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 Berenshtĕın and Zelevinskĭı. Finally, we discuss some computational aspects of the problem, and we give an application to multiplicities in the degenerate Gel’fand-Graev representations of the finite general linear group. 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.