On Minimal Polynomials of Elements in Symmetric and Alternating Groups

2024/12/30 by S, Velmurugan · 1 citation
#05E05 #05E10 #20C15 #20C30 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2412.20894

Abstract

Let (ρ, V) be an irreducible representation of the symmetric group S_n (or the alternating group A_n), and let g be a permutation on n letters with each of its cycle lengths divides the length of its largest cycle. We describe completely the minimal polynomial of ρ(g), showing that, in most cases, it equals x^o(g) - 1 , with a few explicit exceptions. As a by-product, we obtain a new proof (using only combinatorics and representation theory) of a theorem of Swanson that gives a necessary and sufficient condition for the existence of a standard Young tableau of a given shape and major index r mod n, for all r. Thereby, we give a new proof of a celebrated result of Klyachko on Lie elements in a tensor algebra, and of a conjecture of Sundaram on the existence of an invariant vector for n-cycles. We also show that for elements g in S_n or A_n of even order, in most cases, ρ(g) has eigenvalue -1, with a few explicit exceptions.

Cited by

• On the Positivity of Dihedral Branching Coefficients of the Symmetric and Alternating Groups

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