Kostka-Shoji polynomials and Lusztig's convolution diagram

TL;DR

This paper introduces an r-variable extension of Kostka-Shoji polynomials with positive coefficients, connecting them to graded multiplicities in line bundle sections over Lusztig's convolution diagram for cyclic quivers.

Contribution

It defines a new r-variable version of Kostka-Shoji polynomials with positive coefficients, linking them to geometric multiplicities in Lusztig's convolution diagram.

Findings

The new polynomials have positive integral coefficients.

They encode graded multiplicities in geometric spaces.

The approach generalizes classical Kostka-Shoji polynomials.

Abstract

We propose an $r$-variable version of Kostka-Shoji polynomials $K^-_{\lambda\mu}$ for $r$-multipartitions $\lambda,\mu$. Our version has positive integral coefficients and encodes the graded multiplicities in the space of global sections of a line bundle over Lusztig's iterated convolution diagram for the cyclic quiver $\tilde{A}_{r-1}$.