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Representation Theory Representation Theory Andrew Kobin Fall 2014 Contents Contents Contents 1 Introduction 1 1.1 Group Theory Review . .1 1.2 Partitions . .2 2 Group Representations 4 2.1 Representations . .4 2.2 G-homomorphisms . .9 2.3 Schur's Lemma . 12 2.4 The Commutant and Endomorphism Algebras . 13 2.5 Characters . 17 2.6 Tensor Products . 24 2.7 Restricted and Induced Characters . 25 3 Representations of Sn 28 3.1 Young Subgroups of Sn .............................. 28 3.2 Specht Modules . 33 3.3 The Decomposition of M µ ............................ 44 3.4 The Hook Length Formula . 48 3.5 Application: The RSK Algorithm . 49 4 Symmetric Functions 52 4.1 Generating Functions . 52 4.2 Symmetric Functions . 53 4.3 Schur Functions . 57 4.4 Symmetric Functions and Character Representations . 60 i 1 Introduction 1 Introduction The following are notes from a course in representation theory taught by Dr. Frank Moore at Wake Forest University in the fall of 2014. The main topics covered are: group repre- sentations, characters, the representation theory of Sn, Young tableaux and tabloids, Specht modules, the RSK algorithm and some further applications to combinatorics. 1.1 Group Theory Review Definition. A group is a nonempty set G with a binary operation \·00 : G×G ! G satisfying (1) (Associativity) a(bc) = (ab)c for all a; b; c 2 G. (2) (Identity) There exists an identity element e 2 G such that for every a 2 G, ae = ea = a. (3) (Inverses) For every a 2 G there is some b 2 G such that ab = ba = e. Examples. 1 The groups of real- or complex-valued n × n matrices, denoted GLn(R) and GLn(C), respectively, are groups under matrix multiplication. The identity in each case is the 21 0 ··· 03 60 1 ··· 07 6 7 identity matrix I = 6. .. .7. 4. .5 0 0 ··· 1 2 For a vector space V over R (resp. C), the general linear group of V , denoted GL(V ), is defined to be the group of invertible linear transformations V ! V . If V ∼ is finite dimensional over R (resp. C) then GL(V ) = GLn(R) (resp. GLn(C)) where n = dim V . The general linear group is a group under function composition, with the identity function id : V ! V defined by v 7! v and inverses defined by invertibility of the elements of GL(V ). 3 The most important example in these notes is the symmetric group on n symbols, Sn. Explicitly, Sn is defined to be the group of bijective functions π : f1; : : : ; ng ! f1; : : : ; ng. The group law here is function composition and the identity and inverses are defined in the same manner as in 2 . There are three types of notation for an element π 2 Sn which will be used interchangeably: 1 2 3 ··· n Two-line notation: π(1) π(2) π(3) ··· π(n) One-line notation: π(1) π(2) ··· π(n) { this is just the bottom row of the two-line notation Cycle notation, e.g. (1 3)(2 4 5)(6 7) 1 1.2 Partitions 1 Introduction Definition. The cycle type of a permutation π 2 Sn is the number and length of disjoint cycles into which π decomposes. The following theorem from group theory guarantees that the cycle type is well-defined for any permutation. Theorem 1.1.1. For all π 2 Sn, π can be written as the product of disjoint cycles. Example 1.1.2. Consider the permutation π = (2 4 3). In one-line notation, this can be written π = 1 4 2 3. The cycle type of π is 31, whereas the cycle type of σ = (1 4 6 3)(2 5)(7 8) 1 2 in S8 is 4 ; 2 . 1.2 Partitions Definition. A partition of a natural number n is a sequence λ = (λ1; : : : ; λm) such that (1) The sequence of λi is weakly decreasing, that is, λ1 ≥ λ2 ≥ · · · ≥ λm. Pm (2) i=1 λi = n. A partition will often be denoted λ ` n. Definition. Let G be any group. Two elements g; h 2 G are said to be conjugate in G if there is a k 2 G such that g = khk−1, or equivalently if gk = kh. The conjugacy class of −1 an element g is Kg = fh 2 G j g = khk for some k 2 Gg. Conjugacy is an equivalence relation. In general, the equivalence classes of an equivalence relation on a set S form a partition of S, as is the case with the symmetric group: Proposition 1.2.1. The conjugacy classes of G form a partition of G, that is, they are disjoint and contain every element of G. Example 1.2.2. Let G = GLn(C), the group of n × n matrices with complex entries. Suppose M and N are conjugate, n × n matrices, i.e. there exists a matrix K 2 G such that M = KNK−1. This corresponds to a commutative diagram n K n CK Cstd N M n n CK Cstd K (The subscripts on the vector spaces denote different bases, as K may be thought of as a change of basis matrix.) This diagram in fact shows that M and N are also similar matrices. In general, conjugacy and similarity are equivalent in GLn(C). (For more, see the section on Jordan canonical form in any linear algebra text.) 2 1.2 Partitions 1 Introduction Let π 2 Sn and suppose π = (i1 i2 ··· i`) ··· (im im+1 ··· in) is the cycle decomposition −1 −1 of π. Let σ be the permutation sending ij 7! j, so that σ sends j 7! ij. Then σπσ = (1 2 3 ··· `) ··· (m m + 1 ··· n) has the same cycle type as π. This is the main ingredient in the proof of the following theorem. Theorem 1.2.3. In Sn, all permutations of the same cycle type are conjugate. Proposition 1.2.4. There is a one-to-one correspondence between the set of conjugacy classes of Sn and partitions λ of n. Definition. The generating function for a partition is a formal series given by 1 1 1 1 ! X Y 1 Y X p tn = = tni : n 1 − ti n=0 i=1 i=1 n=i Definition. We say a group G acts on a set X provided there is some mapping \·00 : G × X ! X such that (1) For all g; h 2 G and x 2 X, g · (h · x) = gh · x. (2) For the identity element e 2 G, e · x = x for every x 2 X. The group action axioms easily imply that g−1 · (g · x) = x = g · (g−1 · x). Definition. Given a group action G X, the orbit and stabilizer of an element x 2 X are defined by OrbG(x) = Ox = fg · x j g 2 Gg and StabG(x) = Gx = fg 2 G j g · x = xg: Remark. The stabilizer Gx is a subgroup of G for every x 2 X. An easy consequence of this fact and Lagrange's theorem is the following result. Theorem 1.2.5 (Orbit-Stabilizer Theorem). Suppose G acts on X. Then there is a bijection between orbits and quotient groups of G by the stabilizers given by Ox ! G=Gx g · x 7−! gGx: In particular, if G has finite order then for any x 2 X, jGj = jOxj jGxj. Examples. 1 Any group G acts on itself via conjugation: g · h = ghg−1. The orbits of this action are precisely the conjugacy classes of G: Oh = Kh. On the other hand, the stabilizers, −1 called centralizers, are of the form Zh = fg 2 G j ghg = hg. The Orbit-Stabilizer Theorem then gives the important relationship jGj = jKhj jZhj for any h 2 G. m1 m2 mn 2 Let λ = (1 ; 2 ; : : : ; n ) be a partition of n and take g 2 Sn with cycle type λ. De- m1 m2 mn note the order of the centralizer by zλ = jZgj, so that zλ = 1 (m1!)2 (m2!) ··· n (mn!). The Orbit-Stabilizer Theorem then gives us n! jKλj = : zλ 3 2 Group Representations 2 Group Representations 2.1 Representations We begin with several additional concepts which are fundamental to group theory. Definition. A group homomorphism is a map ' : G ! H between two groups G and H such that (1) '(gg0) = '(g)'(g0) for all g; g0 2 G. (2) '(eG) = eH . Remark. It is an easy consequence of the two homomorphism axioms that inverses are preserved, so a homomorphism may be thought of as a map that transfers group structure. There are two important subgroups induced by a homomorphism. Definition. The kernel of a homomorphism ' : G ! H is the subgroup of G defined by ker ' = fg 2 G j '(g) = eH g. On the other hand, the image of ' is the subgroup of H defined by im ' = fh 2 H j '(g) = h for some g 2 Gg. We now introduce the central concept in representation theory. Definition. For a group G, a matrix representation of G is a group homomorphism X : G ! GLd(C), for some integer d, called the degree of the representation. Definition. We say a representation X is faithful if ker X = feGg. Examples. There are several important examples of representations which we will repeat- edly use as examples of new concepts. 1 For an arbitrary group G, the trivial representation is a one-dimensional representation ∗ X : G ! GL1(C) = C which sends each g 2 G to 1 2 C. 2 The sign representation of Sn is another one-dimensional representation defined by ∗ X : Sn −! C ( 1 π is even π 7−! −1 π is odd: 3 The defining representation of Sn is X : Sn −! GLn(C) ( 1 π(j) = i π 7−! (xij) = 0 otherwise.
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