p-Saturations of Welter’s game and the irreducible representations of symmetric groups

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Abstract

We establish a relation between the Sprague–Grundy function [InlineEquation not available: see fulltext.] of p-saturations of Welter’s game and the degrees of the ordinary irreducible representations of symmetric groups. In these games, a position can be regarded as a partition λ. Let ρλ be the irreducible representation of the symmetric group Sym (| λ|) corresponding to λ. For every prime p, we show the following results: (1) sg (λ) ≤ | λ| with equality if and only if the degree of ρλ is prime to p; (2) the restriction of ρλ to Sym (sg (λ)) has an irreducible component with degree prime to p. Further, for every integer p greater than 1, we obtain an explicit formula for sg (λ).

Original languageEnglish
Pages (from-to)247-287
Number of pages41
JournalJournal of Algebraic Combinatorics
Volume48
Issue number2
DOIs
Publication statusPublished - 2018 Sep 1
Externally publishedYes

Keywords

  • Combinatorial game
  • Irreducible representation
  • Sprague–Grundy function
  • Symmetric group
  • p-Core

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics

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