Related papers: Double Kostka polynomials and Hall bimodule
We establish a ring isomorphism between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions (i.e., the ring of symmetric polynomials in two sets of countably many variables, invariant under the…
Littlewood-Richardson (LR) coefficients and Kostka Numbers appear in representation theory and combinatorics related to $GL_n$. It is known that Kostka numbers can be represented as special Littlewood-Rischardson coefficient. In this paper,…
An iterative formula for the Kostka-Foulkes polynomials is given using the vertex operator realization of the Hall-Littlewood polynomials. The operational formula can handle large Kostka-Foulkes polynomials, and a stability property for the…
The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say $k$) of $n$ variables, both as $GL_k$-modules and $S_n$-modules. We construct explicit such modules…
Starting from Jacobi-Trudi's type determinental expressions for the Schur functions corresponding to types $B,C$ and $D,$ we define a natural $q$-analogue of the multiplicity $[V(\lambda):M(\mu)]$ when $M(\mu)$ is a tensor product of row or…
We introduce the quantum multi-Schur functions, quantum factorial Schur functions and quantum Macdonald polynomials. We prove that for restricted vexillary permutations the quantum double Schubert polynomial coincides with some quantum…
Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_\lambda$ arise as partition functions of certain path configurations in the $\mathfrak{sl}_2$ higher spin six vertex models. They are multiparameter generalizations of…
Integral representations are obtained of positive additive functionals on finite products of the space of continuous functions (or of bounded Borel functions) on a compact Hausdorff space. These are shown to yield characterizations of the…
We provide a combinatorial formula for the structure constants of spin Hall--Littlewood functions. This is achieved by representing these functions and the structure constants as the partition function of a lattice model and applying the…
We show that some of the main structural constants for symmetric functions (Littlewood-Richardson coefficients, Kronecker coefficients, plethysm coefficients, and the Kostka--Foulkes polynomials) share symmetries related to the operations…
The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka…
We study a class of representations called ``calibrated representations'' of the degenerate double affine Hecke algebra and those of the rational Cherednik algebra of type ${\mathrm{GL}}_n$. We give a realization of calibrated irreducible…
The fusion rule gives the dimensions of spaces of conformal blocks in the WZW theory. We prove a dimension formula similar to the fusion rulefor spaces of coinvariants of affine Lie algebras g^. An equivalence of filtered spaces is…
We give a new description of the Pieri rule for k-Schur functions using the Bruhat order on the affine type-A Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of…
It is known that the usual Schur $S$- and $P$-polynomials can be described via the Gysin homomorphisms for flag bundles in the ordinary cohomology theory. Recently, P. Pragacz generalized these Gysin formulas to the Hall-Littlewood…
We provide a combinatorial description of the coefficients appearing in the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions. We also give a Littlewood-Richardson rule for Hall-Littlewood polynomials. For…
In our joint paper with W. Fulton (math.AG/9804041) we prove a formula for the cohomology class of a quiver variety. This formula is general enough to give new expressions for all known types of Schubert polynomials. In the present paper we…
Every double coset in $\text{GL}_m(k[[z]])\backslash \text{GL}_m(k((z)))/\text{GL}_m(k((z^2)))$ is uniquely represented by a block diagonal matrix with diagonal blocks in $\{1,z, \begin{pmatrix} 1& z\\ 0 &z^i \end{pmatrix} (i>1)\}$ if…
We introduce a spin analogue of Kostka polynomials and show that these polynomials enjoy favorable properties parallel to the Kostka polynomials. Further connections of spin Kostka polynomials with representation theory are established.
The paper studies quadratic and Koszul duality for modules over positively graded categories. Typical examples are modules over a path algebra, which is graded by the path length, of a not necessarily finite quiver with relations. We…