Generalized Hall-Littlewood polynomials (Macdonald spherical functions) and generalized Kostka-Foulkes polynomials ($q$-weight multiplicities) arise in many places in combinatorics, representation theory, geometry, and mathematical physics. This paper attempts to organize the different definitions of these objects and prove the fundamental combinatorial results from ``scratch'', in a presentation which, hopefully, will be accessible and useful for both the nonexpert and researchers currently working in this very active field. The combinatorics of the affine Hecke algebra plays a central role. The final section of this paper can be read independently of the rest of the paper. It presents, with proof, Lascoux and Schützenberger's positive formula for the Kostka-Foulkes poynomials in the type A case.
An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the Kostka-Foulkes polynomials is shown. We also use our algorithm to give a formula of $K_{λμ}(t)$ for $μ$ being hook-shaped.
We investigate the relationship between Kostka-Foulkes polynomials and certain symmetric functions that arise from Garsia and Haglund's study of the q,t-Catalan series.
We study the specialization of the type A nonsymmetric Macdonald polynomials at $t=0$ based on the combinatorial formula of Haglund, Haiman, and Loehr. We prove that this specialization expands nonnegatively into the fundamental slide polynomials, introduced by the author and Searles. Using this and weak dual equivalence, we prove combinatorially that this specialization is a positive graded sum of Demazure characters. We use stability results for fundamental slide polynomials to show that this specialization stabilizes and to show that the Demazure character coefficients give a refinement of …
We introduce a generalization of the classical Hall-Littlewood and Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove that these Kostka-Foulkes polynomials coincide with the natural generalization of Lusztig's $t$-analog of weight multiplicities, thereby extending a theorem of Kato. For $g$ an affine Kac-Moody algebra, we define $t$-analogs of string functions and use Cherednik's constant term identities to derive explicit product expressions for them.
We use Kashiwara-Nakashima's combinatorics of crystal graphs associated to the roots sytems $B_{n}$ and $D_{n}$ to extend the results of \QCITE{cite}{}{lec3} and \QCITE{cite}{}{Mor} by showing that Morris type recurrence formulas also exist for the orthogonal root systems. We derive from these formulas a statistic on Kashiwara-Nakashima's tableaux of types $B_{n},C_{n}$ and $D_{n}$ generalizing Lascoux-Sch\UNICODE{0xfc}tzenberger's charge and from which it is possible to compute the Kostka-Foulkes polynomials $K_{λ,μ}(q)$ with restrictive conditions on $(λ,μ)$ . This statistic is different fro…
We prove a conjecture of Lecouvey, which proposes a closed, positive combinatorial formula for symplectic Kostka-Foulkes polynomials, in the case of rows of arbitrary weight. To show this, we construct a new algorithm for computing cocyclage in terms of which the conjecture is described. Our algorithm is free of local constraints, which were the main obstacle in Lecouvey's original construction. In particular, we show that our model is governed by the situation in type A. This approach works for arbitrary weight and we expect it to lead to a proof of the conjecture in full generality.
Let $G$ be a semisimple algebraic group and $B$ a Borel subgroup. We consider generalisations of Lusztig's q-analogues of weight multiplicity, where the set of positive roots is replaced with the multiset of weights of a $B$-submodule of an arbitrary finite-dimensional $G$-module.
Macdonald defined two-parameter Kostka functions K_{λμ}(q,t) where λ, μare partitions. The main purpose of this paper is to extend his definition to include all compositions as indices. Following Macdonald, we conjecture that also these more general Kostka functions are polynomials in q and t^{1/2} with non-negative integers as coefficients. If q=0 then our Kostka functions are Kazhdan-Lusztig polynomials of a special type. Therefore, our positivity conjecture combines Macdonald positivity and Kazhdan-Lusztig positivity and hints towards a connection between Macdonald and Kazhdan-Lusztig theor…
For any triple $(i,a,μ)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $μ$, we define a quiver current $H^{(i,a)}_μ$ acting on the tensor power $Λ^Q$ of symmetric functions over the vertices of $Q$. These provide a quiver generalization of parabolic Garsia-Jing creation operators in the theory of Hall-Littlewood symmetric functions. For a triple $(\mathbf{i},\mathbf{a},μ(\bullet))$ of sequences of such data, we define the quiver Hall-Littlewood function $H^{\mathbf{i},\mathbf{a}}_{μ(\bullet)}$ as the result of acting on $1\inΛ^Q$ by the corres…
We find and prove a factorization formula for certain Macdonald Littlewood-Richardson coefficients $c_{λμ}^ν(q,t)$. Namely, we consider the case that the Kostka number $K_{μ, ν-λ}$ is $1$. This settles a particular case of a more general conjecture of Richard Stanely. This conjecture proposes that a factorization formula exists whenever the corresponding regular Littlewood-Richardson coefficient $c^ν_{λμ}$ is $1$.
We introduce Macdonald polynomials indexed by $n$-tuples of partitions and characterized by certain orthogonality and triangularity relations. We prove that they can be explicitly given as products of ordinary Macdonald polynomials depending on special alphabets. With this factorization in hand, we establish their most basic properties, such as explicit formulas for their norm-squared, evaluation and reproducing kernel. Moreover, we show that the $q,t$-Kostka coefficients associated to the multi-Macdonald polynomials are positive and correspond to $q,t$-analogs of the dimensions of the irreduc…
We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\,|\,a,-a,c,-c\,|\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric polynomials $m_{(1^{r})}(x)$. The entries of the matrix $\mathcal{C}$ enjoy a set of three term recursion relations, which can be regarded as a $(a,c,t)$-deformation of the one for the Catalan triangle or ballot numbers. Some transition matrices are studied associated with the type $(C_n,C_n)$ Macdonald polynomials $P^{(C_n,C_n)}_{(1^r)}(x\,|\,b;q,t)= P_{(1^r)}\b…
Introduced by Kawanaka in order to find the unipotent representations of finite groups of Lie type, generalized Gelfand--Graev characters have remained somewhat mysterious. Even in the case of the finite general linear groups, the combinatorics of their decompositions has not been worked out. This paper re-interprets Kawanaka's definition in type $A$ in a way that gives far more flexibility in computations. We use these alternate constructions to show how to obtain generalized Gelfand--Graev representations directly from the maximal unipotent subgroups. We also explicitly decompose the corresp…
Haglund's conjecture states that $\dfrac{\langle J_λ(q,q^k),s_μ\rangle}{(1-q)^{|λ|}} \in \mathbb{Z}_{\geq 0}[q]$ for all partitions $λ,μ$ and all non-negative integers $k$, where $J_λ$ is the integral form Macdonald symmetric function and $s_μ$ is the Schur function. This paper proves Haglund's conjecture in the cases when the pair $(λ,μ)$ satisfies $K_{λ,μ}=1$ or $K_{μ',λ'}=1$ where $K$ denotes the Kostka number. We also obtain some general results about the transition matrix between Macdonald symmetric functions and Schur functions.
The Gini index is a number that attempts to measure how equitably a resource is distributed throughout a population, and is commonly used in economics as a measurement of inequality of wealth or income. The Gini index is often defined as the area between the "Lorenz curve" of a distribution and the line of equality, normalized to be between zero and one. In this fashion, we will define a Gini index on the set of integer partitions and prove some combinatorial results related to it; culminating in the proof of an identity for the expected value of the Gini index. We will then discuss symmetric …
According to a program of Braverman, Kazhdan and Ngô Bao Châu, for a large class of split unramified reductive groups $G$ and representations $ρ$ of the dual group $\hat{G}$, the unramified local $L$-factor $L(s,π,ρ)$ can be expressed as the trace of $π(f_{ρ,s})$ for a suitable function $f_{ρ,s}$ with non-compact support whenever $\mathrm{Re}(s) \gg 0$. Such functions can be plugged into the trace formula to study certain sums of automorphic $L$-functions. It also fits into the conjectural framework of Schwartz spaces for reductive monoids due to Sakellaridis, who coined the term basic functio…
