Related papers: Compact formulas for Macdonald polynomials and qua…
This paper uses the theory of dual equivalence graphs to give explicit Schur expansions for several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified…
The aim of this work is to extend the study of super-coinvariant polynomials, to the case of the generalized symmetric group $G_{n,m}$, defined as the wreath product $C_m\wr\S_n$ of the symmetric group by the cyclic group. We define a…
Sahi, Stokman, and Venkateswaran have constructed, for each positive integer $n$, a family of Laurent polynomials depending on parameters $q$ and $k$ (in addition to $\lfloor n/2\rfloor$ "metaplectic parameters"), such that the $n=1$ case…
Extending Eilenberg-Mac Lane's methods, higher level cohomologies for commutative monoids are introduced and studied. Relationships with pre-existing theories (Leech, Grillet, ...) are stated. The paper includes a cohomological…
We give an explicit formula for an operator that sends a wreath Macdonald polynomial to the delta function at a character associated to its partition. This allows us to prove many new results for wreath Macdonald polynomials, especially…
A new method of composition orthogonality is introduced. It is applied to generate new sequences of orthogonal polynomials and functions. In particular, classical orthogonal polynomials are interpreted in the sense of composition…
This paper is a supplement to Guo-Ram arXiv:2104.02942, containing examples, remarks and additional material that could be useful to researchers working with Type $GL_n$ Macdonald polynomials. In the course of our comparison of the alcove…
We establish a connection between a specialization of the nonsymmetric Macdonald polynomials and the Demazure characters of the corresponding affine Kac-Moody algebra. This allows us to obtain a representation-theoretical interpretation of…
We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables $x_1,x_2,...$ and of two parameters $q,t$ are their eigenfunctions. These operators are defined as limits at…
Let $K(q,t)= \|K_{\la\mu}(q,t)\|_{\la,\mu}$ be the Macdonald q,t-Kostka matrix and $K(t)=K(0,t)$ be the matrix of the Kostka-Foulkes polynomials K_{\la\mu}(t). In this paper we present a new proof of the polynomiality of the q,t-Kostka…
The recently discovered general formulas for perturbative correlators in basic matrix models can be interpreted as the Schur-preservation property of Gaussian measures. Then substitution of Schur by, say, Macdonald polynomials, defines a…
The Cherednik-Orr conjecture expresses the $t\to\infty$ limit of the nonsymmetric Macdonald polynomials in terms of the PBW twisted characters of the affine level one Demazure modules. We prove this conjecture in several special cases.
In this work we study the relationship between several combinatorial formulas for type $A$ spherical Whittaker functions. These are spherical functions on $p$-adic groups, which arise in the theory of automorphic forms. They depend on a…
The Jack polynomials with prescribed symmetry are obtained from the nonsymmetric polynomials via the operations of symmetrization, antisymmetrization and normalization. After dividing out the corresponding antisymmetric polynomial of…
We generalize and prove the recursive relation \[ S_{\lambda}(V) = \sum_{L\subseteq V\text{ line}} S_{\lambda}(V \mathbin{/\mkern-5mu/} L) \] conjectured by I. G. Macdonald for his "7th variation" of the Schur functions. This variation is a…
This paper is a continuation of our papers [EK1, EK2]. In [EK2] we showed that for the root system A_n-1 one can obtain Macdonald's polynomials - a new interesting class of symmetric functions recently defined by I. Macdonald {M1] - as…
In this article we give a combinatorial formula for a certain class of Koornwinder polynomials, also known as Macdonald polynomials of type $\tilde{C}$. In particular, we give a combinatorial formula for the Koornwinder polynomials…
Jack polynomials generalize several classical families of symmetric polynomials, including Schur polynomials, and are further generalized by Macdonald polynomials. In 1989, Richard Stanley conjectured that if the Littlewood-Richardson…
We present formulas for the homogenous multivariate resultant as a quotient of two determinants. They extend classical Macaulay formulas, and involve matrices of considerably smaller size, whose non zero entries include coefficients of the…
Koornwinder polynomials are $q$-orthogonal polynomials equipped with extra five parameters and the $B C_n$-type Weyl group symmetry, which were introduced by Koornwinder (1992) as multivariate analogue of Askey-Wilson polynomials. They are…