Combinatorial Hopf Algebras. YORK Nantel Bergeron York Research - PowerPoint PPT Presentation

Combinatorial Hopf Algebras. YORK Nantel Bergeron York Research Chair in Applied Algebra www.math.yorku.ca/bergeron [with J.Y. Thibon ... ... and many more] U N I V E R S I T ´ E ——————— U N I V E R S I T Y Ottrott Mar 2017

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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

Combinatorial Hopf Algebra � H = H n a graded connected Hopf algebra is CHA if n ≥ 0 (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) Ottrott, Mar 2017 1/20 Combinatorial Hopf Algebra

Combinatorial Hopf Algebra Hopf Monoid Categorification F K � H = H n T rivial Representations n ≥ 0 Realization Character Cauchy ζ : H → Q Kernel Ottrott, Mar 2017 1/20 Combinatorial Hopf Algebra

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 Ottrott, Mar 2017 2/20 Combinatorial Hopf Algebra

Sym is the model CHA Hopf Monoid Categorification � � Sym Realization Character ζ : H → Q Ottrott, Mar 2017 2/20 Combinatorial Hopf Algebra

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 . Ottrott, Mar 2017 3/20 Combinatorial Hopf Algebra

Sym is the model CHA Hopf Monoid Π Categorification { A } A ⊢ J K { h λ } λ ⊢ n � � Sym { m λ } λ ⊢ n Realization Character ζ : H → Q Ottrott, Mar 2017 3/20 Combinatorial Hopf Algebra

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 � Res S n ∆ M = 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 Ottrott, Mar 2017 4/20 Combinatorial Hopf Algebra

Sym is the model CHA Hopf Monoid Π Categorification { S λ } λ ⊢ n { A } A ⊢ J F K { h λ } λ ⊢ n � � Sym { s λ } λ ⊢ n { m λ } λ ⊢ n Realization Character ζ : H → Q Ottrott, Mar 2017 4/20 Combinatorial Hopf Algebra

Realization of Sym → lim n →∞ Q [ x 1 , x 2 , . . . , x n ] Sym ֒ Allows us to understand coproducts, internal coproduct, plethysm, Cauchy kernel, ... Ottrott, Mar 2017 5/20 Combinatorial Hopf Algebra

Sym is the model CHA Hopf Monoid Π Categorification { S λ } λ ⊢ n { A } A ⊢ J F K { h λ } λ ⊢ n � � Sym { s λ } λ ⊢ n { m λ } λ ⊢ n n →∞ Q [ x 1 , x 2 , . . . , x n ] lim Character ζ : H → Q Ottrott, Mar 2017 5/20 Combinatorial Hopf Algebra

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: Sym H ζ ζ 0 Q � ζ ∗ 0 = h n n ≥ 0 1 � � Ω( X ) = h n ( X ) = 1 − x n ≥ 0 x ∈ X Ottrott, Mar 2017 6/20 Combinatorial Hopf Algebra

Sym is the model CHA Hopf Monoid Π Categorification { S λ } λ ⊢ n { A } A ⊢ J F K { h λ } λ ⊢ n � � Sym { s λ } λ ⊢ n { m λ } λ ⊢ n T rivial Representations � hn n →∞ Q [ x 1 , x 2 , . . . , x n ] lim ( Sym, ζ 0 ) Ω( x 1 ,x 2 ,... ) ζ : H → Q Ottrott, Mar 2017 6/20 Combinatorial Hopf Algebra

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 Ottrott, Mar 2017 7/20 Combinatorial Hopf Algebra

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 n � Res A n ∆ M = A k ⊗ A n − k M k =0 • H is a strong CHA if there is an isomorphism F : K 0 ( A ) → H Ottrott, Mar 2017 7/20 Combinatorial Hopf Algebra

Example of Tower of Algebras Q S = � n ≥ 0 Q S n : F : K 0 ( Q S ) → 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? Ottrott, Mar 2017 8/20 Combinatorial Hopf Algebra

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... Ottrott, Mar 2017 9/20 Combinatorial Hopf Algebra

Tower of Supercharacters [... B ... Novelli ... Thibon ...] • Unipotent upper triangular matrices over finite Fields F q : U n ( q ). • Superclasses in U n ( q ): A ∼ ↔ ( A − I ) = M ( B − I ) N = B • Supercharacters χ : characters constant on superclasses: Res U n ( q ) � ∆( χ ) = U | A | ( q ) × U | B | ( q ) χ A + B =[ n ] χ · ψ = Inf U n + m ( q ) U n ( q ) × U m ( q ) χ ⊗ ψ = ( χ ⊗ ψ ) ◦ π where π : U n + m ( q ) → U n ( q ) × U m ( q ) . � � � • F : K 0 U n (2) → NCSym is iso. n ≥ 0 NCSym symmetric functions in non-commutative variables. Ottrott, Mar 2017 10/20 Combinatorial Hopf Algebra

Some open questions (Q-1) Find other examples of Categorification (Can we do NCQsym (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 ). Ottrott, Mar 2017 11/20 Combinatorial Hopf Algebra

About Realization Many CHA are realized: Sym, NSym , QSym, NCSym, • • • Can we described all → Q � x 1 , x 2 , . . . � H ֒ with monomial basis (equivalence classes on words) [Giraldo]. [B-Hohlweg] Monomial basis embeddings → SSym H ֒ (Q-3) Realization Theory: Can we describe monomial embeddings H ֒ → Q M for different monoid M Ottrott, Mar 2017 12/20 Combinatorial Hopf Algebra

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Reverse Lex and Gr¨ obner 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 ] + � : x n =0 G n +1 G n  0 if LT ( g ) | xn =0=0  g ( x 1 , . . . , x n +1 ) ˜ if LT ( g ) | xn =0= LT (˜ g g ) � =0  � B n basis of quotient Q [ x 1 ,...,x n ] � H [ x 1 ,...,x n ] + � : B n +1 B n Ottrott, Mar 2017 14/20 Combinatorial Hopf Algebra

Reverse Lex and Gr¨ obner 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 ] x n =0 G n +1 G n  0 if LT ( g ) | xn =0=0  g ( x 1 , . . . , x n +1 ) g ˜ if LT ( g ) | xn =0= LT (˜ g ) � =0  B n +1 B n mult by x n B n +1 B n mult by x 2 n B n +1 B n mult by x 3 n B n +1 B n • • • Ottrott, Mar 2017 14/20 Combinatorial Hopf Algebra

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About family of Realization (Q-4) Prove previous question about Hilbert series (Q-5) Realized Quotient in general • • • Ottrott, Mar 2017 20/20 Combinatorial Hopf Algebra

3 M E R C I 2 1 3 3 23 2 2 1 1 1 3 T H A N K S 13 2 12 123 3 3 23 2 2 1 1 1 G R A C I A S Ottrott, Mar 2017 561/20 Combinatorial Hopf Algebra