Kostka multiplicity one for multipartitions

If $[λ(j)]$ is a multipartition of the positive integer $n$ (a sequence of partitions with total size $n$), and $μ$ is a partition of $n$, we study the number $K_{[λ(j)]μ}$ of sequences of semistandard Young tableaux of shape $[λ(j)]$ and total weight $μ$. We show that the numbers $K_{[λ(j)] μ}$ occur naturally as the multiplicities in certain permutation representations of wreath products. The main result is a set of conditions on $[λ(j)]$ and $μ$ which are equivalent to $K_{[λ(j)] μ} = 1$, generalizing a theorem of Berenshte\uın and Zelevinski\uı. We also show that the questions of whether $K_{[λ(j)] μ} > 0$ or $K_{[λ(j)] μ} = 1$ 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.

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