A new characterization for the m-quasiinvariants of S_n and explicit basis for two row hook shapes

In 2002, Feigin and Veselov defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of Chalykh and Veselov. While many properties of those spaces were proven from this definition, an explicit computation of a basis was only done in certain cases. In particular, Feigin and Veselov computed bases for the m-quasiinvariants of dihedral groups, including S_3, and Felder and Veselov computed the non-symmetric m-quasiinvariants of lowest degree for general S_n. In this paper, we provide a new characterization of the m-quasiinvariants of S_n, and use this to provide a basis for the isotypic component indexed by the partition [n-1,1]. This builds on a previous paper in which we computed a basis for S_3 via combinatorial methods.Comment: 26 pages, uses youngtab.st

The quasiinvariants of the symmetric group

For $m$ a non-negative integer and $G$ a Coxeter group, we denote by $\mathbf{QI_m}(G)$ the ring of $m$-quasiinvariants of $G$, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with $\mathbf{QI_0}(G)$ the whole polynomial ring, and the limit $\mathbf{QI}_{\infty}(G)$ the usual ring of invariants. Remarkably, the ring $\mathbf{QI_m}(G)$ is freely generated over the ideal generated by the invariants of $G$ without constant term, and the quotient is isomorphic to the left regular representation of $G$. However, even in the case of the symmetric group, no basis for $\mathbf{QI_m}(G)$ is known. We provide a new description of $\mathbf{QI_m}(S_n)$, and use this to give a basis for the isotypic component of $\mathbf{QI_m}(S_n)$ indexed by the shape $[n-1,1]$

A new proof of a theorem of Littlewood

Abstract In this paper we give a new combinatorial proof of a result of Littlewood [3]: , where Sµ denotes the Schur function of the partition µ, n(µ) is the sum of the legs of the cells of µ and h (µ) (s) is the hook number of the cell s ∈ µ

The Murnaghan―Nakayama rule for k-Schur functions

We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene

The Murnaghan-Nakayama rule for k-Schur functions

We prove the Murgnaghan--Nakayama rule for $k$-Schur functions of Lapointe and Morse, that is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a $k$-Schur function in terms of $k$-Schur functions. This is proved using the noncommutative $k$-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.Comment: 23 pages, updated to reflect referee comments, to appear in Journal of Combinatorial Theory, Series

On the uniqueness of promotion operators on tensor products of type A crystals

The affine Dynkin diagram of type $A_n^{(1)}$ has a cyclic symmetry. The analogue of this Dynkin diagram automorphism on the level of crystals is called a promotion operator. In this paper we show that the only irreducible type $A_n$ crystals which admit a promotion operator are the highest weight crystals indexed by rectangles. In addition we prove that on the tensor product of two type $A_n$ crystals labeled by rectangles, there is a single connected promotion operator. We conjecture this to be true for an arbitrary number of tensor factors. Our results are in agreement with Kashiwara's conjecture that all `good' affine crystals are tensor products of Kirillov-Reshetikhin crystals.Comment: 31 pages; 8 figure

Combinatorics of Macdonald polynomials and extensions

The theory of symmetric functions is ubiquitous throughout mathematics. They arise naturally in combinatorics, algebra, and geometry, and as a result have been studied intensively for many years. This classical area was revitalized in 1988, with Ian Macdonald's description of what are now known as the Macdonald polynomials. These are a two parameter basis for the space of symmetric functions, which specialize to many of the well-known one parameter and classical bases. Macdonald conjectured that when a certain normalization of these polynomials were expanded in terms of the classical Schur functions, the coefficients would always be polynomials in N[q,t]. He called these coefficients q, t-Kostka functions, and the conjecture became known as the Macdonald positivity conjecture. It was proved in 2001 by Mark Haiman. While attempting to prove the positivity conjecture, Garsia and Haiman conjectured the existence of a larger class of symmetric functions, satisfying certain properties and indexed by finite subsets of N x N (usually thought of as a collection of 1 x 1 squares in the first quadrant of the plane). Computer generated data strongly suggested the existence of these polynomials in general. In 2003, Jim Haglund proposed a purely combinatorial description of the Macdonald polynomials. This description, the generating function for a particular pair of statistics, was soon proved correct by Haiman, Haglund and Nick Loehr. In this dissertation, we show that the statistics of Haglund allow us to construct the polynomials of Garsia and Haiman for a particular class of diagrams; namely, skew shapes with no column of height greater than two. The proof of this fact involves a new and careful analysis of these statistic