DOCSLIB.ORG

From Root Systems to Dynkin Diagrams

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
From Root Systems to Dynkin Diagrams From root systems to Dynkin diagrams Heiko Dietrich Abstract. We describe root systems and their associated Dynkin diagrams; these notes follow closely the book of Erdman & Wildon (\Introduction to Lie algebras", 2006) and lecture notes of Willem de Graaf (Italy). We briefly describe how root systems arise from Lie algebras. 1. Root systems 1.1. Euclidean spaces. Let V be a finite dimensional Euclidean space, that is, a finite dimensional R-space with inner product ( ; ): V V R, which is bilinear, symmetric, and − − p× ! positive definite. The length of v V is v = (v; v); the angle α between two non-zero 2 jj jj v; w V is defined by cos α = (v;w) . 2 jjvjjjjwjj If v V is non-zero, then the hyperplane perpendicular to v is Hv = w V (w; v) = 0 . 2 f 2 j g The reflection in Hv is the linear map sv : V V which maps v to v and fixes every w Hv; ! − 2 recall that V = Hv Span (v), hence ⊕ R 2(w; v) sv : V V; w w v: ! 7! − (v; v) In the following, for v; w V we write 2 2(w; v) w; v = ; h i (v; v) note that ; is linear only in the first component. An important observation is that each h− −i su leaves the inner product invariant, that is, if v; w V , then (su(v); su(w)) = (v; w). 2 We use this notation throughout these notes. 1.2. Abstract root systems. Definition 1.1. A finite subset Φ V is a root system for V if the following hold: ⊆ (R1)0 = Φ and Span (Φ) = V , 2 R (R2) if α Φ and λα Φ with λ R, then λ 1 , 2 2 2 2 {± g (R3) sα(β) Φ for all α; β Φ, 2 2 (R4) α; β Z for all α; β Φ. h i 2 2 The rank of Φ is dim(V ). Note that each sα permutes Φ, and if α Φ, then α Φ. 2 − 2 Lemma 1.2. If α; β Φ with α = β, then α; β β; α 0; 1; 2; 3 . 2 6 ± h ih i 2 f g Proof. By (R4), the product in question is an integer. If v; w V 0 , then (v; w)2 = v w cos2(θ) where θ is the angle between v and w, thus α; β2 β; αn f=g 4 cos2 θ 4. If jj jjjj jj h ih i ≤ cos2(θ) = 1, then θ is a multiple of π, so α and β are linearly dependent, a contradiction. 1 2 Heiko Dietrich If α; β Φ with α = β and β α , then β; α α; β ; by the previous lemma, all possibilities2 are listed6 ± in Tablejj 1.2.jj ≥ jj jj jh ij ≥ jh ij α; β β; α θ (β,β) h i h i (α,α) 0 0 π=2 { 1 1 π=3 1 -1 -1 2π=3 1 1 2 π=4 2 -1 -2 3π=4 2 1 3 π=6 3 -1 -3 5π=6 3 Table 1. Angles between root vectors. Let α; β Φ with (α; β) = 0 and (β; β) (α; α). Recall that sβ(α) = α α; β β Φ, and α; β = 21, depending on6 whether the angle≥ between α and β is obtuse or− acute, h i see2 Table 1. Thus,h i if the± angle is > π=2, then α + β Φ; if the angle is < π=2, then α β Φ. 2 − 2 Example 1.3. We construct all root systems Φ of R2. Suppose α Φ is of shortest length and choose β Φ such that the angle θ π=2; 2π=3; 3π=4; 5π=6 between2 α and β is as large as possible. This2 gives root systems of type2 f A A , A , B , andg G , respectively: 1 × 1 2 2 2 β β β β α α α α A1 A1 A2 B2 × G2 Definition 1.4. A root system Φ is irreducible if it cannot be partitioned into two non- emtpy subsets Φ and Φ , such that (α; β) = 0 for all α Φ and β Φ . 1 2 2 1 2 2 Lemma 1.5. If Φ is a root system, then Φ = Φ ::: Φk, where each Φi is an irreducible root 1 [ [ system for the space Vi = Span (Φi) V ; in particular, V = V ::: Vk. R ≤ 1 ⊕ ⊕ Proof. For α; β Φ write α β if and only if there exist γ ; : : : ; γs Φ with α = γ , β = γs, 2 ∼ 1 2 1 and (γi; γi+1) = 0 for 1 i < s; then is an equivalence relation on Φ. Let Φ1;:::; Φk be the equivalence6 classes of≤ this relation.∼ Clearly, (R1), (R2), and (R4) are satisfied for each Φk and Vk = Span (Φk). To prove (R3), consider α Φk and β Φk; if (α; β) = 0, then R 2 2 sα(β) = β Φk. If (α; β) = 0, then (α; sα(β)) = 0 since sα leaves the inner product invariant; 2 6 6 thus, sα(β) α, and sα(β) Φk. In particular, Φk is an irreducible root system of Vk. Clearly, ∼ 2 every root appears in some Vi, and the sum of Vi spans V . If v1 + ::: + vk = 0 with each vi Vi, then 0 = (v + ::: + vk; vj) = (vj; vj) for all j, that is, vj = 0 for all j. 2 1 From root systems to Dynkin diagrams 3 1.3. Bases of root systems. Let Φ be a root system for V . Definition 1.6. A subset Π Φ is a base (or root basis) for Φ if the following hold: ⊆ (B1) Π is a vector space basis for V , P (B2) every α Φ can be written as α = kββ with either all kβ N or all kβ N. 2 β2Π 2 − 2 A root α Φ is positive with respect to Π if the coefficients in (B2) are positive; otherwise 2 α is negative. The roots in Π are called simple roots; the reflections sβ with β Π are simple reflections. 2 We need the notion of root orders to prove that every root system has a base. Definition 1.7. A root order is a partial order \>" on V such that every α Φ satisfies α > 0 or α > 0, and \>" is compatible with addition and scalar multiplication.2 − Lemma 1.8. Let Φ be a root system of V . P` a) Let v ; : : : ; v` be a basis of V and write v > 0 if and only if v = kivi and the first f 1 g i=1 non-zero ki is positive; define v > w if v w > 0. Then \>" is the lexicographic root − order with respect to the ordered basis v ; : : : ; v` . f 1 g b) Choose v V outside the (finitely many) hyperplanes Hα, α Φ. For u; v V write 0 2 2 2 u > v if and only if (u; v0) > (v; v0). Then \>" is the root order defined by v0. Let \>" be any root order; call α Φ positive if α > 0, and negative otherwise; α is simple if α > 0 and it cannot be written2 as a sum of two positive roots. The proof of the following theorem shows that the set Π of all simple roots is a base for Φ: Theorem 1.9. Every root system has a base. Proof. Let \>" be a root order with set of simple roots Π; we show that Π is a base. First, let α; β Π with α = β: If α β is a positive root, then α = (α β) + β, and α is not simple; if α β 2is a negative6 root, then− β = (α β) + α is not simple;− thus, α β = Φ. This implies that− the angle of α and β is at least −π=2,− hence (α; β) 0 as seen in Table− 1.2 ≤ Second, Π is linearly independent: if not, then there exist pairwise distinct α1; : : : ; αk Π Pk 2 with α1 = i=2 kiαi = β+ + β−, where β+ and β− are the sums of all kiαi with ki positive and negative, respectively. By construction, (β ; β−) 0 since each (αi; αj) 0. Note that + ≥ ≤ β = 0 since α > 0, thus (β ; β ) > 0 and (α ; β ) = (β ; β ) + (β−; β ) > 0. On the other + 6 1 + + 1 + + + + hand, (α ; αj) 0 and the definition of β imply that (α ; β ) 0, a contradiction. 1 ≤ + 1 + ≤ Finally, we show that every positive root α Φ is a linear combination of Π with coefficients 2 in N. Clearly, this holds if α Π. If α = Π, then α = β + γ for positive roots β; γ Φ with 2 2 2 α > β; γ. By induction, β and γ are linear combinations of Π with coefficients in N. Corollary 1.10. Let Π = α ; : : : ; α` be a root basis of Φ. f 1 g a) If \>" is the lexicographic order on V with respect to the basis Π of V , then Π is the set of simple roots with respect to \>"; thus, every base is defined by some root order. b) If v V with (v ; α) > 0 for all α Π, then Π is the set of simple roots with respect to 0 2 0 2 the root basis defined by v0. 4 Heiko Dietrich Proof. a) This is obvious. b) Denote by \>" the root order defined by v0. Let αj Π; clearly, αj > 0. Suppose αj = β+γ P` 2 P` for some β; γ Φ with β; γ > 0.
Load more

Recommended publications
  • Lie Algebras Associated with Generalized Cartan Matrices
    LIE ALGEBRAS ASSOCIATED WITH GENERALIZED CARTAN MATRICES BY R. V. MOODY1 Communicated by Charles W. Curtis, October 3, 1966 1. Introduction. If 8 is a finite-dimensional split semisimple Lie algebra of rank n over a field $ of characteristic zero, then there is associated with 8 a unique nXn integral matrix (Ay)—its Cartan matrix—which has the properties Ml. An = 2, i = 1, • • • , n, M2. M3. Ay = 0, implies An = 0. These properties do not, however, characterize Cartan matrices. If (Ay) is a Cartan matrix, it is known (see, for example, [4, pp. VI-19-26]) that the corresponding Lie algebra, 8, may be recon­ structed as follows: Let e y fy hy i = l, • • • , n, be any Zn symbols. Then 8 is isomorphic to the Lie algebra 8 ((A y)) over 5 defined by the relations Ml = 0, bifj] = dyhy If or all i and j [eihj] = Ajid [f*,] = ~A„fy\ «<(ad ej)-A>'i+l = 0,1 /,(ad fà-W = (J\ii i 7* j. In this note, we describe some results about the Lie algebras 2((Ay)) when (Ay) is an integral square matrix satisfying Ml, M2, and M3 but is not necessarily a Cartan matrix. In particular, when the further condition of §3 is imposed on the matrix, we obtain a reasonable (but by no means complete) structure theory for 8(04t/)). 2. Preliminaries. In this note, * will always denote a field of char­ acteristic zero. An integral square matrix satisfying Ml, M2, and M3 will be called a generalized Cartan matrix, or g.c.m.
    [Show full text]
  • Hyperbolic Weyl Groups and the Four Normed Division Algebras
    Hyperbolic Weyl Groups and the Four Normed Division Algebras Alex Feingold1, Axel Kleinschmidt2, Hermann Nicolai3 1 Department of Mathematical Sciences, State University of New York Binghamton, New York 13902–6000, U.S.A. 2 Physique Theorique´ et Mathematique,´ Universite´ Libre de Bruxelles & International Solvay Institutes, Boulevard du Triomphe, ULB – CP 231, B-1050 Bruxelles, Belgium 3 Max-Planck-Institut fur¨ Gravitationsphysik, Albert-Einstein-Institut Am Muhlenberg¨ 1, D-14476 Potsdam, Germany – p. Introduction Presented at Illinois State University, July 7-11, 2008 Vertex Operator Algebras and Related Areas Conference in Honor of Geoffrey Mason’s 60th Birthday – p. Introduction Presented at Illinois State University, July 7-11, 2008 Vertex Operator Algebras and Related Areas Conference in Honor of Geoffrey Mason’s 60th Birthday – p. Introduction Presented at Illinois State University, July 7-11, 2008 Vertex Operator Algebras and Related Areas Conference in Honor of Geoffrey Mason’s 60th Birthday “The mathematical universe is inhabited not only by important species but also by interesting individuals” – C. L. Siegel – p. Introduction (I) In 1983 Feingold and Frenkel studied the rank 3 hyperbolic Kac–Moody algebra = A++ with F 1 y y @ y Dynkin diagram @ 1 0 1 − Weyl Group W ( )= w , w , w wI = 2, wI wJ = mIJ F h 1 2 3 || | | | i 2 1 0 − 2(αI , αJ ) Cartan matrix [AIJ ]= 1 2 2 = − − (αJ , αJ ) 0 2 2 − 2 3 2 Coxeter exponents [mIJ ]= 3 2 ∞ 2 2 ∞ – p. Introduction (II) The simple roots, α 1, α0, α1, span the − 1 Root lattice Λ( )= α = nI αI n 1,n0,n1 Z F − ∈ ( I= 1 ) X− – p.
    [Show full text]
  • Semisimple Q-Algebras in Algebraic Combinatorics
    Semisimple Q-algebras in algebraic combinatorics Allen Herman (Regina) Group Ring Day, ICTS Bangalore October 21, 2019 I.1 Coherent Algebras and Association Schemes Definition 1 (D. Higman 1987). A coherent algebra (CA) is a subalgebra CB of Mn(C) defined by a special basis B, which is a collection of non-overlapping 01-matrices that (1) is a closed set under the transpose, (2) sums to J, the n × n all 1's matrix, and (3) contains a subset ∆ of diagonal matrices summing I , the identity matrix. The standard basis B of a CA in Mn(C) is precisely the set of adjacency matrices of a coherent configuration (CC) of order n and rank r = jBj. > B = B =) CB is semisimple. When ∆ = fI g, B is the set of adjacency matrices of an association scheme (AS). Allen Herman (Regina) Semisimple Q-algebras in Algebraic Combinatorics I.2 Example: Basic matrix of an AS An AS or CC can be illustrated by its basic matrix. Here we give the basic matrix of an AS with standard basis B = fb0; b1; :::; b7g: 2 012344556677 3 6 103244556677 7 6 7 6 230166774455 7 6 7 6 321066774455 7 6 7 6 447701665523 7 6 7 d 6 7 X 6 447710665532 7 ibi = 6 7 6 556677012344 7 i=0 6 7 6 556677103244 7 6 7 6 774455230166 7 6 7 6 774455321066 7 6 7 4 665523447701 5 665532447710 Allen Herman (Regina) Semisimple Q-algebras in Algebraic Combinatorics I.3 Example: Basic matrix of a CC of order 10 and j∆j = 2 2 0 123232323 3 6 1 032323232 7 6 7 6 4567878787 7 6 7 6 5476787878 7 d 6 7 X 6 4587678787 7 ibi = 6 7 6 5478767878 7 i=0 6 7 6 4587876787 7 6 7 6 5478787678 7 6 7 4 4587878767 5 5478787876 ∆ = fb0; b6g, δ(b0) = δ(b6) = 1 Diagonal block valencies: δ(b1) = 1, δ(b7) = 4, δ(b8) = 3; Off diagonal block valencies: row valencies: δr (b2) = δr (b3) = 4, δr (b4) = δr (b5) = 1 column valencies: δc (b2) = δc (b3) = 1, δc (b4) = δc (b5) = 4.
    [Show full text]
  • Completely Integrable Gradient Flows
    Commun. Math. Phys. 147, 57-74 (1992) Communications in Maltlematical Physics Springer-Verlag 1992 Completely Integrable Gradient Flows Anthony M. Bloch 1'*, Roger W. Brockett 2'** and Tudor S. Ratiu 3'*** 1 Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA z Division of Applied Sciences, Harvard University, Cambridge, MA 02138, USA 3 Department of Mathematics, University of California, Santa Cruz, CA 95064, USA Received December 6, 1990, in revised form July 11, 1991 Abstract. In this paper we exhibit the Toda lattice equations in a double bracket form which shows they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. Representations for the flows are given and a convexity result associated with a momentum map is proved. Some general properties of the double bracket equations are demonstrated, including a discussion of their invariant subspaces, and their function as a Lie algebraic sorter. O. Introduction In this paper we present details of the proofs and applications of the results on the generalized Toda lattice equations announced in [7]. One of our key results is exhibiting the Toda equations in a double bracket form which shows directly that they are gradient flow equations (on their isospectral set) on an adjoint orbit of a compact Lie group. This system, which is gradient with respect to the normal metric on the orbit, is quite different from the repre- sentation of the Toda flow as a gradient system on N" in Moser's fundamental paper [27]. In our representation the same set of equations is thus Hamiltonian and a gradient flow on the isospectral set.
    [Show full text]
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2
    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
    [Show full text]
  • Symmetries, Fields and Particles. Examples 1
    Michaelmas Term 2016, Mathematical Tripos Part III Prof. N. Dorey Symmetries, Fields and Particles. Examples 1. 1. O(n) consists of n × n real matrices M satisfying M T M = I. Check that O(n) is a group. U(n) consists of n × n complex matrices U satisfying U†U = I. Check similarly that U(n) is a group. Verify that O(n) and SO(n) are the subgroups of real matrices in, respectively, U(n) and SU(n). By considering how U(n) matrices act on vectors in Cn, and identifying Cn with R2n, show that U(n) is a subgroup of SO(2n). 2. Show that for matrices M ∈ O(n), the first column of M is an arbitrary unit vector, the second is a unit vector orthogonal to the first, ..., the kth column is a unit vector orthogonal to the span of the previous ones, etc. Deduce the dimension of O(n). By similar reasoning, determine the dimension of U(n). Show that any column of a unitary matrix U is not in the (complex) linear span of the remaining columns. 3. Consider the real 3 × 3 matrix, R(n,θ)ij = cos θδij + (1 − cos θ)ninj − sin θǫijknk 3 where n = (n1, n2, n3) is a unit vector in R . Verify that n is an eigenvector of R(n,θ) with eigenvalue one. Now choose an orthonormal basis for R3 with basis vectors {n, m, m˜ } satisfying, m · m = 1, m · n = 0, m˜ = n × m. By considering the action of R(n,θ) on these basis vectors show that this matrix corre- sponds to a rotation through an angle θ about an axis parallel to n and check that it is an element of SO(3).
    [Show full text]
  • Contents 1 Root Systems
    Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations.
    [Show full text]
  • The Waldspurger Transform of Permutations and Alternating Sign Matrices
    Séminaire Lotharingien de Combinatoire 78B (2017) Proceedings of the 29th Conference on Formal Power Article #41, 12 pp. Series and Algebraic Combinatorics (London) The Waldspurger Transform of Permutations and Alternating Sign Matrices Drew Armstrong∗ and James McKeown Department of Mathematics, University of Miami, Coral Gables, FL Abstract. In 2005 J. L. Waldspurger proved the following theorem: Given a finite real reflection group G, the closed positive root cone is tiled by the images of the open weight cone under the action of the linear transformations 1 g. Shortly after this − E. Meinrenken extended the result to affine Weyl groups and then P. V. Bibikov and V. S. Zhgoon gave a uniform proof for a discrete reflection group acting on a simply- connected space of constant curvature. In this paper we show that the Waldspurger and Meinrenken theorems of type A give an interesting new perspective on the combinatorics of the symmetric group. In particular, for each permutation matrix g S we define a non-negative integer 2 n matrix WT(g), called the Waldspurger transform of g. The definition of the matrix WT(g) is purely combinatorial but it turns out that its columns are the images of the fundamental weights under 1 g, expressed in simple root coordinates. The possible − columns of WT(g) (which we call UM vectors) biject to many interesting structures including: unimodal Motzkin paths, abelian ideals in the Lie algebra sln(C), Young diagrams with maximum hook length n, and integer points inside a certain polytope. We show that the sum of the entries of WT(g) is half the entropy of the corresponding permutation g, which is known to equal the rank of g in the MacNeille completion of the Bruhat order.
    [Show full text]
  • Full Text (PDF Format)
    Pure and Applied Mathematics Quarterly Volume 13, Number 2, 291–335, 2017 Self-dual Grassmannian, Wronski map, and representations of glN , sp2r, so2r+1 Kang Lu, E. Mukhin, and A. Varchenko Dedicated to Yuri Ivanovich Manin on the occasion of his 80th birthday Abstract: We define a glN -stratification of the Grassmannian of N planes Gr(N,d). The glN -stratification consists of strata ΩΛ labeled by unordered sets Λ =(λ(1),...,λ(n)) of nonzero parti- tions with at most N parts, satisfying a condition depending on ⊗n slN d,andsuchthat( i=1Vλ(i) ) = 0. Here Vλ(i) is the irreducible (i) glN -module with highest weight λ . We show that the closure of (1) (m) astratumΩΛ is the union of the strata ΩΞ, Ξ =(ξ ,...,ξ ), { } { } such that there is a partition I1,...,Im of 1, 2,...,n with (i) ⊗ ∈ (j) HomglN (Vξ , j Ii Vλ =0fori =1,...,m.TheglN -stratification of the Grassmannian agrees with the Wronski map. We introduce and study the new object: the self-dual Grass- mannian sGr(N,d) ⊂ Gr(N,d). Our main result is a similar gN - stratification of the self-dual Grassmannian governed by represen- tation theory of the Lie algebra g2r+1 := sp2r if N =2r +1 andof the Lie algebra g2r := so2r+1 if N =2r. 1. Introduction The Grassmannian Gr(N,d)ofN-dimensional subspaces of the complex d- dimensional vector space has the standard stratification by Schubert cells Ωλ labeled by partitions λ =(d − N λ1 ... λN 0). A Schubert cycle is the closure of a cell Ωλ.ItiswellknownthattheSchubertcycleΩλ is the union of the cells Ωξ such that the Young diagram of λ is inscribed into the Young diagram of ξ.
    [Show full text]
  • Quivers with Relations for Symmetrizable Cartan Matrices and Algebraic Lie Theory
    P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (117–142) QUIVERS WITH RELATIONS FOR SYMMETRIZABLE CARTAN MATRICES AND ALGEBRAIC LIE THEORY C G Abstract We give an overview of our effort to introduce (dual) semicanonical bases in the setting of symmetrizable Cartan matrices. 1 Introduction One of the original motivations of Fomin and Zelevinsky for introducing cluster algebras was “to understand, in a concrete and combinatorial way, G. Lusztig’s theory of total pos- itivity and canonical bases” Fomin [2010]. This raised the question of finding a cluster algebra structure on the coordinate ring of a unipotent cell, and to study its relation with Lusztig’s bases. In a series of works culminating with Geiß, Leclerc, and Schröer [2011] and Geiß, Leclerc, and Schröer [2012], we showed that the coordinate ring of a unipotent cell of a symmetric Kac-Moody group has indeed a cluster algebra structure, whose cluster monomials belong to the dual of Lusztig’s semicanonical basis of the enveloping algebra of the attached Kac-Moody algebra. Since the semicanonical basis is built in terms of constructible functions on the complex varieties of nilpotent representations of the prepro- jective algebra of a quiver, it is not straightforward to extend those results to the setting of symmetrizable Cartan matrices, which appears more natural from the Lie theoretic point of view. The purpose of these notes is to give an overview of Geiß, Leclerc, and Schröer [2017a] - Geiß, Leclerc, and Schröer [2017d], where we are trying to make progress into this direction. The starting point of our project was Hernandez and Leclerc [2016], where they ob- served that certain quivers with potential allowed to encode the q-characters of the Kirillov- Reshetikhin modules of the quantum loop algebra Uq(Lg), where g is a complex simple Lie algebra of arbitrary Dynkin type.
    [Show full text]
  • Cartan Matrices and Brauer's K(B)-Conjecture V
    Cartan matrices and Brauer’s k(B)-Conjecture V Cesare Giulio Ardito∗ and Benjamin Sambale† November 26, 2019 We prove Brauer’s k(B)-Conjecture for the 3-blocks with abelian defect groups of rank at most 5 and for all 3-blocks of defect at most 4. For this purpose we develop a computer algorithm to construct isotypies based on a method of Usami and Puig. This leads further to some previously unknown perfect isometries for the 5-blocks of defect 2. We also investigate basic sets which are compatible under the action of the inertial group. Keywords: number of characters, Brauer’s Conjecture, Usami–Puig method, perfect isometries AMS classification: 20C15, 20C20 1 Introduction This work continues a series of articles the last one being [27]. Before stating the main theorems we briefly explain the strategy behind all papers in this series. Let B be a block of a finite group G with respect to an algebraically closed field F of characteristic p > 0. Let D be a defect group B, and let k(B) := |Irr(B)| and l(B) := |IBr(B)|. To prove Brauer’s k(B)-Conjecture, that k(B) ≤ |D|, we investigate Brauer correspondents of B in local subgroups. More precisely, let z ∈ Z(D) and let bz be a Brauer correspondent of B in CG(z). If we can determine t the Cartan matrix Cz of bz up to basic sets (i. e. up to transformations Cz → SCzS where S ∈ GL(l(bz), Z)), then Brauer’s Conjecture usually follows from [26, Theorem 4.2] or from the much stronger result [31, Theorem A].
    [Show full text]
  • Quantum Spin Chains from Onsager Algebras and Reflection $ K $-Matrices
    QUANTUM SPIN CHAINS FROM ONSAGER ALGEBRAS AND REFLECTION K-MATRICES ATSUO KUNIBA AND VINCENT PASQUIER Abstract (1) (2) (1) We present a representation of the generalized p-Onsager algebras Op(An−1), Op(Dn+1), Op(Bn ), (1) (1) Op(B˜n ) and Op(Dn ) in which the generators are expressed as local Hamiltonians of XXZ type spin chains with various boundary terms reflecting the Dynkin diagrams. Their symmetry is described by the reflection K matrices which are obtained recently by a q-boson matrix product construction related to the 3D integrability and characterized by Onsager coideals of quantum affine algebras. The spectral decomposition of the K matrices with respect to the classical part of the Onsager algebra is described conjecturally. We also include a proof of a certain invariance property of boundary vectors in the q-boson Fock space playing a key role in the matrix product construction. 1. Introduction (1) The generalized p-Onsager algebra Op(An−1) is generated by b1,..., bn with the relations bibj − bj bi = 0 (aij = 0), 2 2 −2 2 (1) bi bj − (p + p )bibj bi + bj bi = bj (aij = −1), where we assume n ≥ 3 and the parameter p is generic. The data (aij )i,j∈Zn is the Cartan matrix of (1) the affine Lie algebra An−1 [15]. The above relation with p = 1 goes back to [29, eqs.(11),(12)]. In what follows, the algebra Op(g) introduced for any affine Lie algebra g [3] will simply be called an Onsager algebra for short. We refer to [28, Rem.
    [Show full text]