Multiplicity free actions of simple algebraic groups
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1 Multiplicity free actions of simple algebraic groups D. Testerman (with M. Liebeck and G. Seitz) EPF Lausanne Edinburgh, April 2016 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
2 Introduction Let H be a group acting on a (possibly infinite dimensional) k-vector space V (where k = k) via a completely reducible representation π. We say that (V, π) is multiplicity free if End H (V ) is commutative. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
3 Introduction Let H be a group acting on a (possibly infinite dimensional) k-vector space V (where k = k) via a completely reducible representation π. We say that (V, π) is multiplicity free if End H (V ) is commutative. If dim V <, multiplicity free any irreducible H-module occurs with multiplicity at most one in a decomposition of V into a direct sum of irreducibles. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
4 In different contexts Stembridge ( ) Multiplicity free tensor products and exterior algebras, multiplicity free restrictions to parabolics, for representations of complex semisimple Lie algebras. Highly combinatorial methods. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
5 In different contexts Stembridge ( ) Multiplicity free tensor products and exterior algebras, multiplicity free restrictions to parabolics, for representations of complex semisimple Lie algebras. Highly combinatorial methods. Kobayashi ( ) Studies action of compact Lie group H on the space O(D, V) of holomorphic sections of an H-equivariant vector bundle V D, where D is a complex manifold equipped with an H action. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
6 In different contexts Stembridge ( ) Multiplicity free tensor products and exterior algebras, multiplicity free restrictions to parabolics, for representations of complex semisimple Lie algebras. Highly combinatorial methods. Kobayashi ( ) Studies action of compact Lie group H on the space O(D, V) of holomorphic sections of an H-equivariant vector bundle V D, where D is a complex manifold equipped with an H action. Obtains geometric criteria on D and the fibers of the vector bundle which imply multiplicity freeness of H on O(D, V). Surprise! Reestablishes Stembridge s tensor product list for GL n. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
7 Multiplicity freeness in classical invariant theory (Howe: Perspectives in Invariant Theory, Schur lectures 1992) Connected reductive G over C, acting rationally on a variety X R(X ), ring of regular functions on X, a representation space for G. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
8 Multiplicity freeness in classical invariant theory (Howe: Perspectives in Invariant Theory, Schur lectures 1992) Connected reductive G over C, acting rationally on a variety X R(X ), ring of regular functions on X, a representation space for G. If each irreducible G-module V satisfies dim Hom G (V, R(X )) 1, Howe says X is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
9 Multiplicity freeness in classical invariant theory (Howe: Perspectives in Invariant Theory, Schur lectures 1992) Connected reductive G over C, acting rationally on a variety X R(X ), ring of regular functions on X, a representation space for G. If each irreducible G-module V satisfies dim Hom G (V, R(X )) 1, Howe says X is multiplicity free. Theorem (Vinberg, Servedio) Let X be an affine G-variety. Then X is multiplicity free (in the above sense) if and only if a Borel subgroup of G has a dense and open orbit on X. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
10 (Sketch of =) Suppose B has a dense orbit in X ; let x 0 represent the dense orbit.. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
11 (Sketch of =) Suppose B has a dense orbit in X ; let x 0 represent the dense orbit. Let Q be a maximal vector for B of weight ψ occurring in the representation space R(X ).. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
12 (Sketch of =) Suppose B has a dense orbit in X ; let x 0 represent the dense orbit. Let Q be a maximal vector for B of weight ψ occurring in the representation space R(X ). For b B, Q(bx 0 ) = (b 1.Q)(x 0 ) = ψ(b 1 )Q(x 0 ).. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
13 (Sketch of =) Suppose B has a dense orbit in X ; let x 0 represent the dense orbit. Let Q be a maximal vector for B of weight ψ occurring in the representation space R(X ). For b B, Q(bx 0 ) = (b 1.Q)(x 0 ) = ψ(b 1 )Q(x 0 ). So Q B.x0 is unique up to multiples, i.e. any other maximal vector of weight ψ acts on B.x 0 as a multiple of Q.. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
14 (Sketch of =) Suppose B has a dense orbit in X ; let x 0 represent the dense orbit. Let Q be a maximal vector for B of weight ψ occurring in the representation space R(X ). For b B, Q(bx 0 ) = (b 1.Q)(x 0 ) = ψ(b 1 )Q(x 0 ). So Q B.x0 is unique up to multiples, i.e. any other maximal vector of weight ψ acts on B.x 0 as a multiple of Q. B.x 0 dense = Q has a unique (regular) extension to X. = the space of maximal vectors of weight ψ in R(X ) is 1-dimensional. So X is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
15 Example: GL n on symmetric matrices V n-dimensional vector space over C. Proposition The natural action of GL(V ) on Sym 2 (V ) is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
16 Example: GL n on symmetric matrices V n-dimensional vector space over C. Proposition The natural action of GL(V ) on Sym 2 (V ) is multiplicity free. Identify Sym 2 (V ) with symmetric n n matrices. GL n action : GL n Sym 2 (V ) Sym 2 (V ) (g, M) gmg t. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
17 Example: GL n on symmetric matrices V n-dimensional vector space over C. Proposition The natural action of GL(V ) on Sym 2 (V ) is multiplicity free. Identify Sym 2 (V ) with symmetric n n matrices. GL n action : GL n Sym 2 (V ) Sym 2 (V ) (g, M) gmg t. Orbit of I n under the upper triangular group is open dense in Sym 2 (V ): D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
18 Example: GL n on symmetric matrices V n-dimensional vector space over C. Proposition The natural action of GL(V ) on Sym 2 (V ) is multiplicity free. Identify Sym 2 (V ) with symmetric n n matrices. GL n action : GL n Sym 2 (V ) Sym 2 (V ) (g, M) gmg t. Orbit of I n under the upper triangular group is open dense in Sym 2 (V ): For M Sym 2 (V ), set δ k to be the determinant of the k k leading minor of M. If we assume δ k 0 for all k, Gaussian elimination yields u GL n, unipotent lower triangular, such that um is upper triangular, so that umu t is still upper triangular and now symmetric, hence diagonal. Acting further by (umu t ) 1/2 transforms umu t to I n. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
19 Example: GL n on symmetric matrices V n-dimensional vector space over C. Proposition The natural action of GL(V ) on Sym 2 (V ) is multiplicity free. Identify Sym 2 (V ) with symmetric n n matrices. GL n action : GL n Sym 2 (V ) Sym 2 (V ) (g, M) gmg t. Orbit of I n under the upper triangular group is open dense in Sym 2 (V ): For M Sym 2 (V ), set δ k to be the determinant of the k k leading minor of M. If we assume δ k 0 for all k, Gaussian elimination yields u GL n, unipotent lower triangular, such that um is upper triangular, so that umu t is still upper triangular and now symmetric, hence diagonal. Acting further by (umu t ) 1/2 transforms umu t to I n. Theorem = Sym(Sym 2 (V )) MF = Sym k (Sym 2 (V )) MF for k 1. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
20 Our setting Linear algebraic groups, defined over an algebraically closed field k. Rational representations Restrictions to closed subgroups of irreducible representations. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
21 Our setting Linear algebraic groups, defined over an algebraically closed field k. Rational representations Restrictions to closed subgroups of irreducible representations. Question 1. For which subgroups and which representations is the restriction multiplicity free? D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
22 Rank one: connection to overgroups of distinguished unipotent elements G simple algebraic group over k u G unipotent Definition u distinguished in G if C G (u) is unipotent (e.g., regular unipotents are distinguished) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
23 Rank one: connection to overgroups of distinguished unipotent elements G simple algebraic group over k u G unipotent Definition u distinguished in G if C G (u) is unipotent (e.g., regular unipotents are distinguished) G classical: distinguished unipotent in SL n, one Jordan block distinguished unipotent in Sp 2n, distinct block sizes, all even distinguished in SO n, distinct block sizes, all odd non distinguished have repeated block D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
24 Rank one: connection to overgroups of distinguished unipotent elements G simple algebraic group over k u G unipotent Definition u distinguished in G if C G (u) is unipotent (e.g., regular unipotents are distinguished) G classical: distinguished unipotent in SL n, one Jordan block distinguished unipotent in Sp 2n, distinct block sizes, all even distinguished in SO n, distinct block sizes, all odd non distinguished have repeated block Question 2 Determine reductive subgroups H G containing a distinguished unipotent element of G. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
25 Translation of problem in case char(k) = 0 Let u H distinguished unipotent and u A an A 1 -type subgroup of H. (Determined up to conjugacy in H.) Now A H G = Cl(V ), a classical group. Blocks of u on V determine irreducible summands of A on V. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
26 Translation of problem in case char(k) = 0 Let u H distinguished unipotent and u A an A 1 -type subgroup of H. (Determined up to conjugacy in H.) Now A H G = Cl(V ), a classical group. Blocks of u on V determine irreducible summands of A on V. So u distinguished in Cl(V ) if V A is irreducible if Cl(V ) = SL(V ) or multiplicity free otherwise. = determine kh-modules V such that V A is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
27 Translation of problem in case char(k) = 0 Let u H distinguished unipotent and u A an A 1 -type subgroup of H. (Determined up to conjugacy in H.) Now A H G = Cl(V ), a classical group. Blocks of u on V determine irreducible summands of A on V. So u distinguished in Cl(V ) if V A is irreducible if Cl(V ) = SL(V ) or multiplicity free otherwise. = determine kh-modules V such that V A is multiplicity free. Which A 1 subgroups of H? u H distinguished in Cl(V ) implies u distinguished in H. = study V A for kh-modules V and A H containing distinguished unipotents of H. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
28 Which A 1 -subgroups: a different characterisation Definition A closed subgroup H in a simple algebraic group G is G-irreducible if H lies in no proper parabolic subgroup of G. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
29 Which A 1 -subgroups: a different characterisation Definition A closed subgroup H in a simple algebraic group G is G-irreducible if H lies in no proper parabolic subgroup of G. An A 1 -subgroup of H is H-irreducible if and only if its nonidentity unipotent elements are distinguished in H. (Recall, char(k) = 0.) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
30 Which A 1 -subgroups: a different characterisation Definition A closed subgroup H in a simple algebraic group G is G-irreducible if H lies in no proper parabolic subgroup of G. An A 1 -subgroup of H is H-irreducible if and only if its nonidentity unipotent elements are distinguished in H. (Recall, char(k) = 0.) = determine pairs (V, A), V a kh-module, A H an H-irreducible A 1 -subgroup with V A multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
31 Which subgroups H Cl(V )? We start by considering H acting irreducibly on the natural module for Cl(V ). D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
32 Which subgroups H Cl(V )? We start by considering H acting irreducibly on the natural module for Cl(V ). ρ : H Cl(V ), irreducible rational representation Question For which V, and for which A H, is V ρ(a) multiplicity free? D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
33 First result Theorem Let H be a simple algebraic group defined over k of characteristic 0. Assume H is of rank at least 2 and A H is an H-irreducible A 1 -type subgroup of H. Let V be an irreducible kh-module of highest weight λ. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
34 First result Theorem Let H be a simple algebraic group defined over k of characteristic 0. Assume H is of rank at least 2 and A H is an H-irreducible A 1 -type subgroup of H. Let V be an irreducible kh-module of highest weight λ. Then V A is multiplicity free if and only if λ and H are as given in the following tables. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
35 H λ (u regular in H) A n ω 1, ω 2, 2ω 1, ω 1 + ω n, ω 3 (5 n 7), 3ω 1 (n 5), 4ω 1 (n 3), 5ω 1 (n 3) A A 2 c1, c0 B n ω 1, ω 2, 2ω 1, ω n (n 8) B 3 101, 002, 300 B 2 b0, 0b (1 b 5), 11, 12, 21 C n ω 1, ω 2, 2ω 1, ω 3 (3 n 5), ω n (n = 4, 5) C C 2 b0, 0b(1 b 5), 11, 12, 21 D n (n 4) ω 1, ω 2 (n = 2k + 1), 2ω 1 (n = 2k), ω n (n 9) E 6 ω 1, ω 2 E 7 ω 1, ω 7 E 8 ω 8 F 4 ω 1, ω 4 G 2 10, 01, 11, 20, 02, 30 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
36 H λ class of u in H (nonregular) B n, C n, D n ω 1 any D n (5 n 7) ω n regular in B n 2 B 1 F 4 ω 4 F 4 (a 1 ) E 6 ω 1 E 6 (a 1 ) E 7 ω 7 E 7 (a 1 ) or E 7 (a 2 ) E 8 ω 8 E 8 (a 1 ) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
37 H λ class of u in H (nonregular) B n, C n, D n ω 1 any D n (5 n 7) ω n regular in B n 2 B 1 F 4 ω 4 F 4 (a 1 ) E 6 ω 1 E 6 (a 1 ) E 7 ω 7 E 7 (a 1 ) or E 7 (a 2 ) E 8 ω 8 E 8 (a 1 ) Method Labelled diagram of unipotent class action of T A (torus of A 1 -subgroup) on simple roots. Deduce T A -weights on V. In irreducible A 1 -modules all weights have multiplicity 1. Reduce to list of possibilities. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
38 Overgroups of distinguished elements Corollary Let H be a simple algebraic group defined over k of characteristic 0 and assume H is of rank at least 2. Let ρ : G Cl(V ) be an irreducible representation with highest weight λ. Let u H be a nonidentity unipotent element and suppose ρ(u) is distinguished in Cl(V ) (the smallest classical group containing ρ(h)). D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
39 Overgroups of distinguished elements Corollary Let H be a simple algebraic group defined over k of characteristic 0 and assume H is of rank at least 2. Let ρ : G Cl(V ) be an irreducible representation with highest weight λ. Let u H be a nonidentity unipotent element and suppose ρ(u) is distinguished in Cl(V ) (the smallest classical group containing ρ(h)). (i) If Cl(V ) = SL(V ), then H = A n, B n, C n, or G 2, and λ = ω 1 (or ω n if H = A n ), and u is regular in H. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
40 Overgroups of distinguished elements Corollary Let H be a simple algebraic group defined over k of characteristic 0 and assume H is of rank at least 2. Let ρ : G Cl(V ) be an irreducible representation with highest weight λ. Let u H be a nonidentity unipotent element and suppose ρ(u) is distinguished in Cl(V ) (the smallest classical group containing ρ(h)). (i) If Cl(V ) = SL(V ), then H = A n, B n, C n, or G 2, and λ = ω 1 (or ω n if H = A n ), and u is regular in H. (ii) If Cl(V) = Sp(V ) or Cl(V ) = SO(V ), then λ and u and H are one of the cases in the above tables for which w 0 (λ) = λ, where w 0 is the longest word of the Weyl group of H. Conversely for each of these cases, ρ(u) is distinguished in Cl(V ).. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
41 Generalisation Take X a simple algebraic group defined over k of characteristic 0, ρ : X SL(W ) an irreducible rational representation. (Usually ρ(x ) is a maximal among connected subgroups of Y := Cl(W ), the smallest classical group containing ρ(x ).) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
42 Generalisation Take X a simple algebraic group defined over k of characteristic 0, ρ : X SL(W ) an irreducible rational representation. (Usually ρ(x ) is a maximal among connected subgroups of Y := Cl(W ), the smallest classical group containing ρ(x ).) Goal Determine all irreducible modules ky -modules V such that V X is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
43 Some well-known examples Natural embeddings Sp(W ) SL(W ), SO(W ) SL(W ). Example X = Sp(W ) Y = SL(W ); Sym a (W ) X is irreducible for all a 1 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
44 Some well-known examples Natural embeddings Sp(W ) SL(W ), SO(W ) SL(W ). Example X = Sp(W ) Y = SL(W ); Sym a (W ) X is irreducible for all a 1 a (W ) X is multiplicity free, for a 1; decomposes as L X (ω a ) L X (ω a 2 ), ending with L X (ω 1 ) or L X (0), according as a is odd or even). D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
45 Some well-known examples Natural embeddings Sp(W ) SL(W ), SO(W ) SL(W ). Example X = Sp(W ) Y = SL(W ); Sym a (W ) X is irreducible for all a 1 a (W ) X is multiplicity free, for a 1; decomposes as L X (ω a ) L X (ω a 2 ), ending with L X (ω 1 ) or L X (0), according as a is odd or even). X = SO(W ) Y = SL(W ), dim W 5 a (W ) X is irreducible for all 1 a ((dim W 1)/2) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
46 Some well-known examples Natural embeddings Sp(W ) SL(W ), SO(W ) SL(W ). Example X = Sp(W ) Y = SL(W ); Sym a (W ) X is irreducible for all a 1 a (W ) X is multiplicity free, for a 1; decomposes as L X (ω a ) L X (ω a 2 ), ending with L X (ω 1 ) or L X (0), according as a is odd or even). X = SO(W ) Y = SL(W ), dim W 5 a (W ) X is irreducible for all 1 a ((dim W 1)/2) Sym a (W ) X is multiplicity free for all a 1 (decomposes as L X (aω 1 ) L X ((a 2)ω 1 ), ending with L X (ω 1 ) or L X (0), according as a is odd or even) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
47 Some well-known examples Natural embeddings Sp(W ) SL(W ), SO(W ) SL(W ). Example X = Sp(W ) Y = SL(W ); Sym a (W ) X is irreducible for all a 1 a (W ) X is multiplicity free, for a 1; decomposes as L X (ω a ) L X (ω a 2 ), ending with L X (ω 1 ) or L X (0), according as a is odd or even). X = SO(W ) Y = SL(W ), dim W 5 a (W ) X is irreducible for all 1 a ((dim W 1)/2) Sym a (W ) X is multiplicity free for all a 1 (decomposes as L X (aω 1 ) L X ((a 2)ω 1 ), ending with L X (ω 1 ) or L X (0), according as a is odd or even) k (W ) is MF when dim W = 2k. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
48 Results Complete result for the case X and Y of type A. Notation: X = A l+1, with fundamental dominant weights {ω 1,..., ω l+1 } and Y of type A n with fundamental dominant weights {λ 1,..., λ n }. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
49 Results Complete result for the case X and Y of type A. Notation: X = A l+1, with fundamental dominant weights {ω 1,..., ω l+1 } and Y of type A n with fundamental dominant weights {λ 1,..., λ n }. Theorem Let X = A l+1 with l 0, let W = L X (δ) and Y = SL(W ) = A n. Suppose V = L Y (λ) is an irreducible ky -module such that V X is multiplicity-free, and assume λ λ 1, λ n and δ ω 1, ω l+1. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
50 Results Complete result for the case X and Y of type A. Notation: X = A l+1, with fundamental dominant weights {ω 1,..., ω l+1 } and Y of type A n with fundamental dominant weights {λ 1,..., λ n }. Theorem Let X = A l+1 with l 0, let W = L X (δ) and Y = SL(W ) = A n. Suppose V = L Y (λ) is an irreducible ky -module such that V X is multiplicity-free, and assume λ λ 1, λ n and δ ω 1, ω l+1. Then λ, δ are as in the Tables below, listed up to duals. Conversely, for each possibility in the tables, V X is multiplicity-free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
51 λ δ 2λ 1, λ 2 ω 1 + cω i cω 1 + ω i ω i + cω i+1 cω i + ω i+1 λ 2 2ω 1 + 2ω l+1 2ω 1 + 2ω 2 λ 3 ω 1 + ω l+1 3λ 1 ω 1 + ω 2 (l = 1) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
52 λ δ λ 1 + λ n cω i λ 1 + λ i (2 i 7) 2ω 1, ω 2 λ 1 + λ n+2 i (2 i 7) λ 2 + λ 3 2λ 1 + λ n 3λ 1 + λ n λ 2 + λ n 1 2λ 1 + λ 2 3λ 1 + λ 2 λ 1 + λ 2 3ω 1, ω 3 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
53 λ δ 2λ 1, λ 2 cω i λ 3 ω i (i 6) cω 1 (c 6) 2ω 2 λ 4 ω i (i 4) cω 1 (c 4) λ 5 cω 1 (c 3) ω 2 λ i (i > 5) 2ω 1 ω 2 3λ 1 cω 1 (c 5) ω i (i 5) 4λ 1 cω 1 (c 3) ω i (i 3) 5λ 1 2ω 1 ω i (i 3) cλ 1 (c > 5) 2ω 1, ω 2 2λ 2, 3λ 2 2ω 1, ω 2 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
54 X λ δ A 1 aλ 1 + λ 2 2ω 1 5λ 1 3ω 1 λ 3 7ω 1 A 3 aλ i ω 2 aλ i + λ j λ 1 + λ 2 + λ 3 λ 1 + λ 2 + λ 5 A 4 aλ 1 + λ 9 ω 2 aλ 1 + λ 2 λ 1 + 2λ 2 2λ 3 2λ 4 A 5 λ i ω 3 λ 1 + λ 18 A 6 λ 5 ω 3 λ 6 A 7 λ 5 ω 3 D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
55 Comparison with Dynkin s list for irreducible triples: Example X = A l+1 Y = A n, via δ = 2ω 1. L Y (λ 2 ) X is irreducible (with highest weight 2ω 1 + ω 2 ) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
56 Comparison with Dynkin s list for irreducible triples: Example X = A l+1 Y = A n, via δ = 2ω 1. L Y (λ 2 ) X is irreducible (with highest weight 2ω 1 + ω 2 ) L Y (λ i ) X is multiplicity free for all i. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
57 Comparison with Dynkin s list for irreducible triples: Example X = A l+1 Y = A n, via δ = 2ω 1. L Y (λ 2 ) X is irreducible (with highest weight 2ω 1 + ω 2 ) L Y (λ i ) X is multiplicity free for all i. X = A l+1, l 2, Y = A n, via δ = ω 2. L Y (λ 2 ) X is irreducible (with highest weight ω 1 + ω 3 ) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
58 Comparison with Dynkin s list for irreducible triples: Example X = A l+1 Y = A n, via δ = 2ω 1. L Y (λ 2 ) X is irreducible (with highest weight 2ω 1 + ω 2 ) L Y (λ i ) X is multiplicity free for all i. X = A l+1, l 2, Y = A n, via δ = ω 2. L Y (λ 2 ) X is irreducible (with highest weight ω 1 + ω 3 ) L Y (λ i ) X is multiplicity free for all i. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
59 Comparison with Dynkin s list for irreducible triples: Example X = A l+1 Y = A n, via δ = 2ω 1. L Y (λ 2 ) X is irreducible (with highest weight 2ω 1 + ω 2 ) L Y (λ i ) X is multiplicity free for all i. X = A l+1, l 2, Y = A n, via δ = ω 2. L Y (λ 2 ) X is irreducible (with highest weight ω 1 + ω 3 ) L Y (λ i ) X is multiplicity free for all i. (Established by Howe.) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
60 Tool : parabolic embeddings Let X = A l+1 Y = SL(W ) = SL n+1, with W = L X (δ). Set V = L Y (λ) for some dominant weight λ. Fix P X = L X Q X X the opposite maximal parabolic corresponding to the root α l+1 ; i.e. L X = T X, U ±αi, 1 i l and Q X = U β β Φ + \ Φ(L X ). Set L X = L X T, where T is a central torus in L X. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
61 Tool : parabolic embeddings Let X = A l+1 Y = SL(W ) = SL n+1, with W = L X (δ). Set V = L Y (λ) for some dominant weight λ. Fix P X = L X Q X X the opposite maximal parabolic corresponding to the root α l+1 ; i.e. L X = T X, U ±αi, 1 i l and Q X = U β β Φ + \ Φ(L X ). Set L X = L X T, where T is a central torus in L X. Let F := W [W, Q X ] [[W, Q X ], Q X ]... 0 Then Stab Y (F) is a parabolic subgroup of Y, say P Y = L Y Q Y satisfying Q X Q Y (and the root groups in Q Y correspond to negative roots), L X L Y = C G (T ), and T X T Y a maximal torus of L Y. [W, QX d ] = [W, Qd Y ] for all d 1. (Here [W, QX d ] = [[W, Qd 1 X ], Q X ], for d 1.) D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
62 Suppose V = L Y (λ) X is multiplicity free. Write V X = r i=1 V i, V i an irreducible kx -module with highest weight θ i. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
63 Suppose V = L Y (λ) X is multiplicity free. Write V X = r i=1 V i, V i an irreducible kx -module with highest weight θ i. Proposition Fix 1 i r. (i) There exists a unique n i Z 0 such that (λ n i α l+1 ) T = θ i T. (ii) n i is maximal subject to V i [V, Q n i Y ]. (iii) (V i + [V, Q n i +1 Y ]/[V, Q n i +1 ]) is irreducible as a kl X -module. Y D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
64 Completeness of list : main inductive tool For d 0, write V d+1 (Q Y ) for the quotient [V, QY d ]/[V, Qd+1 Y ] ( level d, so V /[V, Q Y ] = V 1 (Q Y ) is level 0, etc.) Proposition The following conditions hold. (i) (V /[V, Q Y ]) L X = i,n i =0 V i/[v i, Q X ] is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
65 Completeness of list : main inductive tool For d 0, write V d+1 (Q Y ) for the quotient [V, QY d ]/[V, Qd+1 Y ] ( level d, so V /[V, Q Y ] = V 1 (Q Y ) is level 0, etc.) Proposition The following conditions hold. (i) (V /[V, Q Y ]) L X = i,n i =0 V i/[v i, Q X ] is multiplicity free. (ii) V 2 (Q Y ) L X = i,n i =0 V 2 i (Q X ) + j,n j =1 V j/[v j, Q X ]. Moreover, the second sum is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
66 Completeness of list : main inductive tool For d 0, write V d+1 (Q Y ) for the quotient [V, QY d ]/[V, Qd+1 Y ] ( level d, so V /[V, Q Y ] = V 1 (Q Y ) is level 0, etc.) Proposition The following conditions hold. (i) (V /[V, Q Y ]) L X = i,n i =0 V i/[v i, Q X ] is multiplicity free. (ii) V 2 (Q Y ) L X = i,n i =0 V 2 i (Q X ) + j,n j =1 V j/[v j, Q X ]. Moreover, the second sum is multiplicity free. (iii) V d+1 (Q Y ) L X = i,n i =0 V d+1 i (Q X ) + j,n j =1 V j d (Q X ) + + k,n k =d 1 V k/[v k, Q X ]. The last summand is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
67 Completeness of list : main inductive tool For d 0, write V d+1 (Q Y ) for the quotient [V, QY d ]/[V, Qd+1 Y ] ( level d, so V /[V, Q Y ] = V 1 (Q Y ) is level 0, etc.) Proposition The following conditions hold. (i) (V /[V, Q Y ]) L X = i,n i =0 V i/[v i, Q X ] is multiplicity free. (ii) V 2 (Q Y ) L X = i,n i =0 V 2 i (Q X ) + j,n j =1 V j/[v j, Q X ]. Moreover, the second sum is multiplicity free. (iii) V d+1 (Q Y ) L X = i,n i =0 V d+1 i (Q X ) + j,n j =1 V j d (Q X ) + + k,n k =d 1 V k/[v k, Q X ]. The last summand is multiplicity free. Rank 1 follows from result on multiplicity free A 1 -subgroups. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
68 Proof of MF-ness Main tool: build V as a sum/difference of various tensor products, use the domino method (tilings, skew tableau, Littlewood-Richardson, etc) to study the kx -summands of V.. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
69 Proof of MF-ness Main tool: build V as a sum/difference of various tensor products, use the domino method (tilings, skew tableau, Littlewood-Richardson, etc) to study the kx -summands of V. Example (very easy) X = A 1 Y = A 2 via δ = 2ω. Show L Y (cλ 1 + λ 2 ) X is MF. L Y (cλ 1 ) L Y (λ 2 ) = L Y (cλ 1 + λ 2 ) L Y ((c 1)λ 1 ).. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
70 Proof of MF-ness Main tool: build V as a sum/difference of various tensor products, use the domino method (tilings, skew tableau, Littlewood-Richardson, etc) to study the kx -summands of V. Example (very easy) X = A 1 Y = A 2 via δ = 2ω. Show L Y (cλ 1 + λ 2 ) X is MF. L Y (cλ 1 ) L Y (λ 2 ) = L Y (cλ 1 + λ 2 ) L Y ((c 1)λ 1 ). L Y (dλ 1 ) = Sym d (L Y (λ 1 )).. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
71 Proof of MF-ness Main tool: build V as a sum/difference of various tensor products, use the domino method (tilings, skew tableau, Littlewood-Richardson, etc) to study the kx -summands of V. Example (very easy) X = A 1 Y = A 2 via δ = 2ω. Show L Y (cλ 1 + λ 2 ) X is MF. L Y (cλ 1 ) L Y (λ 2 ) = L Y (cλ 1 + λ 2 ) L Y ((c 1)λ 1 ). L Y (dλ 1 ) = Sym d (L Y (λ 1 )). Check Sym c (L X (2ω)) = L X (2cω) L X ((2c 4)ω), ending with L X (2ω) or L X (0), depending on c odd or even.. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
72 Proof of MF-ness Main tool: build V as a sum/difference of various tensor products, use the domino method (tilings, skew tableau, Littlewood-Richardson, etc) to study the kx -summands of V. Example (very easy) X = A 1 Y = A 2 via δ = 2ω. Show L Y (cλ 1 + λ 2 ) X is MF. L Y (cλ 1 ) L Y (λ 2 ) = L Y (cλ 1 + λ 2 ) L Y ((c 1)λ 1 ). L Y (dλ 1 ) = Sym d (L Y (λ 1 )). Check Sym c (L X (2ω)) = L X (2cω) L X ((2c 4)ω), ending with L X (2ω) or L X (0), depending on c odd or even. Tensor with L X (2ω) and remove the summands coming from Sym c 1 (L X (2ω)).. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
73 Further directions H simple algebraic group over k = C, acting irreducibly on W via ρ : H SL(W ). Determine irreducible ksl(w )-modules V such that V H is multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
74 Further directions H simple algebraic group over k = C, acting irreducibly on W via ρ : H SL(W ). Determine irreducible ksl(w )-modules V such that V H is multiplicity free. H G := Isom(W ), W, vector space over k = C, equipped with nondegenerate bilinear form. Determine the irreducible kg-modules V with V H multiplicity free. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
75 Further directions H simple algebraic group over k = C, acting irreducibly on W via ρ : H SL(W ). Determine irreducible ksl(w )-modules V such that V H is multiplicity free. H G := Isom(W ), W, vector space over k = C, equipped with nondegenerate bilinear form. Determine the irreducible kg-modules V with V H multiplicity free. Consider the above two questions over fields k = k of positive characteristic; count occurrences of composition factors. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
76 Further directions H simple algebraic group over k = C, acting irreducibly on W via ρ : H SL(W ). Determine irreducible ksl(w )-modules V such that V H is multiplicity free. H G := Isom(W ), W, vector space over k = C, equipped with nondegenerate bilinear form. Determine the irreducible kg-modules V with V H multiplicity free. Consider the above two questions over fields k = k of positive characteristic; count occurrences of composition factors. Consider overgroups of distinguished unipotent elements in simple algebraic groups defined over fields of positive characteristic; currently under investigation by Korhonen. D. Testerman (with M. Liebeck and G. Seitz) (EPF Lausanne) Multiplicity free actions Edinburgh, April
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