Related papers: Pieri rules for the Jack polynomials in superspace…

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert…

The Pieri rule expresses the product of a Schur function and a single row Schur function in terms of Schur functions. We extend the classical Pieri rule by expressing the product of a skew Schur function and a single row Schur function in…

The determinantal form of the partition function of the 6-vertex model with domain wall boundary conditions was given by Izergin. It is known that for a special value of the crossing parameter the partition function reduces to a Schur…

In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend…

The Clifford-Hermite and the Clifford-Gegenbauer polynomials of standard Clifford analysis are generalized to the new framework of Clifford analysis in superspace in a merely symbolic way. This means that one does not a priori need an…

We explore the codimension one strata in the degree-one cohomology jumping loci of a finitely generated group, through the prism of the multivariable Alexander polynomial. As an application, we give new criteria that must be satisfied by…

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace.…

Macdonald superpolynomials provide a remarkably rich generalization of the usual Macdonald polynomials. The starting point of this work is the observation of a previously unnoticed stability property of the Macdonald superpolynomials when…

Answering a question of Frank Calegari, we extend some of our earlier results on dimension of fixed point spaces of elements in irreducible linear groups. We consider characteristic polynomials rather than just fixed spaces.

A recent novel derivation of the representation of Virasoro singular vectors in terms of Jack polynomials is extended to the supersymmetric case. The resulting expression of a generic super-Virasoro singular vector is given in terms of a…

We prove that Euler supercharacters for orthosymplectic Lie superalgebras can be obtained as a certain specialization of super Jacobi polynomials. A new version of Weyl type formula for super Schur functions and specialized super Jacobi…

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…

In their 1987 paper Kra\'skiewicz and Pragacz defined certain modules, which we call KP modules, over the upper triangular Lie algebra whose characters are Schubert polynomials. In a previous work the author showed that the tensor product…

We develop several notions of multiplicity for linear factors of multivariable polynomials over different arithmetics (hyperfields). The key example is multiplicities over the hyperfield of signs, which encapsulates the arithmetic of…

We give an explicit formula for the power-sum expansion of Jack polynomials. We deduce it from a more general formula, which we provide here, that interprets Jack characters in terms of bipartite maps. We prove Lassalle's conjecture from…

Let $k_i\ (i=1,2,\ldots,t)$ be natural numbers with $k_1>k_2>\cdots>k_t>0$, $k_1\geq 2$ and $t<k_1.$ Given real numbers $\alpha_{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape…

In the present paper, we treat multidimensional polynomial Euler products with complex coefficients on ${\mathbb{R}}^d$. We give necessary and sufficient conditions for the multidimensional polynomial Euler products to generate infinitely…

We analyze conditions under which a projection from the vector-valued Jack or Macdonald polynomials to scalar polynomials has useful properties, especially commuting with the actions of the symmetric group or Hecke algebra, respectively,…

We propose a new formulation of Hall polynomials in terms of honeycombs, which were previously introduced in the context of the Littlewood--Richardson rule. We prove a Pieri rule and associativity for our honeycomb formula, thus showing…

An overview of the basic results on Macdonald(-Koornwinder) polynomials and double affine Hecke algebras is given. We develop the theory in such a way that it naturally encompasses all known cases. Among the basic properties of the…