2015/06/23 by James Janopaul‐Naylor, James Janopaul-Naylor, Janopaul-Naylor, James +2
Mathematics · #05A17 #05E10 #68Q17 #Advanced Topology and Set Theory #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Representation Theory (math.RT) #math.CO #math.RT #msc:05A17 #msc:05E10 #msc:68Q17
paper · pdf · doi:10.48550/arxiv.1506.07022
24 pages
openalex publication_date 2015/06/23 · arxiv created 2015/07/08 · arxiv updated 2015/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.