Introduction Plethystic substitution Substitution operation in the - PowerPoint PPT Presentation

A simplicial groupoid for plethysm ∗ Alex Cebrian Universitat Aut` onoma de Barcelona July 11, 2018 ∗ arXiv:1804.09462

Introduction Plethystic substitution Substitution operation in the ring of power series in infinitely many variables, G ( x 1 , x 2 , . . . ) ⊛ F ( x 1 , x 2 , . . . ) = G ( F 1 , F 2 , . . . ) , where F k ( x 1 , x 2 , . . . ) = F ( x k , x 2 k , . . . ) . ◮ P´ olya (1937): unlabelled enumeration. Given a species F : B − → Set , its cycle index series is a power series Z F ( x 1 , x 2 , . . . ). It satisfies Z F ◦ G = Z F ⊛ Z G ◮ Littlewood (1944): representation theory of GL. The character of the composition of polynomial representations is the plethysm of their characters.

Introduction ◮ Nava–Rota (1985): combinatorial interpretation of plethystic substitution based on partitions. Goal Recover ⊛ from the coproduct of Q π 0 T 1 S for an explicit simplicial groupoid T S : ∆ op − → Grpd . homotopy T S : ∆ op − → Grpd incidence � Grpd / T 1 S cardinality � Q π 0 T 1 S bialgebra

Contents Introduction Fa` a di Bruno Segal groupoids Incidence bialgebras Homotopy cardinality Plethystic substitution The simplicial groupoid T S

Fa` a di Bruno One-variable power series Bell polynomials B n , k x n � F ( x ) = n ! ∈ Q [[ x ]] f n n n =1 � ∆( A n ) = B n , k ( A 1 , A 2 , . . . ) ⊗ A k x n k =1 � G ( x ) = n ! ∈ Q [[ x ]] g n n =1 B n , k counts the number of surjections Fa` a di Bruno bialgebra F Free algebra Q [ A 1 , A 2 , . . . ], n ։ k ։ 1 A n : Q [[ x ]] − → Q ∼ up to k − → k . F �− → f n , Example with coproduct B 6 , 2 = 6 A 1 A 5 +15 A 2 A 4 +10 A 3 A 3 ∆( A n )( F ⊗ G ) = A n ( G ◦ F )

� ✤ Fa` a di Bruno Theorem (Joyal,1981) F is isomorphic to the homotopy cardinality of the incidence bialgebra of the fat nerve of the category of finite sets and surjections N S : ∆ op − → Grpd . This isomorphisms takes A n to n ։ 1, and A n 1 · · · A n ℓ to ( n = n 1 + · · · + n ℓ ։ ℓ ). The coproduct is given by � ∆( n ։ ℓ ) = ( n ։ k ) ⊗ ( k ։ ℓ ) . n ։ k ։ ℓ Remark ✤ ( d 2 , d 0 ) � ( n ։ k , k ։ ℓ ) d 1 ( n ։ ℓ ) ( n ։ k ։ ℓ )

� � Segal groupoids A simplicial groupoid X : ∆ op − → Grpd is Segal if for all n > 0 d 0 X n +1 X n d n +1 � d n � X n − 1 . X n d 0 Example The fat nerve of a category. Remark There is an up to equivalence “composition” given by ∼ d 1 X 1 × X 0 X 1 ← − − − X 2 − − − → X 1 , which is actual composition when X is the fat nerve of a category.

� � � � � � Incidence bialgebras ( d 2 , d 0 ) � X 1 × X 1 d 1 X 1 X 2 A A ∆: Grpd / X 1 − → Grpd / X 1 × X 1 A s ( d 2 , d 0 ) ! ◦ d ∗ − → X 1 �− → 1 ( s ) In a similar way we obtain a functor ǫ : Grpd / X 1 − → Grpd . aaaaaaa Theorem (G´ alvez, Kock, Tonks) If X is a Segal space the functors ∆ and ǫ are respectively coassociative and counital, up to homotopy.

� � � Incidence bialgebras Definition The slice groupoid Grpd / X 1 together with ∆ and ǫ is the incidence coalgebra of X . CULF monoidal structure Product X n × X n → X n compatible with face and degeneracy maps and such that g × g X n × X n X 1 × X 1 � X 1 , X n g with g induced by the endpoint preserving map [1] → [ n ]. Most of times in combinatorics the monoidal structure is disjoint union. Incidence bialgebra If X is CULF monoidal the incidence coalgebra becomes a bialgebra.

Homotopy cardinality Homotopy cardinality of a groupoid 1 � | · | : Grpd − → Q , | A | := | Aut( a ) | ∈ Q a ∈ π 0 A p Homotopy cardinality of a finite map of groupoids A − → B | A b | � | · | : Grpd / B − → Q π 0 B , | p | := | Aut( b ) | δ b ∈ Q π 0 B , b ∈ π 0 B where A b is homotopy fibre. Remark | 1 b | | 1 � b � − − → B | = | Aut( b ) | δ b = δ b

� � Homotopy cardinality The homotopy cardinality of the incidence bialgebra of X gives a bialgebra structure on Q π 0 X 1 . ǫ : Grpd / X 1 − → Grpd ∆: Grpd / X 1 − → Grpd / X 1 × X 1 ǫ : Q π 0 X 1 − → Q ∆: Q π 0 X 1 − → Q π 0 X 1 ⊗ Q π 0 X 1

Plethystic substitution Notation ◮ λ = ( λ 1 , λ 2 , . . . ), nonzero infinite vector of natural numbers with finite number of nonzero entries, ◮ aut( λ ) = 1! λ 1 λ 1 ! · 2! λ 2 λ 2 ! · · · , ◮ x λ = x λ 1 1 x λ 2 2 · · · . n th Verschiebung operator Shifts the k th entry λ k of λ to the nk th position. For example V 2 (5 , 9 , 2 , 0 . . . ) = (0 , 5 , 0 , 9 , 0 , 2 , 0 . . . ) .

� � Plethystic substitution ii) aut( λ ) = | Aut( a ։ b ) | Remark ∼ � a a i) λ represents the isomorphism class of a surjection of finite sets ∼ � b b a ։ b iii) V n λ is the class of n × a ։ b with λ k fibers of size k . Example V 2 (1 , 0 , 2) = (0 , 1 , 0 , 0 , 0 , 2) Example corresponds to (1 , 0 , 2) corresponds to : : : : : : : . . . . . . . . . . . . .

Plethystic substitution Infinitely many variables power series x µ x λ � � F ( x ) = f µ aut( µ ) ∈ Q [[ x ]] , G ( x ) = g λ aut( λ ) ∈ Q [[ x ]] µ λ Plethystic substitution ( G ⊛ F )( x ) = G ( F 1 , F 2 , . . . ) , with x V k µ � F k ( x 1 , x 2 , . . . ) = F ( x k , x 2 k , . . . ) = f µ aut( µ ) . µ

Plethystic substitution Plethystic bialgebra P Polynomials P σ,λ ( { A µ } ) Free algebra Q [ { A λ } λ ], A σ : Q [[ x ]] − → Q � ∆( A σ ) = P σ,λ ( { A µ } ) ⊗ A λ F �− → f σ , λ with coproduct What does P σ,λ count? ∆( A σ )( F ⊗ G ) = A σ ( G ⊛ F ) Example 6! 2 2!4! 6! 2 2!4! 4!3! 2 2! 2 2! A (0 , 0 , 0 , 1) A 2 P (0 , 0 , 0 , 1 , 0 , 2) , (1 , 2) = (0 , 0 , 1) + 6!3!2!2! 2 2! 2 A (0 , 0 , 0 , 0 , 0 , 1) A (0 , 0 , 1) A (0 , 1) (0 , 0 , 0 , 1 , 0 , 2) = V 1 (0 , 0 , 0 , 0 , 0 , 1) + V 2 (0 , 0 , 1) + V 2 (0 , 1) (0 , 0 , 0 , 1 , 0 , 2) = V 1 (0 , 0 , 0 , 1) + V 2 (0 , 0 , 1) + V 2 (0 , 0 , 1)

�� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� �� The simplicial groupoid T S : ∆ op − → Grpd t 02 t 01 t 01 t 12 � � t 11 , t 00 � � t 11 � � t 22 t 00 t 03 t 02 t 13 t 01 t 12 t 23 � � t 11 � � t 22 � � t 33 t 00 ◮ t ij are finite sets, ◮ ։ are surjections, ◮ every square is a pullback of finite sets.

�� �� �� �� �� �� �� �� �� �� �� �� �� The simplicial groupoid T S : ∆ op − → Grpd Face maps d i removes all the elements containing an i index: t 02   t 01     t 01 t 12 = d 0   � � t 11   t 00   � � t 11 � � t 22 , t 00 Degeneracy maps s i repeats all the elements containing an i index: t 01   t 01  = s 1 t 01 t 11  � � t 11 t 00 � � t 11 t 00 t 11

�� �� �� �� �� �� �� �� �� �� �� �� �� The simplicial groupoid T S : ∆ op − → Grpd Face maps d i removes all the elements containing an i index: t 02   t 12     t 01 t 12 = d 2   � � t 22   t 11   � � t 11 � � t 22 , t 00 Degeneracy maps s i repeats all the elements containing an i index: t 01   t 01  = s 1 t 01 t 11  � � t 11 t 00 � � t 11 t 00 t 11

�� �� �� �� �� �� �� �� �� �� �� �� �� The simplicial groupoid T S : ∆ op − → Grpd Face maps d i removes all the elements containing an i index: t 02   t 02     t 01 t 12 = d 1   � � t 22   t 00   � � t 11 � � t 22 , t 00 Degeneracy maps s i repeats all the elements containing an i index: t 01   t 01  = s 1 t 01 t 11  � � t 11 t 00 � � t 11 t 00 t 11

�� �� �� �� �� �� �� �� �� �� �� �� �� The simplicial groupoid T S : ∆ op − → Grpd Face maps d i removes all the elements containing an i index: t 02   t 02     t 01 t 12 = d 1   � � t 22   t 00   � � t 11 � � t 22 , t 00 Degeneracy maps s i repeats all the elements containing an i index: t 01   t 01  = s 0 t 00 t 01  � � t 11 t 00 � � t 11 t 00 t 00

� � � � �� �� �� �� �� � �� �� �� �� �� � The simplicial groupoid T S : ∆ op − → Grpd Proposition. T S is a Segal groupoid. ∼ Equivalence T 1 S × T 0 S T 1 S − − − → T 2 S : t 01 × t 11 t 12 t 01 t 12 � � �− → t 01 t 12 � � t 11 � � t 22 t 00 � � � � t 11 � � t 22 t 00 Proposition. T S is CULF monoidal with disjoint union (+) . t ′ t 01 + t ′ t 01 01 01 + = � � � � t 11 � � t ′ � � t 11 + t ′ t 00 t ′ t 00 + t ′ 00 11 00 11

�� �� The simplicial groupoid T S : ∆ op − → Grpd Corollary Q π 0 T S has a bialgebra structure given the homotopy cardinality of the incidence bialgebra of T S . Remark The isomorphism classes of connected elements t 01 � � 1 t 00 of T 1 S form a basis of Q π 0 T 1 S .