Cross products, invariants, and centralizers

Transcription

1 Cross products, invariants, and centralizers Alberto Elduque (joint work with Georgia Benkart)

2 1 Schur-Weyl duality 2 3-tangles 3 7-dimensional cross product 4 3-dimensional cross product 5 A (1 2)-dimensional cross product 2/34

3 1 Schur-Weyl duality 2 3-tangles 3 7-dimensional cross product 4 3-dimensional cross product 5 A (1 2)-dimensional cross product 3/34

4 Schur-Weyl duality. General linear group F: algebraically closed field, char F = 0. V finite-dimensional vector space over F. GL(V) V n S n End GL(V) ( V n ) = alg action of S n, End Sn ( V n ) = alg action of GL(V). 4/34

5 Orthogonal group Assume that now V is endowed with a nondegenerate quadratic form. Then: End O(V) ( V n ) = alg action of S n and of the c ij s, where the contractions are given by c ij (v 1 v n ) = ( ) r v i v j v 1 e l f l v n, l=1 where { e l } and { fl } are dual bases. 5/34

6 Brauer algebra Br(x) is the algebra with basis consisting of diagrams of the form: α = with multiplication given by bordism, and by multiplying by the parameter x each time we get a circle: β = α β = x 6/34

7 Orthogonal and symplectic groups Orthogonal group: O(V) V n Br(dim V) ( End O(V) V n ) = alg action of Br(dim V), ( End Br(dim V) V n ) = alg action of O(V). Symplectic group: Sp(V) V n Br( dim V) End Sp(V) ( V n ) = alg action of Br( dim V), End Br( dim V) ( V n ) = alg action of Sp(V). What about G 2 and its natural representation? G 2 V n?? 7/34

8 1 Schur-Weyl duality 2 3-tangles 3 7-dimensional cross product 4 3-dimensional cross product 5 A (1 2)-dimensional cross product 8/34

9 3-tangles A 3-tangle is an equivalence class of graphs with nodes of valence 1, 2 or 3, together with an orientation on the edges incident to each node of valence 3: γ = The boundary consists of the nodes of valence 1: γ. Two such graphs are said to be equivalent if they have the same boundary, and admit a common refinement. Refinements are obtained by splitting edges adding valence two nodes : refinement. 9/34

10 3-tangles [n] [m] For n N, let [n] = {1,..., n} with [0] =. Then, for n, m N, a 3-tangle γ : [n] [m] is a 3-tangle γ with γ = [n] [m] (disjoint union, which may thought of as {1,..., n, 1,..., m }) γ = (The orientation of a valency 3 node is given by clockwise order.) 10/34

11 Operations on 3-tangles γ = γ t = transpose 1 2 γ = γ γ = disjoint union γ γ = composition /34

12 Category T Objects: Morphisms: [n], n N (0 N). linear combinations of 3-tangles [n] [m]. induces a tensor product : T T T. The transpose induces bijections Mor T ([n], [m]) Mor T ([m], [n]), γ γ t, such that (γ γ) t = γ t (γ ) t whenever this makes sense. There are natural maps Φ n,m : Mor T ([n], [m]) Mor T ([n + m], [0]) and Ψ n,m : Mor T ([n + m], [0]) Mor T ([n], [m]). 12/34

13 Basic morphisms The morphisms I 1 = β = µ = τ = β t = µ t = are called basic. They constitute the alphabet of T. 13/34

14 Category T Γ In addition to generators, some relations can be imposed in the category T. Let Γ = {γ i Mor T ([n i ], [m i ]) : i = 1,..., k} be a finite set of morphisms in T. For each n, m N, the set Γ generates, through compositions and tensor products with arbitrary 3-tangles, a subspace R Γ ([n], [m]) of Mor T ([n], [m]), and we define a new category T Γ with the same objects and with Mor TΓ ([n], [m]) = Mor T ([n], [m])/ R Γ ([n], [m]). T Γ is the 3-tangle category associated with the set of relations Γ. 14/34

15 The functor R V V = (V, b, m) finite-dimensional nonassociative algebra with multiplication m, endowed with an associative, nondegenerate, symmetric bilinear form b : V V F. Let V be the category of finite-dimensional vector spaces over F with linear maps as morphisms. Denote by τ the switch map τ : V 2 V 2, x y y x. Identify b with a linear map V 2 F and m with a linear map V 2 V. Let 1 V be the identity map on V. Theorem (Boos, Cadorin, Knus, Rost ) There exists a unique functor R V : T V such that: 1. R V ([0]) = F and R V ([n]) = V n, for any n R V (I 1 ) = 1 V and R V (τ) = τ. 3. R V (β) = b, R V (µ) = m, R V (γ t ) = R V (γ) t, and R V (γ δ) = R V (γ) R V (δ), for morphisms γ and δ in T. 15/34

16 The functor R V Remark Let R V (β t β) = dim F V F = Hom F (F, F). Γ V = {c i Hom F (V n i, V m i ) : i = 1,..., k} be a finite set of homomorphisms. Definition The algebra V is said to be of tensor type Γ V if the c i s are identities for V, i.e., if c i (x 1 x ni ) = 0 for all i = 1,..., k, and all x 1,..., x ni V. 16/34

17 The functor R V Corollary V = (V, b, m) be an algebra of tensor type Γ V = {c i : i = 1,..., k}, for tensors c i expressible in terms of the alphabet, 1 V, m, m t, b, b t, and τ. Γ = {γ i : i = 1,..., k} with R V (γ i ) = c i for all i = 1,..., k. Then there is a unique functor R Γ : T Γ V such that R V = R Γ P, where P is the natural projection T T Γ. 17/34

18 1 Schur-Weyl duality 2 3-tangles 3 7-dimensional cross product 4 3-dimensional cross product 5 A (1 2)-dimensional cross product 18/34

19 Cross products Let (V, b) be a vector space V over F equipped with a nondegenerate symmetric bilinear form b. A cross product on (V, b) is a bilinear multiplication V V V, (u, v) u v, such that: for any u, v V. u u = 0, b(u v, u) = 0, b(u v, u v) = b(u, u) b(u, v) b(v, u) b(v, v), A nonzero cross product exists only if dim F V = 3 or 7. 19/34

20 Cross products Anticommutativity of the cross product corresponds, through R V to the identity γ 1 : + = 0. The last condition on the definition of cross product corresponds to γ 2 : = 0. The dimension corresponds to: γ 0 : β t β (dim F V)1 = (dim F V)1 = 0. 20/34

21 Cross products Proposition If V = (V, b, ) for a vector space V endowed with a nonzero cross product relative to the nondegenerate symmetric bilinear form b, then the functor R V induces a functor R Γ : T Γ V, with Γ = {γ 0, γ 1, γ 2 }. 21/34

22 7-dimensional cross product Goal To prove that is a bijection. R Γ : Mor TΓ ([n], [m]) Hom Aut(V, ) (V n, V m ) Several steps will be followed: = = Therefore, in T Γ we have = 0 22/34

23 7-dimensional cross product Relation γ 2 gives: = + 2 = = (1 dim F V) = 6. so = 6 23/34

24 7-dimensional cross product Again with relation γ 2, we get: + = 2 = 3 so we can get rid of triangles: = 3 24/34

25 7-dimensional cross product [ ] [ ] = [ = [ ] ] 25/34

26 7-dimensional cross product Theorem Let n, m N, and Γ = {γ 0, γ 1, γ 2 }, (a) The classes modulo Γ of the 3-tangles [n] [m] without crossings and without any of the subgraphs:,,,,,, form a basis of Mor TΓ ([n], [m]). (b) The functor R Γ gives a linear isomorphism Mor TΓ ([n], [m]) Hom Aut(V, ) ( V n, V m). (c) The 3-tangles [n] [n] as in part (a) give a basis of the centralizer algebra: End Aut(V, ) ( V n ) Mor TΓ ([n], [n]). 26/34

27 1 Schur-Weyl duality 2 3-tangles 3 7-dimensional cross product 4 3-dimensional cross product 5 A (1 2)-dimensional cross product 27/34

28 3-dimensional cross product V = (V, b, ) a 3-dimensional vector space over F, endowed with a nonzero cross product u v, relative to a nondegenerate symmetric bilinear form b. Then we have for any u, v, w V. u u = 0, b(u v, u) = 0, (u v) w = b(u, w)v b(v, w)u, We must replace Γ = {γ 0, γ 1, γ 2 } above by Γ = { γ 0, γ 1 = γ 1, γ 2 }: γ 0 = β t β 3 : 3 γ 1 = + γ 2 = + 28/34

29 3-dimensional cross product Theorem Let n, m N, and Γ = { γ 0, γ 1, γ 2 }. (a) The classes modulo Γ of normalized 3-tangles [n] [m] form a basis of Mor T Γ([n], [m]). (b) R Γ gives a linear isomorphism Mor T Γ([n], [m]) Hom SO(V,b) (V n, V m ). (c) The normalized 3-tangles [n] [n] give a basis of the centralizer algebra End SO(V,b) ( V n ) Mor T Γ([n], [n]), and dim End SO(V,b) ( V n ) equals the number a(2n) of Catalan partitions. 29/34

30 3-dimensional cross product For the proof, as for the 7-dimensional case, we can get rid of crossings, circles, and here we get rid also of all cycles. Relation γ 2 = 0 can be thought as: = which allows to prove, for instance, = integral linear combination of tree 3-tangles with a lower number of trivalent nodes. (normalized) 30/34

31 1 Schur-Weyl duality 2 3-tangles 3 7-dimensional cross product 4 3-dimensional cross product 5 A (1 2)-dimensional cross product 31/34

32 for all u, v V 1. 32/34 Kaplansky superalgebra with V 0 = Fe, V 1 = Fp Fq, e e = e, e u = u e = 1 u u V 1, 2 p p = q q = 0, p q = q p = e. Consider the even nondegenerate supersymmetric bilinear form b : V V F such that Then b(e, e) = 1, b(p, q) = 1. 2 e e = e, e u = u e = 1 u, u v = b(u, v)e, 2

33 (1 2)-dimensional cross product We must replace here Γ = {γ 0, γ 1, γ 2 } by Γ s = {γ s 0, γs 1, γs 2, γs 3 }, with γ s 0 = + 1 γ s 1 = γ s 2 = γ s 3 = 33/34

34 (1 2)-dimensional cross product Theorem Let V be the 3-dimensional Kaplansky superalgebra over a field F of characteristic 0. Let n, m N. (a) The classes modulo Γ s of the normalized 3-tangles [n] [m] form a basis of Mor TΓ s ([n], [m]). (b) There is a natural functor R Γ s that gives a linear isomorphism Mor TΓ s ([n], [m]) Hom osp(v,b) (V n, V m ). (Caution: the switch map is now u v ( 1) uv v u.) (c) The normalized 3-tangles [n] [n] give a basis of the centralizer algebra End osp(v,b) ( V n ) Mor TΓ s ([n], [n]), whose dimension is the number a(2n) of Catalan partitions. 34/34