Combinatorial Hopf Algebras. YORK

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1 Combinatorial Hopf Algebras. YORK Nantel Bergeron York Research Chair in Applied Algebra [with J.Y. Thibon and many more] U N I V E R S I T É U N I V E R S I T Y Ottrott Mar 2017

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4 Outline What would be a good gift for a mathematician? What is a Combinatorial Hopf Algebra? Sym is a strong, realizable CHA with character. On strong CHA (categorification) On realizable CHA (word combinatorics and quotients). Mar 2017 Lotharingien outline

5 Combinatorial Hopf Algebra H = n 0 H n a graded connected Hopf algebra is CHA if (weak) There is a distinguished (combinatorial) basis with positive integral structure coefficients (from Hopf monoid). (strong) The structure is obtained from representation operation (from categorification). (real.) It can be realized in a space of series in variables. (it is realizable) (char.) It has a distinguished character. (with character) 1/20 Combinatorial Hopf Algebra

6 Combinatorial Hopf Algebra Hopf Monoid Categorification K F H = n 0 H n T rivial Representations Realization Cauchy Kernel Character ζ : H Q 1/20 Combinatorial Hopf Algebra

7 Sym is the model CHA Sym is the space of symmetric functions Z[h 1, h 2,...], with deg(h k ) = k and k (h k ) = h i h k i. i=0 2/20 Combinatorial Hopf Algebra

8 Sym is the model CHA Hopf Monoid Categorification ( Sym ) Realization Character ζ : H Q 2/20 Combinatorial Hopf Algebra

9 Sym is the model CHA Sym is the space of symmetric functions Z[h 1, h 2,...], with deg(h k ) = k and k (h k ) = h i h k i. i=0 It is the functorial image of a Hopf Monoid Π: For any finite set J let Π[J] = {A : A J} the set partitions of J. Product and Coproduct: combinatorial constructions on set partitions It correspond to flats of the hyperplane arrangement of type A. 3/20 Combinatorial Hopf Algebra

10 Sym is the model CHA Hopf Monoid Π Categorification {A} A J K {h λ } λ n {m λ } λ n ( Sym ) Realization Character ζ : H Q 3/20 Combinatorial Hopf Algebra

11 Hopf structure on n 0 K 0(S n ) K 0 (S) = n 0 K 0(S n ) is the space of S n -modules up to isomorphism Basis: Irreducible modules S λ Structure: M N = Ind S n+m S n S m M N n M = Res S n S k S n k M k=0 F : K 0 (S) Sym is an isomorphism of graded Hopf algebra where F(S λ ) = s λ 4/20 Combinatorial Hopf Algebra

12 Sym is the model CHA Hopf Monoid Π Categorification {A} A J K F {S λ } λ n {h λ } λ n {m λ } λ n ( Sym ) {s λ } λ n Realization Character ζ : H Q 4/20 Combinatorial Hopf Algebra

13 Realization of Sym Sym lim n Q[x 1, x 2,..., x n ] Allows us to understand coproducts, internal coproduct, plethysm, Cauchy kernel,... 5/20 Combinatorial Hopf Algebra

14 Sym is the model CHA Hopf Monoid Π Categorification {A} A J K F {S λ } λ n {h λ } λ n {m λ } λ n ( Sym ) {s λ } λ n lim Q[x 1, x 2,..., x n ] n Character ζ : H Q 5/20 Combinatorial Hopf Algebra

15 Sym with a Hopf Character ζ 0 : Sym Q f(x 1, x 2,...) f(1, 0,...) (Sym, ζ 0 ) is a terminal object for (H, ζ) cocommutative: H Sym ζ ζ 0 Q ζ 0 = n 0 h n Ω(X) = n 0 h n (X) = x X 1 1 x 6/20 Combinatorial Hopf Algebra

16 Sym is the model CHA Hopf Monoid Π Categorification {A} A J K F {S λ } λ n {h λ } λ n {m λ } λ n ( Sym ) {s λ } λ n T rivial Representations h n lim Q[x 1, x 2,..., x n ] (Sym, ζ n 0 ) Ω(x 1,x 2,...) ζ : H Q 6/20 Combinatorial Hopf Algebra

17 Toward Categorification Consider a graded algebra A = n 0 A n Each A n is an algebra. dim A 0 =1 and dim A n <. ρ n,m : A n A m A n+m ; injective algebra homomorphism A n+m is projective bilateral submodule of A m A m. Right and left projective structure of A n+m are compatible. There is a Mackey formula linking induction and restriction [ A is a tower of algebra ] 7/20 Combinatorial Hopf Algebra

18 Toward Categorification Consider a tower of algebras A = n 0 A n Let K 0 (A) = n 0 K 0(A n ) is the space of (projective) A n -modules up to isomorphism and modulo short exact sequences K 0 (A) is a graded Hopf algebra: M N = Ind A n+m A n A m M N M = n k=0 Res A n A k A n k M H is a strong CHA if there is an isomorphism F : K 0 (A) H 7/20 Combinatorial Hopf Algebra

19 Example of Tower of Algebras QS = n 0 QS n: F : K 0 (QS) Sym H(0) = n 0 H n(0): [Krob-Thibon] F : K 0 (H(0)) NSym F : G 0 (H(0)) QSym HC(0) = n 0 HC n(0): [B-Hivert-Thibon]... Peak algebras... seams rare? 8/20 Combinatorial Hopf Algebra

20 Obstruction to Tower of algebras? Consider a tower of algebras A = n 0 A n where K 0 (A) and G 0 (A) are graded dual Hopf algebra: THEOREM[B-Lam-Li] [ if A is a tower of algebras, then dim(a n ) = r n n! ] this is very restrictive... 9/20 Combinatorial Hopf Algebra

21 Tower of Supercharacters [... B... Novelli... Thibon...] Unipotent upper triangular matrices over finite Fields F q : U n (q). Superclasses in U n (q): A = B (A I) = M(B I)N Supercharacters χ: characters constant on superclasses: (χ) = Res U n(q) U A (q) U B (q) χ A+B=[n] F : K 0 ( n 0 χ ψ = Inf U n+m(q) U n (q) U m (q) χ ψ = (χ ψ) π ) U n (2) NCSym is iso. where π : U n+m (q) U n (q) U m (q). N CSym symmetric functions in non-commutative variables. 10/20 Combinatorial Hopf Algebra

22 Some open questions (Q-1) Find other examples of Categorification (Can we do N CQsym (quasi-symmetric in non commutative variables)? (Q-2) Tower of algebra A (axiomatization with superclasses/ supermodules and Harish-Chandra induction: Ind Inf and Def Res ). 11/20 Combinatorial Hopf Algebra

23 About Realization Many CHA are realized: Sym, NSym, QSym, NCSym, Can we described all H Q x 1, x 2,... with monomial basis (equivalence classes on words) [Giraldo]. [B-Hohlweg] Monomial basis embeddings H SSym (Q-3) Realization Theory: Can we describe monomial embeddings H QM for different monoid M 12/20 Combinatorial Hopf Algebra

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25 Reverse Lex and Gröbner basis x n =0 Q[x 1,..., x n+1 ] Q[x 1,..., x n ] x n =0 H[x 1,..., x n+1 ] H[x 1,..., x n ] G n G-basis of ideal H[x 1,..., x n ] + : G n+1 g(x 1,..., x n+1 ) x n =0 G n 0 if LT (g) x n=0 =0 g if LT (g) x =LT ( g) 0 n=0 / B n basis of quotient Q[x 1,...,x n ] H[x 1,...,x n ] + : B n+1 B n 14/20 Combinatorial Hopf Algebra

26 Reverse Lex and Gröbner basis x n =0 Q[x 1,..., x n+1 ] Q[x 1,..., x n ] x n =0 H[x 1,..., x n+1 ] H[x 1,..., x n ] G n+1 g(x 1,..., x n+1 ) x n =0 G n 0 if LT (g) x n=0 =0 g if LT (g) x =LT ( g) 0 n=0 B n+1 B n+1 B n+1 B n+1 mult by x n mult by x 2 n mult by x 3 n B n B n B n B n 14/20 Combinatorial Hopf Algebra

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32 About family of Realization (Q-4) Prove previous question about Hilbert series (Q-5) Realized Quotient in general 20/20 Combinatorial Hopf Algebra

33 M E R C I T H A N K S G R A C I A S 561/20 Combinatorial Hopf Algebra