Related papers: Supersymmetric polynomials and algebro-combinatori…
We prove a combinatorial formula for Macdonald cumulants which generalizes the celebrated formula of Haglund for Macdonald polynomials. We provide several applications of our formula. Firstly, it gives a new, constructive proof of a strong…
We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$ but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain $m$-symmetric…
Let $\mathbb D=G/K$ be a complex bounded symmetric domain of tube type in a Jordan algebra $V_{\mathbb C}$, and let $D=H/L =\mathbb D\cap V$ be its real form in a Jordan algebra $V\subset V_{\mathbb C}$. The analytic continuation of the…
In this paper, we introduce the notion of super-immanants for supermatrices over a supercommutative algebra. Using the super Schur-Weyl duality we show that the super immanants play a significant role in covariant tensor representations of…
We give an elementary proof of the development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions.
The quasisymmetic Macdonald polynomials $G_{\gamma}(X; q, t)$ were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials $P_{\lambda}(X; q, t)$ with the…
Kantor and Trishin described the algebra of polynomial invariants of the adjoint representation of the Lie supergalgebra $gl(m|n)$ and a related algebra $A_s$ of what they called pseudosymmetric polynomials over an algebraically closed…
The branching coefficients in the expansion of the elementary symmetric function multiplied by a symmetric Macdonald polynomial $P_\kappa(z)$ are known explicitly. These formulas generalise the known $r=1$ case of the Pieri-type formulas…
We investigate the zonal polynomials, a family of symmetric polynomials that appear in many mathematical contexts, such as multivariate statistics, differential geometry, representation theory, and combinatorics. We present two computer…
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,…
Generalized Macdonald polynomials (GMP) are eigenfunctions of specifically-deformed Ruijsenaars Hamiltonians and are built as triangular polylinear combinations of Macdonald polynomials. They are orthogonal with respect to a modified scalar…
Quantum analogues of the homogeneous spaces $\GL(n)/\SO(n)$ and $\GL(2n)/\Sp(2n)$ are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials…
This paper introduces new structural decompositions for almost symmetric numerical semigroups through the combinatorial lens of Young diagrams. To do that, we use the foundational correspondence between numerical sets and Young diagrams,…
Vector-valued Jack polynomials associated to the symmetric group ${\mathfrak S}_N$ are polynomials with multiplicities in an irreducible module of ${\mathfrak S}_N$ and which are simultaneous eigenfunctions of the Cherednik-Dunkl operators…
A one-parameter generalisation R_{\lambda}(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry,…
The rational Cherednik algebra $\HH$ is a certain algebra of differential-reflection operators attached to a complex reflection group $W$. Each irreducible representation $S^\lambda$ of $W$ corresponds to a standard module $M(\lambda)$ for…
Multivariate extensions of the Krawtchouk polynomials have been studied by numerous authors in recent decades by exploring new connections to probability, representation theory and quantum integrability. We develop a theory of multivariate…
We study the Young graph with edge multiplicities arising in a Pieri-type formula for Jack symmetric polynomials $P_\mu(x;a)$ with a parameter $a$. Starting with the empty diagram, we define recurrently the `dimensions' $\dim_a$ in the same…
In the intersection of the theories of nonsymmetric Jack polynomials in $N$ variables and representations of the symmetric groups $\mathcal{S}_{N}$ one finds the singular polynomials. For certain values of the parameter $\kappa$ there are…
We consider semisimple triangular operators acting in the symmetric component of the group algebra over the weight lattice of a root system. We present a determinantal formula for the eigenbasis of such triangular operators. This…