Kostka multiplicity one for multipartitions

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\uın and Zelevinski\uı. 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.

24 pages

References in corpus (1)

• Wreath Product Symmetric Functions