 5.2.2. In the interpretation, the standard flag PJ corresponds to the subspace E = {e1, e2, e3, e4}, and w0PJ corresponds to the subspace F = {e−1, e−2, e−3, e−4}. Proposition 5.4.1. A subspace W of V of type J is opposite to E if and only if W ∩ E = {0}. Proof. First assume that W is a four dimensional totally isotropic subspace of V opposite to E. In the geometric interpretation of the building, W corresponds to a coset gPJ of G. Recall that gPJ is opposite to E when gPJ ⊆ PJ w0PJ , and w0PJ corresponds to F . Suppose there exists a nonzero v ∈ E ∩ W . Then v is in the PJ -orbit of F , so v = g u, for some u ∈ F , 65 −1 −1 g ∈ PJ , but then u = g v = v since v ∈ E, and g ∈ PJ , which stabilizes E. Note that v ∈ E ∩ F = {0}, and thus E ∩ W = {0}. Conversely, suppose that W is a subspace of V such that W is a four-dimensional totally isotropic subspace and W ∩ E = {0}. Consider the subspace E ⊕ W ≤ V . We can find elements g1, g2, g3, g4 ∈ W such that hgi , ej i = δij for the following reason. We claim that W ∩ E ⊥ = {0}. Otherwise, if there exists nonzero w ∈ W ∩ E ⊥, then hw, Ei is a 5-dimensional isotropic subspace of V . Thus W ∩ E ⊥ = {0}. Thus we have an ⊥ ∗ injective mapping W ,→ V /E =∼ E , so there exist dual elements gi to the basis of E in W . Thus there exists an isometry g ∈ G such that g : E ⊕ W → E ⊕ F , g : ei 7→ ei , and g : gi 7→ e−i . By Witt’s Lemma, this lifts to an isometry g of V . Since g fixes e1, e2, e3, and e4, g ∈ Stab(E) and thus g ∈ PJ . Thus there exists g ∈ PJ such that gW = F , i.e. W is in the PJ -orbit of w0PJ , and so W and E are opposite objects. Corollary 5.4.2. Let W and W 0 be two four dimensional totally isotropic subspaces of V . W , W 0 are opposite if and only if W ∩ W 0 = {0}. Recall Theorem 4.2.1 on the characters of simple modules for algebraic groups of type B4. This example has examined oppositeness in such groups, and now we present an application of Theorem 4.2.1. The following proposition uses the results to determine the ranks of incidence matrices for oppositeness. Proposition 5.4.3. Let A be the incidence matrix for the relation of oppositeness in a t Chevalley group of type B4 over Fq for q = p . 1. If p = 2, then rank A = 16t . 2. If p = 3, then rank A = 126t . 3. If p = 5, then rank A = 2719t . 4. If p ≥ 7, then 1 t rank(A) = p(p + 1)(p + 2)(17p4 51p3 + 175p2 108p + 72) 630 − − 66 Proof. Recall Weyl’s dimension formula in Corollary 2.2.7. The theorem holds in all characteristics, so we use the dimension formula with Theorem 4.2.1, which describes the simple modules L(rω4) in a group of type B4, and Equation 5–1. From Steinberg’s Tensor Product Theorem, note that [1] [t] L((q − 1)ω4) =∼ L((p − 1)ω4) ⊗ L((p − 1)ω4) ⊗ · · · ⊗ L((p − 1)ω4) . t Thus the dimension of L((q − 1)ω4) is simply (dim L((p − 1)ω4)) . This reduces our calculations to computing dim L((p − 1)ω4). From Theorem 4.2.1, we get the following character formulas for L((p − 1)ω4). 1. If p = 2, then Ch L(ω4) = Ch V (ω4). 2. If p = 3, then Ch L(2ω4) = Ch V (2ω4). 3. If p = 5, then Ch L(4ω4) = Ch V (4ω4) − Ch V (ω1) − Ch V (2ω1). 4. If p ≥ 7, then Ch L((p−1)ω4) = Ch V ((p−1)ω4)−Ch V (2ω1+(p−5)ω4)+Ch V (ω1+ω3+(p−7)ω4) − Ch V (ω2 + (p − 7)ω4). To calculate these dimensions, we program Weyl’s dimension formula in Sage [13]. 5.4.2 Groups of Type D4 2 Let G be a twisted group of type D4 constructed from a Chevalley group over k = Fq, as a group of fixed points of an automorphism induced by an isometry ρ of the Dynkin ∗ diagram of an overlying Chevalley group G of type D4, for example, the adjoint group G ∗ = P ( , f ) G ∗ G = P ( , f ) 8 Fq D and the twisted group from is 8 Fq0 , which is shown in detail in [4]. In this example, the underlying 8-dimensional vector space V has a basis e1, e2, e3, e4, e−4, e−3, e−2, e−1 and the quadratic form f given by X f (x) = x1x−1 + x2x−2 + x3x−3 + (x4 − αx−4)(x4 − αx−4), for x = xi ei . i 67 k k = In the form, α is a generator of over 0 Fq0 , the fixed field under the field automorphism used to define G. α represents the image of α under the field automorphism. Let I ∗ = {1, 2, 3, 4} be the index set of the simple roots for the overlying Chevalley group ∗ of type D4, with fundamental dominant weights ωi , i ∈ I . Let I = {1, 2, 3} be the index set for G labeling the ρ orbits of I ∗. For any subset J of I , let J∗ be the union of the ρ orbits of J. Recall X λopp = (q0 − 1)ωi . i∈I ∗\J∗ Let J = {3}, so that J∗ = {3, 4}. We wish to interpret oppositeness in G for objects of type J. Objects of type J are 3-dimensional (maximal) isotropic subspaces of V . The Weyl 2 group of D4 is isomorphic to the Weyl group of B3 [4], so w0 = −1 and so J is opposite 2 to itself. Proposition 5.4.1 holds for the groups of type D4 with W a three-dimensional totally isotropic subspace of V in this case. The proof of this proceeds similarly to the proof of Proposition 5.4.1. Similarly to Corollary 5.4.2, for three-dimensional totally isotropic subspaces W and W 0 of V , W and W 0 are opposite if and only if W ∩ W 0 = {0}. Again, we demonstrate an application of the results on simple modules for algebraic groups of type D4 here. The following proposition is an application of Theorem 4.3.1 in the subject of oppositeness in buildings. Proposition 5.4.4. Let A be the incidence matrix for the relation of oppositeness in a twisted 2D q = q2 q = pt Chevalley group of type 4 over Fq for 0 , 0 . 1. If p = 2, then rank A = 48t . 2. If p = 3, then rank A = 518t . 3. If p = 5, then rank A = 10, 800t . 4. If p ≥ 7, then 1 t rank(A) = p2(7p2 + 3p + 2)(7p2 3p + 2) 72 − 68 APPENDIX A TABLES FOR CALCULATIONS IN GROUPS OF TYPE B4 The following tables are those referenced in Theorem 4.2.1. Table A-1. B4, p = 2. Noncontributors for λ = ω4. mα λ + ρ − mpα β (1, 0, 0, 0)(2, 3, 2, 1)(1, 0, −1, 0) (2, 0, 0, 0)(0, 3, 2, 1)(1, 0, 0, 0) (3, 0, 0, 0)(−2, 3, 2, 1)(1, 0, 1, 0) (1, 1, 0, 0)(2, 1, 2, 1)(0, 1, 0, −1) (2, 2, 0, 0)(0, −1, 2, 1)(0, 1, 0, 1) (3, 3, 0, 0)(−2, −3, 2, 1)(1, 0, 1, 0) (1, 0, 1, 0)(2, 3, 0, 1)(0, 0, 1, 0) (2, 0, 2, 0)(0, 3, −2, 1)(1, 0, 0, 0) (1, 0, 0, −1)(2, 3, 2, 3)(0, 1, 0, −1) (1, 0, 0, 1)(2, 3, 2, −1)(1, 0, −1, 0) (2, 0, 0, 2)(0, 3, 2, −3)(0, 1, 0, 1) (0, 1, 0, 0)(4, 1, 2, 1)(0, 1, 0, −1) (0, 2, 0, 0)(4, −1, 2, 1)(0, 1, 0, 1) (0, 1, 1, 0)(4, 1, 0, 1)(0, 0, 1, 0) (0, 2, 2, 0)(4, −1, −2, 1)(0, 1, 0, 1) (0, 1, 0, 1)(4, 1, 2, −1)(0, 1, 0, 1) (0, 0, 1, 0)(4, 3, 0, 1)(0, 0, 1, 0) (0, 0, 1, 1)(4, 3, 0, −1)(0, 0, 1, 0) Table A-2. B4, p = 3. Noncontributors for λ = ω4. mα λ + ρ − mpα β (1, 0, 0, 0)(1, 3, 2, 1)(1, 0, 0, −1) (2, 0, 0, 0)(−2, 3, 2, 1)(1, 0, 1, 0) (1, 1, 0, 0)(1, 0, 2, 1)(0, 1, 0, 0) (2, 2, 0, 0)(−2, −3, 2, 1)(1, 0, 1, 0) (1, 0, 1, 0)(1, 3, −1, 1)(0, 0, 1, 1) (1, 0, 0, 1)(1, 3, 2, −2)(0, 0, 1, 1) (0, 1, 0, 0)(4, 0, 2, 1)(0, 1, 0, 0) (0, 1, 1, 0)(4, 0, −1, 1)(0, 0, 1, 1) (0, 1, 0, 1)(4, 0, 2, −2)(0, 0, 1, 1) (0, 0, 1, 0)(4, 3, −1, 1)(0, 0, 1, 1) 69 Table A-3. B4, p = 3. Noncontributors for λ = 2ω4. mα λ + ρ − mpα β 1, 0, 0, 0 3 , 7 , 5 , 3 1, 0, 0, 1 ( ) 2 2 2 2 ( − ) 2, 0, 0, 0 3 , 7 , 5 , 3 1, 0, 0, 1 ( ) − 2 2 2 2 ( ) 1, 1, 0, 0 3 , 1 , 5 , 3 1, 0, 0, 1 ( ) 2 2 2 2 ( − ) 2, 2, 0, 0 3 , 5 , 5 , 3 0, 1, 1, 0 ( ) − 2 − 2 2 2 ( ) 1, 0, 1, 0 3 , 7 , 1 , 3 1, 0, 0, 1 ( ) 2 2 − 2 2 ( − ) 2, 0, 2, 0 3 , 7 , 7 , 3 0, 1, 1, 0 ( ) − 2 2 − 2 2 ( ) 1, 0, 0, 1 3 , 7 , 5 , 3 1, 0, 0, 1 ( ) 2 2 2 − 2 ( ) 0, 2, 0, 0 9 , 5 , 5 , 3 0, 1, 1, 0 ( ) 2 − 2 2 2 ( ) 0, 1, 1, 0 9 , 1 , 1 , 3 0, 1, 1, 0 ( ) 2 2 − 2 2 ( ) Table A-4. B4, p = 3. Contributors for λ = 2ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 0, 1, 0, 0 9 , 1 , 5 , 3 1 1, 0, 0, 0 ( ) 2 2 2 2 ( ) 0, 1, 0, 1 9 , 1 , 5 , 3 1 1, 0, 0, 0 ( ) 2 2 2 − 2 − ( ) 0, 0, 1, 0 9 , 7 , 1 , 3 1 0, 1, 0, 0 ( ) 2 2 − 2 2 ( ) 0, 0, 1, 1 9 , 7 , 1 , 3 1 0, 1, 0, 0 ( ) 2 2 − 2 − 2 − ( ) Table A-5. B4, p = 5. Noncontributors for λ = ω4. mα λ + ρ − mpα β (1, 0, 0, 0)(−1, 3, 2, 1)(1, 0, 0, 1) (1, 1, 0, 0)(−1, −2, 2, 1)(0, 1, 1, 0) (1, 0, 1, 0)(−1, 3, −3, 1)(0, 1, 1, 0) (0, 1, 0, 0)(4, −2, 2, 1)(0, 1, 1, 0) Table A-6. B4, p = 5. Noncontributors for λ = 2ω4. mα λ + ρ − mpα β 1, 1, 0, 0 1 , 3 , 5 , 3 0, 1, 0, 1 ( ) − 2 − 2 2 2 ( ) 1, 0, 0, 1 1 , 7 , 5 , 7 0, 1, 0, 1 ( ) − 2 2 2 − 2 ( ) 0, 1, 0, 0 9 , 3 , 5 , 3 0, 1, 0, 1 ( ) 2 − 2 2 2 ( ) 0, 1, 1, 0 9 , 3 , 5 , 3 0, 1, 0, 1 ( ) 2 − 2 − 2 2 ( ) Table A-7. B4, p = 5. Contributors for λ = 2ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 , 7 , 5 , 3 1 0, 0, 0, 0 ( ) − 2 2 2 2 ( ) 1, 0, 1, 0 1 , 7 , 5 , 3 1 0, 0, 0, 0 ( ) − 2 2 − 2 2 − ( ) 70 Table A-8. B4, p = 5. Noncontributors for λ = 3ω4. mα λ + ρ − mpα β (1, 0, 0, 0)(0, 4, 3, 2)(1, 0, 0, 0) (1, 1, 0, 0)(0, −1, 3, 2)(1, 0, 0, 0) (1, 0, 1, 0)(0, 4, −2, 2)(0, 0, 1, 1) (1, 0, 0, 1)(0, 4, 3, −3)(0, 0, 1, 1) (0, 1, 1, 0)(5, −1, −2, 2)(0, 0, 1, 1) (0, 1, 0, 1)(5, −1, 3, −3)(0, 0, 1, 1) (0, 0, 1, 0)(5, 4, −2, 2)(0, 0, 1, 1) Table A-9. B4, p = 5. Contributors for λ = 3ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (0, 1, 0, 0)(5, −1, 3, 2) −1 (1, 0, 0, 1) Table A-10. B4, p = 5. Noncontributors for λ = ω1 + ω4. mα λ + ρ − mpα β (1, 0, 0, 0)(0, 3, 2, 1)(1, 0, 0, 0) (1, 1, 0, 0)(0, −2, 2, 1)(0, 1, 1, 0) (1, 0, 1, 0)(0, 3, −3, 1)(0, 1, 1, 0) (1, 0, 0, 1)(0, 3, 2, −4)(1, 0, 0, 0) (0, 1, 0, 0)(5, −2, 2, 1)(0, 1, 1, 0) Table A-11. B4, p = 5. Noncontributors for λ = 4ω4. mα λ + ρ − mpα β 2, 0, 0, 0 9 , 9 , 7 , 5 1, 1, 0, 0 ( ) − 2 2 2 2 ( ) 1, 1, 0, 0 1 , 1 , 7 , 5 1, 1, 0, 0 ( ) 2 − 2 2 2 ( ) Table A-12. B4, p = 5. Contributors for λ = 4ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 , 9 , 7 , 5 1 0, 0, 1, 0 ( ) 2 2 2 2 − ( ) 1, 0, 1, 0 1 , 9 , 3 , 5 1 1, 0, 0, 0 ( ) 2 2 − 2 2 − ( ) 1, 0, 0, 1 1 , 9 , 7 , 5 1 0, 0, 1, 0 ( ) 2 2 2 − 2 ( ) 0, 1, 0, 0 11 , 1 , 7 , 5 1 1, 0, 1, 0 ( ) 2 − 2 2 2 − ( ) 0, 1, 1, 0 11 , 1 , 3 , 5 1 2, 0, 0, 0 ( ) 2 − 2 − 2 2 − ( ) 0, 1, 0, 1 11 , 1 , 7 , 5 1 1, 0, 1, 0 ( ) 2 − 2 2 − 2 ( ) 0, 0, 1, 0 11 , 9 , 3 , 5 1 0, 1, 0, 2 ( ) 2 2 − 2 2 ( ) 0, 0, 1, 1 11 , 9 , 3 , 5 1 0, 1, 0, 2 ( ) 2 2 − 2 − 2 − ( ) Table A-13. B4, p = 5. Noncontributors for λ = ω1. mα λ + ρ − mpα β 1, 0, 0, 0 1 , 5 , 3 , 1 1, 0, 0, 1 ( ) − 2 2 2 2 ( ) 1, 1, 0, 0 1 , 5 , 3 , 1 1, 0, 0, 1 ( ) − 2 − 2 2 2 ( ) 1, 0, 1, 0 1 , 5 , 7 , 1 1, 0, 0, 1 ( ) − 2 2 − 2 2 ( ) 71 Table A-14. B4, p = 5. Noncontributors for λ = 2ω1. mα λ + ρ − mpα β 1, 0, 0, 0 1 , 5 , 3 , 1 1, 0, 0, 1 ( ) 2 2 2 2 ( − ) 1, 1, 0, 0 1 , 5 , 3 , 1 1, 0, 0, 1 ( ) 2 − 2 2 2 ( − ) 1, 0, 1, 0 1 , 5 , 7 , 1 1, 0, 0, 1 ( ) 2 2 − 2 2 ( − ) Table A-15. B4, p = 5. Contributors for λ = 2ω1. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 2, 0, 0, 0 9 , 5 , 3 , 1 1 1, 0, 0, 0 ( ) − 2 2 2 2 − ( ) 1, 0, 0, 1 1 , 5 , 3 , 9 1 1, 0, 0, 0 ( ) 2 2 2 − 2 ( ) Table A-16. B4, p ≥ 7. Noncontributors for λ = (p − 5)ω4. mα λ + ρ − mpα β 1, 0, 0, 0 1 p + 1, 1 p, 1 p 1, 1 p 2 1, 0, 1, 0 ( ) − 2 2 2 − 2 − ( ) 1, 1, 0, 0 1 p + 1, 1 p, 1 p 1, 1 p 2 1, 0, 1, 0 ( ) − 2 − 2 2 − 2 − ( ) Table A-17. B4, p ≥ 7. Noncontributors for λ = (p − 4)ω4. mα λ + ρ − mpα β 1, 0, 0, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 3 1, 0, 0, 1 ( ) − 2 2 2 2 2 − 2 2 − 2 ( ) 1, 1, 0, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 3 1, 0, 0, 1 ( ) − 2 2 − 2 2 2 − 2 2 − 2 ( ) 1, 0, 1, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 3 1, 0, 0, 1 ( ) − 2 2 2 2 − 2 − 2 2 − 2 ( ) 0, 1, 0, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 3 0, 1, 1, 0 ( ) 2 2 − 2 2 2 − 2 2 − 2 ( ) Table A-18. B4, p ≥ 7. Noncontributors for λ = (p − 3)ω4. mα λ + ρ − mpα β 1, 1, 0, 0 1 p + 2, 1 p + 1, 1 p, 1 p 1 (0, 1, 0, 1) ( ) − 2 − 2 2 2 − 0, 1, 0, 0 1 p + 2, 1 p + 1, 1 p, 1 p 1 (0, 1, 0, 1) ( ) 2 − 2 2 2 − 1, 0, 0, 1 1 p + 2, 1 p + 1, 1 p, 1 p 1 (0, 1, 0, 1) ( ) − 2 2 2 − 2 − 0, 1, 1, 0 1 p + 2, 1 p + 1, 1 p, 1 p 1 (0, 1, 0, 1) ( ) 2 − 2 − 2 2 − Table A-19. B4, p ≥ 7. Contributors for λ = (p − 3)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 2, 1 p + 1, 1 p, 1 p 1 1 0, 0, 0, p 5 ( ) − 2 2 2 2 − ( − ) 1, 0, 1, 0 1 p + 2, 1 p + 1, 1 p, 1 p 1 1 0, 0, 0, p 5 ( ) − 2 2 − 2 2 − − ( − ) Table A-20. B4, p ≥ 7. Noncontributors for λ = (p − 2)ω4. mα λ + ρ − mpα β 1, 0, 1, 0 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 0, 0, 1, 1 ( ) − 2 2 2 2 − 2 2 2 − 2 ( ) 1, 0, 0, 1 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 0, 0, 1, 1 ( ) − 2 2 2 2 2 2 − 2 − 2 ( ) 0, 1, 1, 0 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 0, 0, 1, 1 ( ) 2 2 − 2 2 − 2 2 2 − 2 ( ) 0, 1, 0, 1 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 0, 0, 1, 1 ( ) 2 2 − 2 2 2 2 − 2 − 2 ( ) 0, 0, 1, 0 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 0, 0, 1, 1 ( ) 2 2 2 2 − 2 2 2 − 2 ( ) 72 Table A-21. B4, p ≥ 7. Contributors for λ = (p − 2)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 1 0, 0, 1, p 6 ( ) − 2 2 2 2 2 2 2 − 2 ( − ) 1, 1, 0, 0 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 1 0, 0, 0, p 6 ( ) − 2 2 − 2 2 2 2 2 − 2 − ( − ) 0, 1, 0, 0 1 p + 5 , 1 p + 3 , 1 p + 1 , 1 p 1 1 1, 0, 0, p 4 ( ) 2 2 − 2 2 2 2 2 − 2 − ( − ) Table A-22. B4, p ≥ 7. Noncontributors for λ = ω3 + (p − 6)ω4. mα λ + ρ − mpα β 1, 1, 0, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 5 0, 1, 1, 0 ( ) − 2 2 − 2 2 2 − 2 2 − 2 ( ) 1, 0, 1, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 5 0, 1, 1, 0 ( ) − 2 2 2 2 − 2 − 2 2 − 2 ( ) 0, 1, 0, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 5 0, 1, 1, 0 ( ) 2 2 − 2 2 2 − 2 2 − 2 ( ) Table A-23. B4, p ≥ 7.Contributors for λ = ω3 + (p − 6)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 3 , 1 p + 1 , 1 p 1 , 1 p 5 1 0, 0, 0, p 6 ( ) − 2 2 2 2 2 − 2 2 − 2 − ( − ) Table A-24. B4, p ≥ 7. Noncontributors for λ = ω1 + (p − 4)ω4. mα λ + ρ − mpα β 1, 1, 0, 0 1 p + 5 , 1 p + 1 , 1 p 1 , 1 p 3 0, 1, 1, 0 ( ) − 2 2 − 2 2 2 − 2 2 − 2 ( ) 1, 0, 1, 0 1 p + 5 , 1 p + 1 , 1 p 1 , 1 p 3 0, 1, 1, 0 ( ) − 2 2 2 2 − 2 − 2 2 − 2 ( ) 0, 1, 0, 0 1 p + 5 , 1 p + 1 , 1 p 1 , 1 p 3 0, 1, 1, 0 ( ) 2 2 − 2 2 2 − 2 2 − 2 ( ) Table A-25. B4, p ≥ 7. Contributors for λ = ω1 + (p − 4)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 5 , 1 p + 1 , 1 p 1 , 1 p 3 1 0, 0, 0, p 6 ( ) − 2 2 2 2 2 − 2 2 − 2 ( − ) 1, 0, 0, 1 1 p + 5 , 1 p + 1 , 1 p 1 , 1 p 3 1 0, 0, 1, p 6 ( ) − 2 2 2 2 2 − 2 − 2 − 2 − ( − ) Table A-26. B4, p ≥ 7. Contributors for λ = (p − 1)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 0, 0, 2, p 7 ( ) − 2 2 2 2 ( − ) 1, 1, 0, 0 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 0, 1, 0, p 7 ( ) − 2 − 2 2 2 − ( − ) 1, 0, 1, 0 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 1, 0, 1, p 7 ( ) − 2 2 − 2 2 ( − ) 1, 0, 0, 1 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 0, 0, 2, p 7 ( ) − 2 2 2 − 2 − ( − ) 0, 1, 0, 0 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 1, 0, 1, p 5 ( ) 2 − 2 2 2 − ( − ) 0, 1, 1, 0 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 2, 0, 0, p 5 ( ) 2 − 2 − 2 2 − ( − ) 0, 1, 0, 1 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 1, 0, 1, p 5 ( ) 2 − 2 2 − 2 ( − ) 0, 0, 1, 0 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 0, 1, 0, p 3 ( ) 2 2 − 2 2 ( − ) 0, 0, 1, 1 1 p + 3, 1 p + 2, 1 p + 1, 1 p 1 0, 1, 0, p 3 ( ) 2 2 − 2 − 2 − ( − ) Table A-27. B4, p ≥ 7. Contributors for λ = ω2 + (p − 7)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 1, 1 p, 1 p 2, 1 p 3 1 0, 0, 0, p 7 ( ) − 2 2 2 − 2 − ( − ) 1, 1, 0, 0 1 p + 1, 1 p, 1 p 2, 1 p 3 1 0, 0, 0, p 7 ( ) − 2 − 2 2 − 2 − − ( − ) 73 Table A-28. B4, p ≥ 7. Contributors for λ = ω1 + ω3 + (p − 7)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 2, 1 p, 1 p 1, 1 p 3 1 0, 0, 0, p 7 ( ) − 2 2 2 − 2 − − ( − ) 1, 1, 0, 0 1 p + 2, 1 p, 1 p 1, 1 p 3 1 0, 0, 0, p 7 ( ) − 2 − 2 2 − 2 − ( − ) 1, 0, 1, 0 1 p + 2, 1 p, 1 p 1, 1 p 3 1 0, 1, 0, p 7 ( ) − 2 2 − 2 − 2 − − ( − ) Table A-29. B4, p ≥ 7. Contributors for λ = 2ω1 + (p − 5)ω4. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ 1, 0, 0, 0 1 p + 3, 1 p, 1 p 1, 1 p 2 1 0, 0, 0, p 7 ( ) − 2 2 2 − 2 − ( − ) 1, 1, 0, 0 1 p + 3, 1 p, 1 p 1, 1 p 2 1 0, 0, 0, p 7 ( ) − 2 − 2 2 − 2 − − ( − ) 1, 0, 1, 0 1 p + 3, 1 p, 1 p 1, 1 p 2 1 0, 1, 0, p 7 ( ) − 2 2 − 2 − 2 − ( − ) 1, 0, 0, 1 1 p + 3, 1 p, 1 p 1, 1 p 2 1 1, 0, 1, p 7 ( ) − 2 2 2 − − 2 − − ( − ) 74 APPENDIX B TABLES FOR CALCULATIONS IN GROUPS OF TYPE D4 The following tables are those referenced in Theorem 4.2.1. Table B-1. D4, p = 2. Noncontributors for λ = µ. mα λ + ρ − mpα β (1, 1, 0, 0)(2, 1, 2, 0)(1, 0, −1, 0) (2, 2, 0, 0)(0, −1, 2, 0)(1, 0, 0, 1) (3, 3, 0, 0)(−2, −3, 2, 0)(1, 0, 1, 0) (1, 0, 1, 0)(2, 3, 0, 0)(0, 0, 1, 1) (2, 0, 2, 0)(0, 3, −2, 0)(1, 0, 0, 1) (1, 0, 0, −1)(2, 3, 2, 2)(0, 0, 1, −1) (1, 0, 0, 1)(2, 3, 2, −2)(0, 0, 1, 1) (0, 1, 1, 0)(4, 1, 0, 0)(0, 0, 1, 1) (0, 1, 0, −1)(4, 1, 2, 2)(0, 0, 1, −1) (0, 1, 0, 1)(4, 1, 2, −2)(0, 0, 1, 1) Table B-2. D4, p = 2. Contributors for λ = µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (0, 2, 2, 0)(4, −1, −2, 0) −1 (1, 0, 0, 0) Table B-3. D4, p = 2. Noncontributors for λ = ω1. mα λ + ρ − mpα β (1, 1, 0, 0)(2, 0, 1, 0)(0, 1, 0, 1) (2, 2, 0, 0)(0, −2, 1, 0)(1, 0, 0, 1) (1, 0, −1, 0)(2, 2, 3, 0)(1, −1, 0, 0) (1, 0, 1, 0)(2, 2, −1, 0)(1, −1, 0, 0) (2, 0, 2, 0)(0, 2, −3, 0)(1, 0, 0, 1) (1, 0, 0, −1)(2, 2, 1, 2)(0, 1, 0, −1) (1, 0, 0, 1)(2, 2, 1, −2)(0, 1, 0, 1) (0, 1, 1, 0)(4, 0, −1, 0)(0, 1, 0, 1) 75 Table B-4. D4, p = 3. Noncontributors for λ = µ. mα λ + ρ − mpα β (1, 1, 0, 0)(1, 0, 2, 0)(0, 1, 0, 1) (2, 2, 0, 0)(−2, −3, 2, 0)(1, 0, 1, 0) (1, 0, 1, 0)(1, 3, −1, 0)(1, 0, 1, 0) (1, 0, 0, −1)(1, 3, 2, 3)(0, 1, 0, −1) (1, 0, 0, 1)(1, 3, 2, −3)(0, 1, 0, 1) (0, 1, 1, 0)(4, 0, −1, 0)(0, 1, 0, 1) Table B-5. D4, p = 3. Noncontributors for λ = 2µ. mα λ + ρ − mpα β (1, 0, 1, 0)(2, 4, 0, 0)(0, 0, 1, 1) (1, 0, 0, −1)(2, 4, 3, 3)(0, 0, 1, −1) (1, 0, 0, 1)(2, 4, 3, −3)(0, 0, 1, 1) (0, 1, 1, 0)(5, 1, 0, 0)(0, 0, 1, 1) (0, 1, 0, −1)(5, 1, 3, 3)(0, 0, 1, −1) (0, 1, 0, 1)(5, 1, 3, −3)(0, 0, 1, 1) Table B-6. D4, p = 3. Contributors for λ = 2µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (1, 1, 0, 0)(2, 1, 3, 0) 1 (0, 0, 0, 0) (2, 2, 0, 0)(−1, −2, 3, 0) −1 (0, 0, 0, 0) (2, 0, 2, 0)(−1, 4, −3, 0) 1 (0, 1, 0, 0) (0, 2, 2, 0)(5, −2, −3, 0) −1 (1, 0, 1, 1) Table B-7. D4, p = 3. Noncontributors for λ = ω2. mα λ + ρ − mpα β (1, 1, 0, 0)(1, 0, 1, 0)(0, 1, 0, 1) (1, 0, 0, −1)(1, 3, 1, 3)(0, 1, 0, −1) (1, 0, 0, 1)(1, 3, 1, −3)(0, 1, 0, 1) (0, 1, 1, 0)(4, 0, −2, 0)(0, 1, 0, 1) Table B-8. D4, p = 3. Contributors for λ = ω2. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (2, 2, 0, 0)(−2, −3, 1, 0) −1 (0, 0, 0, 0) (1, 0, 1, 0)(1, 3, −2, 0) 1 (0, 0, 0, 0) Table B-9. D4, p = 3. Noncontributors for λ = ω1 + µ. mα λ + ρ − mpα β (1, 1, 0, 0)(2, 0, 2, 0)(0, 1, 0, 1) (1, 0, 0, −1)(2, 3, 2, 3)(0, 1, 0, −1) (1, 0, 0, 1)(2, 3, 2, −3)(0, 1, 0, 1) (0, 1, 1, 0)(5, 0, −1, 0)(0, 1, 0, 1) 76 Table B-10. D4, p = 3. Contributors for λ = ω1 + µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (2, 2, 0, 0)(−1, −3, 2, 0) 1 (0, 0, 0, 0) (1, 0, 1, 0)(2, 3, −1, 0) −1 (0, 0, 0, 0) (2, 0, 2, 0)(−1, 3, −4, 0) −1 (0, 1, 0, 0) Table B-11. D4, p = 5. Noncontributors for λ = µ. mα λ + ρ − mpα β (1, 1, 0, 0)(−1, −2, 2, 0)(0, 1, 1, 0) (1, 0, 1, 0)(−1, 3, −3, 0)(0, 1, 1, 0) Table B-12. D4, p = 5. Noncontributors for λ = 2µ. mα λ + ρ − mpα β (1, 1, 0, 0)(0, −1, 3, 0)(1, 0, 0, 1) (1, 0, 1, 0)(0, 4, −2, 0)(1, 0, 0, 1) Table B-13. D4, p = 5. Contributors for λ = 2µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (0, 1, 1, 0)(5, −1, −2, 0) −1 (2, 0, 0, 0) Table B-14. D4, p = 5. Noncontributors for λ = 2ω1. mα λ + ρ − mpα β (1, 1, 0, 0)(0, −3, 1, 0)(1, 0, 0, 1) (1, 0, 1, 0)(0, 2, −4, 0)(1, 0, 0, 1) Table B-15. D4, p = 5. Noncontributors for λ = 3µ,. mα λ + ρ − mpα β (1, 1, 0, 0)(1, 0, 4, 0)(0, 1, 0, 1) (2, 2, 0, 0)(−4, −5, 4, 0)(1, 0, 1, 0) (1, 0, 1, 0)(1, 5, −1, 0)(1, 0, 1, 0) (1, 0, 0, −1)(1, 5, 4, 5)(0, 1, 0, −1) (1, 0, 0, 1)(1, 5, 4, −5)(0, 1, 0, 1) (0, 1, 1, 0)(6, 0, −1, 0)(0, 1, 0, 1) Table B-16. D4, p = 5. Noncontributors for λ = 4µ. mα λ + ρ − mpα β (1, 0, 1, 0)(2, 6, 0, 0)(0, 0, 1, 1) (1, 0, 0, −1)(2, 6, 5, 5)(0, 0, 1, −1) (1, 0, 0, 1)(2, 6, 5, −5)(0, 0, 1, 1) (0, 1, 1, 0)(7, 1, 0, 0)(0, 0, 1, 1) (0, 1, 0, −1)(7, 1, 5, 5)(0, 0, 1, −1) (0, 1, 0, 1)(7, 1, 5, −5)(0, 0, 1, 1) 77 Table B-17. D4, p = 5. Contributors for λ = 4µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (1, 1, 0, 0)(2, 1, 5, 0) 1 (2, 0, 0, 0) (2, 2, 0, 0)(−3, −4, 5, 0) −1 (0, 0, 2, 2) (2, 0, 2, 0)(−3, 6, −5, 0) 1 (0, 1, 2, 2) (0, 2, 2, 0)(7, −4, −5, 0) −1 (1, 0, 3, 3) Table B-18. D4, p = 5. Noncontributors for λ = ω2 + 2µ. mα λ + ρ − mpα β (1, 1, 0, 0)(1, 0, 3, 0)(0, 1, 0, 1) (1, 0, 0, −1)(1, 5, 3, 5)(0, 1, 0, −1) (1, 0, 0, 1)(1, 5, 3, −5)(0, 1, 0, 1) (0, 1, 1, 0)(6, 0, −2, 0)(0, 1, 0, 1) Table B-19. D4, p = 5. Contributors for λ = ω2 + 2µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (2, 2, 0, 0)(−4, −5, 3, 0) −1 (0, 0, 2, 2) (1, 0, 1, 0)(1, 5, −2, 0) 1 (2, 0, 0, 0) Table B-20. D4, p = 5. Noncontributors for λ = ω1 + 3µ. mα λ + ρ − mpα β (1, 1, 0, 0)(2, 0, 4, 0)(0, 1, 0, 1) (1, 0, 0, −1)(2, 5, 4, 5)(0, 1, 0, −1) (1, 0, 0, 1)(2, 5, 4, −5)(0, 1, 0, 1) (0, 1, 1, 0)(7, 0, −1, 0)(0, 1, 0, 1) Table B-21. D4, p = 5. Contributors for λ = ω1 + 3µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (2, 2, 0, 0)(−3, −5, 4, 0) 1 (0, 0, 2, 2) (1, 0, 1, 0)(2, 5, −1, 0) −1 (2, 0, 0, 0) (2, 0, 2, 0)(−3, 5, −6, 0) −1 (0, 1, 2, 2) D p 7 = p−3 Table B-22. 4, ≥ . Noncontributors for λ 2 µ. mα λ + ρ − mpα β 1, 1, 0, 0 p + 3 , p + 1 , p 1 , 0 0, 1, 1, 0 ( ) − 2 2 − 2 2 2 − 2 ( ) 1, 0, 1, 0 p + 3 , p + 1 , p 1 , 0 0, 1, 1, 0 ( ) − 2 2 2 2 − 2 − 2 ( ) D p 7 = r p−1 r p 4 Table B-23. 4, ≥ . Contributors for λ µ, for 2 ≤ ≤ − . mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (1, 1, 0, 0)(−p + r + 3, −p + r + 2, r + 1, 0) −1 (−p + 2r + 2, 0, p − r − 4, p − r − 4) (1, 0, 1, 0)(−p + r + 3, r + 2, −p + r + 1, 0) 1 (−p + 2r + 2, 1, p − r − 4, p − r − 4) (0, 1, 1, 0)(r + 3, −p + r + 2, −p + r + 1, 0) −1 (−p + 2r + 3, 0, p − r − 3, p − r − 3) 78 Table B-24. D4, p ≥ 7. Contributors for λ = (−p + 2r + 2)ω1 + ω2 + (p − r − 4)µ, for p−1 r p 4 2 ≤ ≤ − . mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (1, 1, 0, 0) (−p + r + 2, −r − 1, p − r − 3, 0) −1 (−p + 2r + 2, 0, p − r − 4, p − r − 4) Table B-25. D4, p ≥ 7. Contributors for λ = (−p + 2r + 3)ω1 + (p − r − 3)µ, for p−1 r p 4 2 ≤ ≤ − . mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (1, 1, 0, 0) (−p + r + 3, −r − 1, p − r − 2, 0) 1 (−p + 2r + 2, 0, p − r − 4, p − r − 4) (1, 0, 1, 0) (−p + r + 3, p − r − 1, −r − 2, 0) −1 (−p + 2r + 2, 1, p − r − 4, p − r − 4) Table B-26. D4, p ≥ 7. Noncontributors for λ = (p − 3)µ. mα λ + ρ − mpα β (1, 1, 0, 0) (0, −1, p − 2, 0) (1, 0, 0, 1) (1, 0, 1, 0) (0, p − 1, −2, 0) (1, 0, 0, 1) Table B-27. D4, p ≥ 7. Contributors for λ = (p − 3)µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (0, 1, 1, 0) (p, −1, −2, 0) −1 (p − 3, 0, 0, 0) Table B-28. D4, p ≥ 7. Noncontributors for λ = (p − 3)ω1. mα λ + ρ − mpα β (1, 1, 0, 0) (0, −p + 2, 1, 0) (1, 0, 0, 1) (1, 0, 1, 0) (0, 2, −p + 1, 0) (1, 0, 0, 1) Table B-29. D4, p ≥ 7. Noncontributors for λ = (p − 2)µ. mα λ + ρ − mpα β (1, 1, 0, 0)(1, 0, p − 1, 0)(0, 1, 0, 1) (2, 2, 0, 0)(−p + 1, −p, p − 1, 0)(1, 0, 1, 0) (1, 0, 1, 0)(1, p, −1, 0)(1, 0, 1, 0) (1, 0, 0, −1)(1, p, p − 1, p)(0, 1, 0, −1) (1, 0, 0, 1)(1, p, p − 1, −p)(0, 1, 0, 1) (0, 1, 1, 0)(p + 1, 0, −1, 0)(0, 1, 0, 1) Table B-30. D4, p ≥ 7. Noncontributors for λ = (p − 1)µ. mα λ + ρ − mpα β (1, 0, 1, 0)(2, p + 1, 0, 0)(0, 0, 1, 1) (1, 0, 0, −1)(2, p + 1, p, p)(0, 0, 1, −1) (1, 0, 0, 1)(2, p + 1, p, −p)(0, 0, 1, 1) (0, 1, 1, 0)(p + 2, 1, 0, 0)(0, 0, 1, 1) (0, 1, 0, −1)(p + 2, 1, p, p)(0, 0, 1, −1) (0, 1, 0, 1)(p + 2, 1, p, −p)(0, 0, 1, 1) 79 Table B-31. D4, p ≥ 7. Contributors for λ = (p − 1)µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (1, 1, 0, 0)(2, 1, p, 0) 1 (p − 3, 0, 0, 0) (2, 2, 0, 0)(−p + 2, −p + 1, p, 0) −1 (0, 0, p − 3, p − 3) (2, 0, 2, 0)(−p + 2, p + 1, −p, 0) 1 (0, 1, p − 3, p − 3) (0, 2, 2, 0)(p + 2, −p + 1, −p, 0) −1 (1, 0, p − 2, p − 2) Table B-32. D4, p ≥ 7. Noncontributors for λ = ω2 + (p − 3)µ. mα λ + ρ − mpα β (1, 1, 0, 0)(1, 0, p − 2, 0)(0, 1, 0, 1) (1, 0, 0, −1)(1, p, p − 2, p)(0, 1, 0, −1) (1, 0, 0, 1)(1, p, p − 2, −p)(0, 1, 0, 1) (0, 1, 1, 0)(p + 1, 0, −2, 0)(0, 1, 0, 1) Table B-33. D4, p ≥ 7. Contributors for λ = ω2 + (p − 3)µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (2, 2, 0, 0)(−p + 1, −p, p − 2, 0) −1 (0, 0, p − 3, p − 3) (1, 0, 1, 0)(1, p, −2, 0) 1 (p − 3, 0, 0, 0) Table B-34. D4, p ≥ 7. Noncontributors for λ = ω1 + (p − 2)µ. mα λ + ρ − mpα β (1, 1, 0, 0)(2, 0, p − 1, 0)(0, 1, 0, 1) (1, 0, 0, −1)(2, p, p − 1, p)(0, 1, 0, −1) (1, 0, 0, 1)(2, p, p − 1, −p)(0, 1, 0, 1) (0, 1, 1, 0)(p + 2, 0, −1, 0)(0, 1, 0, 1) Table B-35. D4, p ≥ 7. Contributors for λ = ω1 + (p − 2)µ. mα λ + ρ − mpα sgn(w) w(λ + ρ − mpα) − ρ (2, 2, 0, 0)(−p + 2, −p, p − 1, 0) 1 (0, 0, p − 3, p − 3) (1, 0, 1, 0)(2, p, −1, 0) −1 (p − 3, 0, 0, 0) (2, 0, 2, 0)(−p + 2, p, −p − 1, 0) −1 (0, 1, p − 3, p − 3) 80 REFERENCES [1] Andersen, H. H., Jantzen, J. 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Sage: Open Source Mathematical Software (Version 2.10.2). The Sage Group, 2008. http://www.sagemath.org. [14] Steinberg, R. Representations of algebraic groups. Nagoya Math. J. 22 (1963), 33–56. [15] Steinberg, R. Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. 81 [16] Tits, J. Buildings of spherical type and finite BN-pairs. Lecture Notes in Mathematics, Vol. 386. Springer-Verlag, Berlin-New York, 1974. [17] Ye, J., and Zhou, Z. Irreducible characters for algebraic groups in characteristic three. II. Comm. Algebra 30, 1 (2002), 273–306. 82 BIOGRAPHICAL SKETCH Elizabeth grew up in Centerville, Ohio. Because of her interest in the equine industry, she decided to do her undergraduate education at Georgetown College, located close to Lexington, Kentucky. She received her Bachelor of Science degree from Georgetown in 2010, where she studied mathematics and French. She moved to Gainesville, Florida in 2010 to begin her studies in mathematics at the University of Florida. Elizabeth received her Master of Science in mathematics in May 2012 and her Ph.D. in mathematics in December 2015 from the University of Florida. 83