Related papers: Wreath Macdonald polynomials, a survey

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath…

We show that the wreath Macdonald polynomials for $\mathbb{Z}/\ell\mathbb{Z}\wr\Sigma_n$, when naturally viewed as elements in the vertex representation of the quantum toroidal algebra…

We construct a novel family of difference-permutation operators and prove that they are diagonalized by the wreath Macdonald $P$-polynomials; the eigenvalues are written in terms of elementary symmetric polynomials of arbitrary degree. Our…

We study the $K$-theoretic enumerative geometry of cyclic Nakajima quiver varieties, with particular focus on $\text{Hilb}^{m}([\mathbb{C}^{2}/\mathbb{Z}_{l}])$, the equivariant Hilbert scheme of points on $\mathbb{C}^2$. The direct sum…

Using the theory of Macdonald, Gordon showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg associated to the symmetric group $\mathfrak{S}_n$ are given by…

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…

The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…

Let W be the complex reflection group G(e,1,n). In the author's previous paper, Hall-Littlewood functions associated to W were introduced. In the special case where W is a Weyl group of type B_n, they are closely related to Green…

We describe a way to study and compute Pieri rules for wreath Macdonald polynomials using the quantum toroidal algebra. The Macdonald pairing can be naturally generalized to the wreath setting, but the wreath Macdonald polynomials are no…

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…

This paper defines and investigates nonsymmetric Macdonald polynomials with values in an irreducible module of the Hecke algebra of type $A_{N-1}$. These polynomials appear as simultaneous eigenfunctions of Cherednik operators. Several…

The super-Macdonald polynomials, introduced by Sergeev and Veselov, generalise the Macdonald polynomials to (arbitrary numbers of) two kinds of variables, and they are eigenfunctions of the deformed Macdonald-Ruijsenaars operators…

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on…

We formulate a precise conjecture relating integral form partially-symmetric Macdonald polynomials and the parabolic flag Hilbert schemes of Carlsson, Gorsky, and Mellit. This extends, in an explicit fashion, Haiman's realization of…

Intermediate Macdonald polynomials for an affine root system $S$ with fixed origin and finite Weyl group $W_0$ are orthogonal polynomials invariant under a parabolic subgroup $W_J\le W_0$. The extreme cases of $W_J=1$ and $W_J=W_0$…

From the irreducible decompositions' point of view, the structure of the cyclic $GL_n$-module generated by the $\alpha$-determinant degenerates when $\alpha=\pm \frac1k (1\leq k\leq n-1)$. In this paper, we show that $-\frac1k$-determinant…

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal…

We introduce a new family of operators as multi-parameter deformation of the one-row Macdonald polynomials. The matrix coefficients of these operators acting on the space of symmetric functions with rational coefficients in two parameters…

In this paper we present a formula for Macdonald's polynomials for the root system A(n-1) which arises from the representation theory of quantum sl(n). This formula expresses Macdonald's polynomials via (weighted) traces of intertwining…

In this paper, we introduce higher rank generalizations of Macdonald polynomials. The higher rank non-symmetric Macdonald polynomials are Laurent polynomials in several sets of variables which form weight bases for higher rank polynomial…