相关论文: Science Fiction and Macdonald's Polynomials

Parametric families in the centre ${\bf Z}({\bf C}[S_n])$ of the group algebra of the symmetric group are obtained by identifying the indeterminates in the generating function for Macdonald polynomials as commuting Jucys-Murphy elements.…

This is the first in a series of papers in which we describe explicit structural properties of spaces of diagonal rectangular harmonic polynomials in $k$ sets of $n$ variables, both as $GL_k$-modules and $S_n$-modules, as well as some of…

There are representations of the type-A Hecke algebra on spaces of polynomials in anti-commuting variables. Luque and the author [S\'em. Lothar. Combin. 66 (2012), Art. B66b, 68 pages, arXiv:1106.0875] constructed nonsymmetric Macdonald…

We derive combinatorial formulae for the modified Macdonald polynomial $H_{\lambda}(x;q,t)$ using coloured paths on a square lattice with quasi-cylindrical boundary conditions. The derivation is based on an integrable model associated to…

We construct type A partially-symmetric Macdonald polynomials $P_{(\lambda \mid \gamma)}$, where $\lambda \in \mathbb{Z}_{\geq 0}^{n-k}$ is a partition and $\gamma \in \mathbb{Z}_{\geq 0}^k$ is a composition. These are polynomials which are…

Given two Schubert classes $\sigma_{\lambda}$ and $\sigma_{\mu}$ in the quantum cohomology of a Grassmannian, we construct a partition $\nu$, depending on $\lambda$ and $\mu$, such that $\sigma_{\nu}$ appears with coefficient 1 in the…

A conformal partition function ${\cal P}_n^m(s)$, which arose in the theory of Diophantine equations supplemented with additional restrictions, is concerned with {\it self-dual symmetric polynomials} -- reciprocal ${\sf R}^{\{m\}}_ {S_n}$…

In a previous paper J.-G. Luque and the author (Sem. Loth. Combin. 2011) developed the theory of nonsymmetric Macdonald polynomials taking values in an irreducible module of the Hecke algebra of the symmetric group $\mathcal{S}_{N}$. The…

Singular nonsymmetric Macdonald polynomials are constructed by use of the representation theory of the Hecke algebras of the symmetric groups. These polynomials are labeled by quasistaircase partitions and are associated to special…

We describe a method, based on the theory of Macdonald-Koornwinder polynomials, for proving bounded Littlewood identities. Our approach provides an alternative to Macdonald's partial fraction technique and results in the first examples of…

We introduce a family of ideals $I_{n,\lambda,s}$ in $\mathbb{Q}[x_1,\dots,x_n]$ for $\lambda$ a partition of $k\leq n$ and an integer $s \geq \ell(\lambda)$. This family contains both the Tanisaki ideals $I_\lambda$ and the ideals…

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 $K$ be an infinite field of characteristic $p>0$ and let $\lambda, \mu$ be partitions, where $\mu$ has two parts. We find sufficient arithmetic conditions on $p, \lambda, \mu$ for the existence of a nonzero homomorphism $\Delta(\lambda)…

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 establish relations between Frobenius parts and between flat-dominant dimensions of algebras linked by Frobenius bimodules. This is motivated by the Nakayama conjecture and an approach of Martinez-Villa to the Auslander-Reiten conjecture…

We study the isospectral Hilbert scheme X_n, defined as the reduced fiber product of C^2n with the Hilbert scheme H_n of points in the plane, over the symmetric power S^n C^2. We prove that X_n is normal, Cohen-Macaulay, and Gorenstein, and…

We make progress towards understanding the structure of Littlewood-Richardson coefficients $g_{\lambda,\mu}^{\nu}$ for products of Jack symmetric functions. Building on recent results of the second author, we are able to prove new cases of…

The plethysm coefficient $p(\nu, \mu, \lambda)$ is the multiplicity of the Schur function $s_\lambda$ in the plethysm product $s_\nu \circ s_\mu$. In this paper we use Schur--Weyl duality between wreath products of symmetric groups and the…

Let $\nabla^\lambda$ denote the Schur functor labelled by the partition $\lambda$ and let $E$ be the natural representation of $\mathrm{SL}_2(\mathbb{C})$. We make a systematic study of when there is an isomorphism $\nabla^\lambda…

We propose to parametrize the configuration space of one-dimensional quantum systems of N identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates. It is shown that in this parametrization the…