Transition matrices between Young's natural and seminormal representations

2020/12/07 by Sam Armon, Armon, Sam, Tom Halverson +1
Mathematics · #05E10 #05E18 #0C30 #20C08 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2012.03828

openalex publication_date 2020/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive a formula for the entries in the change-of-basis matrix between Young's seminormal and natural representations of the symmetric group. These entries are determined as sums over weighted paths in the weak Bruhat graph on standard tableaux, and we show that they can be computed recursively as the weighted sum of at most two previously-computed entries in the matrix. We generalize our results to work for affine Hecke algebras, Ariki-Koike algebras, Iwahori-Hecke algebras, and complex reflection groups given by the wreath product of a finite cyclic group with the symmetric group.

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