A bijection on core partitions and a parabolic quotient of the affine symmetric group

Let $\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\ell$-cores with first part equal to $k$ and $(\ell-1)$-cores with first part less than or equal to $k$. This paper gives several new interpretations of that bijection. The $\ell$-cores index minimal length coset representatives for $\widetilde{S_{\ell}} / S_{\ell}$ where $\widetilde{S_{\ell}}$ denotes the affine symmetric group and $S_{\ell}$ denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type $A_{\ell-1}$. We also show that the bijection has a natural description in terms of another correspondence due to Lapointe and Morse.Comment: 23 page

Dominant Shi regions with a fixed separating wall: bijective enumeration

We present a purely combinatorial proof by means of an explicit bijection, of the exact number of dominant regions having as a separating wall the hyperplane associated to the longest root in the m-extended Shi hyperplane arrangement of type A and dimension n-1

Counting Shi regions with a fixed separating wall

Athanasiadis introduced separating walls for a region in the extended Shi arrangement and used them to generalize the Narayana numbers. In this paper, we fix a hyperplane in the extended Shi arrangement for type A and calculate the number of dominant regions which have the fixed hyperplane as a separating wall; that is, regions where the hyperplane supports a facet of the region and separates the region from the origin.Comment: To appear in Annals of Combinatoric

Bijections of dominant regions in the mm-Shi arrangements of type AA, BB and CC

International audienceIn the present paper, the relation between the dominant regions in the $m$-Shi arrangement of types $B_n/C_n$, and those of the $m$-Shi arrangement of type $A_{n-1}$ is investigated. More precisely, it is shown explicitly how the sets $R^m(B_n)$ and $R^m(C_n)$, of dominant regions of the $m$-Shi arrangement of types $B_n$ and $C_n$ respectively, can be projected to the set $R^m(A_{n-1})$ of dominant regions of the $m$-Shi arrangement of type $A_{n-1}$. This is done by using two different viewpoints for the representative alcoves of these regions: the Shi tableaux and the abacus diagrams. Moreover, bijections between the sets $R^m(B_n)$, $R^m(C_n)$, and lattice paths inside a rectangle $n\times{mn}$ are provided.Dans cet article, nous étudions la relation entre les régions dominantes du $m$-arrangement de Shi de types $B_n/C_n$ et ceux du $m$-arrangement de Shi de type $A_{n-1}$. Plus précisément, nous montrons comment les ensembles $R^m(B_n)$ et $R^m(C_n)$, des régions dominantes du $m$ -arrangement de Shi de types $B_n$ et $C_n$ respectivement, peuvent être projetés sur l’ensemble $R^m(A_{n-1})$ des régions dominantes du $m$-arrangement de Shi de types $A_{n-1}$. Pour cela nous utilisons deux points de vue différents sur les alcôves représentatives de ces régions: les tableaux de Shi et les diagrammes d’abaques. De plus, nous fournissons des bijections entre les ensembles $R^m(B_n)$, $R^m(C_n)$, et les chemins à l’intérieur d’un rectangle $n\times{mn}$

A bijection between dominant Shi regions and core partitions

It is well-known that Catalan numbers $C_n = \frac{1}{n+1} \binom{2n}{n}$ count the number of dominant regions in the Shi arrangement of type $A$, and that they also count partitions which are both $n$-cores as well as $(n+1)$-cores. These concepts have natural extensions, which we call here the $m$-Catalan numbers and $m$-Shi arrangement. In this paper, we construct a bijection between dominant regions of the $m$-Shi arrangement and partitions which are both $n$-cores as well as $(mn+1)$-cores. The bijection is natural in the sense that it commutes with the action of the affine symmetric group

The enumeration of fully commutative affine permutations

We give a generating function for the fully commutative affine permutations enumerated by rank and Coxeter length, extending formulas due to Stembridge and Barcucci--Del Lungo--Pergola--Pinzani. For fixed rank, the length generating functions have coefficients that are periodic with period dividing the rank. In the course of proving these formulas, we obtain results that elucidate the structure of the fully commutative affine permutations.Comment: 18 pages; final versio

Abacus models for parabolic quotients of affine Weyl groups

We introduce abacus diagrams that describe minimal length coset representatives in affine Weyl groups of types B, C, and D. These abacus diagrams use a realization of the affine Weyl group of type C due to Eriksson to generalize a construction of James for the symmetric group. We also describe several combinatorial models for these parabolic quotients that generalize classical results in affine type A related to core partitions.Comment: 28 pages, To appear, Journal of Algebra. Version 2: Updated with referee's comment

Fixed points in smooth Calogero-Moser spaces

We prove that every irreducible component of the fixed point variety under the action of $d$-th roots of unity in a smooth Caloger-Moser space is isomorphic to a Calogero-Moser space associated with another reflection group.Comment: 29 pages, more details about combinatoric. Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, In pres

Affine dual equivalence and k-Schur functions

The k-Schur functions were first introduced by Lapointe, Lascoux and Morse (2003) in the hopes of refining the expansion of Macdonald polynomials into Schur functions. Recently, an alternative definition for k-Schur functions was given by Lam, Lapointe, Morse, and Shimozono (2010) as the weighted generating function of starred strong tableaux which correspond with labeled saturated chains in the Bruhat order on the affine symmetric group modulo the symmetric group. This definition has been shown to correspond to the Schubert basis for the affine Grassmannian of type A by Lam (2008), and, at t = 1, it is equivalent to the k-tableaux characterization of Lapointe and Morse (2007). In this paper, we extend Haiman's (1992) dual equivalence relation on standard Young tableaux to all starred strong tableaux. The elementary equivalence relations can be interpreted as labeled edges in a graph which share many of the properties of Assaf's dual equivalence graphs. These graphs display much of the complexity of working with k-Schur functions and the interval structure on affine Symmetric Group modulo the Symmetric Group. We introduce the notions of flattening and squashing skew starred strong tableaux in analogy with jeu da taquin slides in order to give a method to find all isomorphism types for affine dual equivalence graphs of rank 4. Finally, we make connections between k-Schur functions and both LLT and Macdonald polynomials by comparing the graphs for these functions.Comment: 49 pages, 14 figure

A bijection on core partitions and a parabolic quotient of the affine symmetric group

Let $\ell,k$ be fixed positive integers. In an earlier work, the first and third authors established a bijection between $\ell$-cores with first part equal to $k$ and $(\ell-1)$-cores with first part less than or equal to $k$. This paper gives several new interpretations of that bijection. The $\ell$-cores index minimal length coset representatives for $\widetilde{S_{\ell}} / S_{\ell}$ where $\widetilde{S_{\ell}}$ denotes the affine symmetric group and $S_{\ell}$ denotes the finite symmetric group. In this setting, the bijection has a beautiful geometric interpretation in terms of the root lattice of type $A_{\ell-1}$. We also show that the bijection has a natural description in terms of another correspondence due to Lapointe and Morse