Kronecker Coefficients For Some Near-Rectangular Partitions

We give formulae for computing Kronecker coefficients occurring in the expansion of $s_{\mu}*s_{\nu}$, where both $\mu$ and $\nu$ are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study $s_{(n,n-1,1)}*s_{(n,n)}$, $s_{(n-1,n-1,1)}*s_{(n,n-1)}$, $s_{(n-1,n-1,2)}*s_{(n,n)}$, $s_{(n-1,n-1,1,1)}*s_{(n,n)}$ and $s_{(n,n,1)}*s_{(n,n,1)}$. Our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group, mainly employing the Pieri rule and a useful identity of Littlewood. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height, in terms of the Motzkin and Catalan numbers

Quasisymmetric Schur functions and modules of the 0-Hecke algebra

Abstract We define a 0-Hecke action on composition tableaux, and then use it to derive 0-Hecke modules whose quasisymmetric characteristic is a quasisymmetric Schur function. We then relate the modules to the weak Bruhat order and use them to derive a new basis for quasisymmetric functions. We also classify those modules that are tableau-cyclic and likewise indecomposable. Finally, we develop a restriction rule that reflects the coproduct of quasisymmetric Schur functions. Résumé Nous définissons une action 0-Hecke sur les tableaux de composition, et ensuite nous l’utilisons pour dériver les modules 0-Hecke dont la caractéristique quasi-symétrique est une fonction de Schur quasi-symétrique. Nous mettons les modules en relation avec l’ordre de Bruhat faible et les utilisons pour dériver une nouvelle base pour les fonctions quasi-symétriques. Nous classons aussi ces modules qui sont tableau-cycliques et aussi indécomposable. Enfin, nous développons une règle de restriction qui reflète le coproduit des fonctions de Schur quasi-symétriques

Asymmetric function theory

The classical theory of symmetric functions has a central position in algebraic combinatorics, bridging aspects of representation theory, combinatorics, and enumerative geometry. More recently, this theory has been fruitfully extended to the larger ring of quasisymmetric functions, with corresponding applications. Here, we survey recent work extending this theory further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the Schubert calculus conference in Guangzhou, Nov. 201

[Perm_n] via QSym_n^+

International audienceWe give a combinatorial interpretation for the expansion of the cohomology class of the permutahedral variety $Perm_n$ in Schubert classes. This requires understanding Schubert polynomials modulo the ideal $QSym_n^+$ of positive degree quasisymmetric polynomials. We introduce a new basis for the polynomial ring that we call forest polynomials, together with a bijective correspondence. Both constructions are of independent interest.Nous donnons une interprétation combinatoire au développement de la classe de cohomologie de la variété permutaédrale Permn sur les classes de Schubert. Pour cela, il faut comprendre la réduction des polynômes de Schubert modulo l’idéal $QSym_n^+$ des polynômes quasi-symétriques de degré positif. Nous introduisons une nouvelle base pour l’anneau de polynômes que nous appelons polynômes forestiers, ainsi qu’une correspondance bijective, toutes deux ayant un intérêt pour elles-mêmes

MODULES OF THE 0-HECKE ALGEBRA AND QUASISYMMETRIC SCHUR FUNCTIONS

Abstract. We begin by deriving an action of the 0-Hecke alegbra on standard reverse composition tableaux and use it to discover 0-Hecke modules whose quasisymmetric characteristic are the natural refinements of Schur functions known as quasisymmetric Schur functions. Furthermore, we classify combinatorially which of these 0-Hecke modules are indecomposable. From here, we establish that the natural equivalence relation arising from our 0-Hecke action has equivalence classes that are isomorphic to subintervals of the weak Bruhat order on the symmetric group. Focussing on the equivalence classes containing a canonical tableau we discover a new basis for the Hopf algebra of quasisymmetric functions, and use the cardinality of these equivalence classes to establish new enumerative results on truncated shifted reverse tableau studied by Panova and Adin-King-Roichman. Generalizing our 0-Hecke action to one on skew standard reverse composition tableaux, we derive 0-Hecke modules whose quasisymmetric characteristic are the skew quasisymmetric Schur functions of Bessenrodt et al. This enables us to prove a restriction rule that reflects the coproduct fromula for quasisymmetric Schur functions, which in turn yields a quasisymmetric branching rule analogous to the classical branching rule for Schur functions