Quasisymmetric functions and Young diagrams

Transcription

1 Quasisymmetric functions and Young diagrams Valentin Féray joint work with Jean-Christophe Aval (LaBRI), Jean-Christophe Novelli and Jean-Yves Thibon (IGM) LaBRI, CNRS, Bordeaux Séminaire Lotharingien de Combinatoire Otrott (Alsace), 27 mars 2012 V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

2 General presentation What is this talk about? Representations of symmetric groups irreducible representations Young diagrams Partition Ô4, 2, 2, 2Õ V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

3 General presentation What is this talk about? Representations of symmetric groups irreducible representations Young diagrams Kerov s and Olshanski s approach : Partition Ô4, 2, 2, 2Õ Consider normalized character values χ λ ÔσÕ tr ÔσÕ ρλ dimôv λ Õ as functions on Young diagrams λ χ λ ÔσÕ (σ fixed). V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

4 General presentation What is this talk about? Representations of symmetric groups irreducible representations Young diagrams Kerov s and Olshanski s approach : Partition Ô4, 2, 2, 2Õ Consider normalized character values χ λ ÔσÕ tr ÔσÕ ρλ dimôv λ Õ as functions on Young diagrams λ χ λ ÔσÕ (σ fixed). In this talk: we explain that these functions live in the ring of quasisymmetric functions Æ. Æ a natural extension of symmetric functions. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

5 General presentation Outline of the talk 1 Existing theory: symmetric functions on Young diagrams 2 An extension: quasisymmetric functions on Young diagrams V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

6 Symmetric functions on Young diagrams Definitions Kerov s and Olshanski s approach Fix µ k. Let us define Ch µ : Y Q; λ nôn 1Õ... Ôn k 1Õχ λ ÔσÕ, where n λ, k µ and σ is a permutation in S n of cycle type µ1 n k. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

7 Symmetric functions on Young diagrams Definitions Kerov s and Olshanski s approach Fix µ k. Let us define Ch µ : Y Q; λ nôn 1Õ... Ôn k 1Õχ λ ÔσÕ, where n λ, k µ and σ is a permutation in S n of cycle type µ1 n k. Examples: Ch µ ÔλÕ 0 as soon as λ µ Ch 1 k ÔλÕ nôn 1Õ... Ôn k 1Õ Ch Ô2Õ ÔλÕ nôn 1Õχ λ Ô1 2Õ ô i for any λ n Ôλ i Õ 2 Ôλ ½ iõ 2 Ch µ 1 ÔλÕ Ôn µ ÕCh µ ÔλÕ for any λ n V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

8 Symmetric functions on Young diagrams Definitions Kerov s and Olshanski s approach Fix µ k. Let us define Ch µ : Y Q; λ nôn 1Õ... Ôn k 1Õχ λ ÔσÕ, where n λ, k µ and σ is a permutation in S n of cycle type µ1 n k. Proposition (Kerov and Olshanski, 1994) The functions Ch µ, when µ runs over all partitions, are linearly independent. Moreover, they span a subalgebra Λ Æ of functions on Young diagrams. Example: Ch Ô2Õ Ch Ô2Õ 4 Ch Ô3Õ Ch Ô2,2Õ 2Ch Ô1,1Õ. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

9 Symmetric functions on Young diagrams Shifted symmetric functions Description of elements of Λ Æ Theorem (Kerov and Olshanski, 1994) Functions in Λ Æ are exactly polynomials in λ 1, λ 2,... which are symmetric in λ 1 1, λ 2 2,... These are called shifted symmetric functions. Widely studied by Olshanski, Okounkov,... V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

10 Symmetric functions on Young diagrams Shifted symmetric functions Description of elements of Λ Æ Theorem (Kerov and Olshanski, 1994) Functions in Λ Æ are exactly polynomials in λ 1, λ 2,... which are symmetric in λ 1 1, λ 2 2,... These are called shifted symmetric functions. Widely studied by Olshanski, Okounkov,... No geometric interpretation of the shifted symmetry. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

11 Symmetric functions on Young diagrams Interlacing coordinates Kerov s interlacing coordinates Alternative description of Young diagrams x-coordinates of lower corners x 0 4, x 1 1, x 2 4 x-coordinates of higher corners y 1 2, y 2 3 x 0 y 1 x 1 y 2 x 2 V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

12 Symmetric functions on Young diagrams Interlacing coordinates Kerov s interlacing coordinates x 0 y 1 x 1 y 2 x 2 Theorem (Kerov, 1999) Λ Æ admits the following algebraic basis: Alternative description of Young diagrams λ ô x k i y k i λ-ring notation: p k X Y : x k i y k i. x-coordinates of lower corners x 0 4, x 1 1, x 2 4 x-coordinates of higher corners y 1 2, y 2 3 k 2. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

13 Symmetric functions on Young diagrams Interlacing coordinates λ-rings in one slide Reminder (k 1): p k ÔXÕ x k i, e k ÔXÕ i 1 i k x i1 x ik. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

14 Symmetric functions on Young diagrams Interlacing coordinates λ-rings in one slide Reminder (k 1): p k ÔXÕ xi k, e k ÔXÕ i 1 i k x i1 x ik. Consider two (finite) alphabets X and Y. We denote X Y their union. ô p k ÔX YÕ p k ÔXÕ p k ÔYÕ, e k ÔX YÕ e i ÔXÕe j ÔYÕ. i j k V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

15 Symmetric functions on Young diagrams Interlacing coordinates λ-rings in one slide Reminder (k 1): p k ÔXÕ xi k, e k ÔXÕ i 1 i k x i1 x ik. Consider two (finite) alphabets X and Y. We denote X Y their union. ô p k ÔX YÕ p k ÔXÕ p k ÔYÕ, e k ÔX YÕ e i ÔXÕe j ÔYÕ. i j k Empty alphabet is neutral. Imagine that Y has an inverse Y. p k ÔY YÕ p k ÔÀÕ 0 p k Ô YÕ p k ÔYÕ

16 Symmetric functions on Young diagrams Interlacing coordinates λ-rings in one slide Reminder (k 1): p k ÔXÕ xi k, e k ÔXÕ i 1 i k x i1 x ik. Consider two (finite) alphabets X and Y. We denote X Y their union. ô p k ÔX YÕ p k ÔXÕ p k ÔYÕ, e k ÔX YÕ e i ÔXÕe j ÔYÕ. i j k Empty alphabet is neutral. Imagine that Y has an inverse Y. p k ÔY YÕ p k ÔÀÕ 0 p k Ô YÕ p k ÔYÕ for 1 l k,e l ÔY YÕ 0 e k Ô YÕ Ô 1Õ k h k ÔYÕ (by induction) V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

17 Symmetric functions on Young diagrams Interlacing coordinates λ-rings in one slide Reminder (k 1): p k ÔXÕ x k i, e k ÔXÕ i 1 i k x i1 x ik. Consider two (finite) alphabets X and Y. We denote X Y their union. ô p k ÔX YÕ p k ÔXÕ p k ÔYÕ, e k ÔX YÕ e i ÔXÕe j ÔYÕ. i j k Empty alphabet is neutral. Imagine that Y has an inverse Y. p k ÔY YÕ p k ÔÀÕ 0 p k Ô YÕ p k ÔYÕ for 1 l k,e l ÔY YÕ 0 e k Ô YÕ Ô 1Õ k h k ÔYÕ (by induction) Y does not exist but we can define f Ô YÕ for any symmetric function f! (and it is compatible with multiplication fg Ô YÕ f Ô YÕg Ô YÕ). V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

18 Symmetric functions on Young diagrams Interlacing coordinates λ-rings in one slide Reminder (k 1): p k ÔXÕ x k i, e k ÔXÕ i 1 i k x i1 x ik. Consider two (finite) alphabets X and Y. We denote X Y their union. ô p k ÔX YÕ p k ÔXÕ p k ÔYÕ, e k ÔX YÕ e i ÔXÕe j ÔYÕ. i j k Empty alphabet is neutral. Imagine that Y has an inverse Y. p k ÔY YÕ p k ÔÀÕ 0 p k Ô YÕ p k ÔYÕ for 1 l k,e l ÔY YÕ 0 e k Ô YÕ Ô 1Õ k h k ÔYÕ (by induction) Y does not exist but we can define f Ô YÕ for any symmetric function f! (and it is compatible with multiplication fg Ô YÕ f Ô YÕg Ô YÕ). Finally p k ÔX YÕ p k ÔXÕ p k Ô YÕ p k ÔXÕ p k ÔYÕ as claimed. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

19 Symmetric functions on Young diagrams Interlacing coordinates Back to Kerov s interlacing coordinates x 0 y 1 x 1 y 2 x 2 Theorem (Kerov, 1999) Λ Æ admits the following algebraic basis: Other way to describe a Young diagram λ ô x k i y k i x-coordinates of lower corners x 0 4, x 1 1, x 2 4 x-coordinates of higher corners y 1 2, y 2 3 k 2. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

20 Symmetric functions on Young diagrams Interlacing coordinates Back to Kerov s interlacing coordinates Other way to describe a Young diagram x-coordinates of lower corners x 0 4, x 1 1, x 2 4 x-coordinates of higher corners y 1 2, y 2 3 x 0 y 1 x 1 y 2 x 2 Theorem (Kerov, 1999) Λ Æ SymÔX YÕ V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

21 Symmetric functions on Young diagrams Interlacing coordinates Back to Kerov s interlacing coordinates x 0 y 1 x 1 y 2 x 2 Theorem (Kerov, 1999) Other way to describe a Young diagram Λ Æ SymÔX YÕ nice compact description. no geometric interpretation of the symmetry. x-coordinates of lower corners x 0 4, x 1 1, x 2 4 x-coordinates of higher corners y 1 2, y 2 3 V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

22 Quasisymmetric functions on Young diagrams Why an extension? And what? Theorem (F. 2006, conjectured by Stanley) Let µ k, π È S k a permutation of type µ. For any Young diagram λ, ô Ch µ ÔλÕ N σ,τ ÔλÕ, σ,τ ÈS k στ π for some nice functions N σ,τ on all Young diagrams. Here, nice means: has a combinatorial description; polynomial with respect to interlacing coordinates. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

23 Quasisymmetric functions on Young diagrams Why an extension? And what? Theorem (F. 2006, conjectured by Stanley) Let µ k, π È S k a permutation of type µ. For any Young diagram λ, ô Ch µ ÔλÕ N σ,τ ÔλÕ, σ,τ ÈS k στ π for some nice functions N σ,τ on all Young diagrams. Here, nice means: has a combinatorial description; polynomial with respect to interlacing coordinates. In general, N σ,τ Ê Λ Æ. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

24 Quasisymmetric functions on Young diagrams Nice functions How to construct a bigger algebra? 2 approaches: Consider the algebra generated by the N σ,τ ; Consider the algebra of nice functions (for some definition of nice). V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

25 Quasisymmetric functions on Young diagrams Nice functions How to construct a bigger algebra? 2 approaches: Consider the algebra generated by the N σ,τ ; Consider the algebra of nice functions (for some definition of nice). In fact, both lead to the same algebra! We will explain the second one. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

26 Quasisymmetric functions on Young diagrams Nice functions What is a nice function? Behave polynomially in interlacing coordinates. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

27 Quasisymmetric functions on Young diagrams Nice functions What is a nice function? Behave polynomially in interlacing coordinates. But different lists of interlacing coordinates represent the same diagram. x 0 y 1 x 1 y 2 x 2 x 0 y 1 x1 y 2 x 2 y 3 x 3 V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

28 Quasisymmetric functions on Young diagrams Nice functions What is a nice function? Behave polynomially in interlacing coordinates. But different lists of interlacing coordinates represent the same diagram. x 0 y 1 x 1 y 2 x x 0 y 2 x 2 y 3 x 3 2 y1 x 1 V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

29 Quasisymmetric functions on Young diagrams Nice functions What is a nice function? Question Which polynomials (in infinitely many variables) fulfill f Ôx 0,y 1,...,x i 1,y i,x i,y i 1,... Õ f Ôx 0,y 1,...,x i 1,y i 1,... Õ; xi y i f Ôx 0,y 1,...,y i,x i,y i 1,x i 1,... Õ f Ôx 0,y 1,...,y i,x i 1,...Õ? xi y i 1 V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

30 Quasisymmetric functions on Young diagrams Nice functions What is a nice function? Question Which polynomials (in infinitely many variables) fulfill f Ôx 0,y 1,...,x i 1,y i,x i,y i 1,... Õ f Ôx 0,y 1,...,x i 1,y i 1,... Õ; xi y i f Ôx 0,y 1,...,y i,x i,y i 1,x i 1,... Õ f Ôx 0,y 1,...,y i,x i 1,...Õ? xi y i 1 Answer The algebra of solutions of the functional equation above is QSymÔÔx 0 Õ Ôy 1 Õ Ôx 1 Õ Ôy 2 Õ Ôx 2 Õ... Õ QSym : quasisymmetric function ring (extension of symmetric function ring) V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

31 Quasisymmetric functions on Young diagrams Nice functions Explanation Example of quasisymmetric function : M 1,2 Ôa 1,a 2,... Õ a i aj 2. i j V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

32 Quasisymmetric functions on Young diagrams Nice functions Explanation Example of quasisymmetric function : M 1,2 Ôa 1,a 2,... Õ a i aj 2. If X, Y and Z are three lists of variables, denote X Y Z their concatenation. M 1,2 ÔX Y ZÕ M 1,2 ÔXÕ M 1,2 ÔYÕ M 1,2 ÔZÕ i j M 1 ÔXÕM 2 ÔYÕ M 1 ÔXÕM 2 ÔZÕ M 1 ÔYÕM 2 ÔZÕ. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

33 Quasisymmetric functions on Young diagrams Nice functions Explanation Example of quasisymmetric function : M 1,2 Ôa 1,a 2,... Õ a i aj 2. If X, Y and Z are three lists of variables, denote X Y Z their concatenation. M 1,2 ÔX Y ZÕ M 1,2 ÔXÕ M 1,2 ÔYÕ Hence we define: M 1,2 ÔZÕ i j M 1 ÔXÕM 2 ÔYÕ M 1 ÔXÕM 2 ÔZÕ M 1 ÔYÕM 2 ÔZÕ. M 1,2 ÔÔx 0 Õ Ôy 1 Õ Ôx 1 ÕÕ M 1,2 Ôx 0 Õ M 1,2 Ô Ôy 1 ÕÕ M 1,2 Ôx 1 Õ M 1 Ôx 0 ÕM 2 Ô Ôy 1 ÕÕ M 1 Ôx 0 ÕM 2 Ôx 1 Õ M 1 ÔÔy 1 ÕÕM 2 Ôx 1 Õ V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

34 Quasisymmetric functions on Young diagrams Nice functions Explanation Example of quasisymmetric function : M 1,2 Ôa 1,a 2,... Õ a i aj 2. If X, Y and Z are three lists of variables, denote X Y Z their concatenation. M 1,2 ÔX Y ZÕ M 1,2 ÔXÕ M 1,2 ÔYÕ Hence we define: M 1,2 ÔZÕ i j M 1 ÔXÕM 2 ÔYÕ M 1 ÔXÕM 2 ÔZÕ M 1 ÔYÕM 2 ÔZÕ. M 1,2 ÔÔx 0 Õ Ôy 1 Õ Ôx 1 ÕÕ 0 M 1,2 Ô Ôy 1 ÕÕ 0 It remains to define M I Ô Ôy 1 ÕÕ. x 0 M 2 Ô Ôy 1 ÕÕ x 0 x 2 1 M 1 ÔÔy 1 ÕÕx 2 1 V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

35 Quasisymmetric functions on Young diagrams Nice functions Explanation (2/2) M I Ô Ôy 1 ÕÕ is computed as for symmetric functions. M 1 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 Õ M 1 Ô Ôy 1 ÕÕ. Thus M 1 Ô Ôy 1 ÕÕ M 1 Ôy 1 Õ y 1. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

36 Quasisymmetric functions on Young diagrams Nice functions Explanation (2/2) M I Ô Ôy 1 ÕÕ is computed as for symmetric functions. M 1 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 Õ M 1 Ô Ôy 1 ÕÕ. Thus M 1 Ô Ôy 1 ÕÕ M 1 Ôy 1 Õ y 1. Similarly, M 2 Ô Ôy 1 ÕÕ M 2 Ôy 1 Õ y 2 1. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

37 Quasisymmetric functions on Young diagrams Nice functions Explanation (2/2) M I Ô Ôy 1 ÕÕ is computed as for symmetric functions. M 1 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 Õ M 1 Ô Ôy 1 ÕÕ. Thus M 1 Ô Ôy 1 ÕÕ M 1 Ôy 1 Õ y 1. Similarly, M 2 Ô Ôy 1 ÕÕ M 2 Ôy 1 Õ y 2 1. Let now consider M 1,2 Ô Ôy 1 Õ 1 Õ: M 1,2 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 ÕM 2 Ô Ôy 1 ÕÕ M 1,2 Ô Ôy 1 ÕÕ Then M 1,2 Ô Ôy 1 ÕÕ M 1 Ôy 1 ÕM 2 Ô Ôy 1 ÕÕ y 3 1. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

38 Quasisymmetric functions on Young diagrams Nice functions Explanation (2/2) M I Ô Ôy 1 ÕÕ is computed as for symmetric functions. M 1 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 Õ M 1 Ô Ôy 1 ÕÕ. Thus M 1 Ô Ôy 1 ÕÕ M 1 Ôy 1 Õ y 1. Similarly, M 2 Ô Ôy 1 ÕÕ M 2 Ôy 1 Õ y 2 1. Let now consider M 1,2 Ô Ôy 1 Õ 1 Õ: M 1,2 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 ÕM 2 Ô Ôy 1 ÕÕ M 1,2 Ô Ôy 1 ÕÕ Then M 1,2 Ô Ôy 1 ÕÕ M 1 Ôy 1 ÕM 2 Ô Ôy 1 ÕÕ y 3 1. Finally M 1,2 ÔÔx 0 Õ Ôy 1 Õ Ôx 1 ÕÕ y 3 1 x 0y 2 1 x 0x 2 1 y 1x 2 1. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

39 Quasisymmetric functions on Young diagrams Nice functions Explanation (2/2) M I Ô Ôy 1 ÕÕ is computed as for symmetric functions. M 1 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 Õ M 1 Ô Ôy 1 ÕÕ. Thus M 1 Ô Ôy 1 ÕÕ M 1 Ôy 1 Õ y 1. Similarly, M 2 Ô Ôy 1 ÕÕ M 2 Ôy 1 Õ y 2 1. Let now consider M 1,2 Ô Ôy 1 Õ 1 Õ: M 1,2 ÔÔy 1 Õ Ôy 1 ÕÕ 0 M 1 Ôy 1 ÕM 2 Ô Ôy 1 ÕÕ M 1,2 Ô Ôy 1 ÕÕ Then M 1,2 Ô Ôy 1 ÕÕ M 1 Ôy 1 ÕM 2 Ô Ôy 1 ÕÕ y 3 1. Finally M 1,2 ÔÔx 0 Õ Ôy 1 Õ Ôx 1 ÕÕ y 3 1 x 0y 2 1 x 0x 2 1 y 1x 2 1. Rk: if we start from M 1,2 Ô Ôy 1 Õ Ôy 1 ÕÕ, we get the same result! V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

40 Quasisymmetric functions on Young diagrams Nice functions Back to the result Theorem The algebra of nice functions on Young diagrams is Painful to compute, but QSymÔÔx 0 Õ Ôy 1 Õ Ôx 1 Õ Ôy 2 Õ Ôx 2 Õ... Õ easy to implement (there are explicit expression for M I Ô YÕ); it gives the algebraic structure of the space of nice functions ( QSym). V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

41 Conclusion Conclusion There is a natural algebra of functions on Young diagrams which is isomorphic to QSym and contains Kerov s and Olshanski s algebra; helps to reprove some results. V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16

42 Conclusion Conclusion There is a natural algebra of functions on Young diagrams which is isomorphic to QSym and contains Kerov s and Olshanski s algebra; helps to reprove some results. Question: as symmetric functions on YD shifted symmetric functions can we consider shifted quasisymmetric functions? Work in progress... V. Féray (+JCA,JCN,JYT) QSym et (LaBRI, digrammes CNRS, de Bordeaux) Young SLC, / 16