Kostka multiplicity one for multipartitions

TL;DR

This paper characterizes when Kostka multiplicities for multipartitions are equal to one, relates them to permutation representations, and explores computational complexity aspects of these multiplicities in representation theory.

Contribution

It provides a set of conditions for Kostka multiplicities to be one, links these multiplicities to permutation representations, and analyzes the computational complexity of related problems.

Findings

Conditions for Kostka multiplicity one are established.

Polynomial-time algorithms for positivity and multiplicity-one questions are developed.

Determining nonzero multiplicities in certain representations is NP-complete.

Abstract

If $[\lambda(j)]$ is a multipartition of the positive integer $n$ (a sequence of partitions with total size $n$), and $\mu$ is a partition of $n$, we study the number $K_{[\lambda(j)]\mu}$ of sequences of semistandard Young tableaux of shape $[\lambda(j)]$ and total weight $\mu$. We show that the numbers $K_{[\lambda(j)] \mu}$ occur naturally as the multiplicities in certain permutation representations of wreath products. The main result is a set of conditions on $[\lambda(j)]$ and $\mu$ which are equivalent to $K_{[\lambda(j)] \mu} = 1$, generalizing a theorem of Berenshte\u{\i}n and Zelevinski\u{\i}. We also show that the questions of whether $K_{[\lambda(j)] \mu} > 0$ or $K_{[\lambda(j)] \mu} = 1$ can be answered in polynomial time, expanding on a result of Narayanan. Finally, we give an application to multiplicities in the degenerate Gel'fand-Graev representations of the finite general linear group, and we show that the problem of determining whether a given irreducible representation of the finite general linear group appears with nonzero multiplicity in a given degenerate Gel'fand-Graev representation, with their partition parameters as input, is $NP$-complete.