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Character degrees in $2$-blocks of $\mathfrak{S}_n$ and $\mathfrak{A}_n$

Bim Gustavsson

TL;DR

The paper resolves when $2$-blocks of $\mathfrak{S}_n$ and $\mathfrak{A}_n$ contain a $p$-divisible ordinary irreducible character for odd primes $p$, proving this for all blocks once $n$ is large enough and providing a detailed construction via $2$-core/beta-set combinatorics. It employs the abacus model and core/weight analysis to produce partitions with prescribed $2$-core while ensuring $p^k$-divisibility of degrees, along with case analyses that cover the relevant range of $c$. For $\mathfrak{A}_n$, Clifford theory transfers the results from $\mathfrak{S}_n$ to the appropriate blocks, yielding a criterion for rational-valued $p$-divisible constituents and showing that, under a large bound such as $n\ge 8p^2+2p-4$, every $2$-block contains a $p$-divisible irreducible character. The findings extend previous work on principal blocks and nonprincipal blocks and provide a concrete framework for understanding $p$-divisibility in the representation theory of symmetric and alternating groups.

Abstract

Let $p$ be an odd prime. We show that for sufficiently large $n$, every $2$-block of $\mathfrak{S}_n$ and $\mathfrak{A}_n$ contains an ordinary irreducible character of degree divisible by $p$. For almost all $2$-blocks of $\mathfrak{A}_n$, we classify whether it contains a rational valued ordinary irreducible character of degree divisible by $p$.

Character degrees in $2$-blocks of $\mathfrak{S}_n$ and $\mathfrak{A}_n$

TL;DR

The paper resolves when -blocks of and contain a -divisible ordinary irreducible character for odd primes , proving this for all blocks once is large enough and providing a detailed construction via -core/beta-set combinatorics. It employs the abacus model and core/weight analysis to produce partitions with prescribed -core while ensuring -divisibility of degrees, along with case analyses that cover the relevant range of . For , Clifford theory transfers the results from to the appropriate blocks, yielding a criterion for rational-valued -divisible constituents and showing that, under a large bound such as , every -block contains a -divisible irreducible character. The findings extend previous work on principal blocks and nonprincipal blocks and provide a concrete framework for understanding -divisibility in the representation theory of symmetric and alternating groups.

Abstract

Let be an odd prime. We show that for sufficiently large , every -block of and contains an ordinary irreducible character of degree divisible by . For almost all -blocks of , we classify whether it contains a rational valued ordinary irreducible character of degree divisible by .
Paper Structure (6 sections, 9 theorems, 20 equations, 3 figures)

This paper contains 6 sections, 9 theorems, 20 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of the symmetric groups
  4. $\beta$-sets and cores
  5. Character degrees in $2$-blocks of $\mathfrak{S}_n$
  6. In the case of $\mathfrak{A}_n$

Key Result

Theorem 1.1

Let $p$ be an odd prime, $n\geq p$ be a natural number and let $r,a,k\in\mathbb{N}_0$ be such that $n=ap^k+r$, $a < p$ and $r<p^k$. Let $G\in\left\{\mathfrak{S}_n,\mathfrak{A}_n\right\}$ and let $B\in\operatorname{Bl}_2(G)$ be a $2$-block of $G$. If $G=\mathfrak{S}_n$ then suppose that $B=B_c$ and i then there exists some $p$-divisible $\chi\in \operatorname{Irr}(B)$. Moreover, if $G = \mathfrak{A

Figures (3)

  • Figure 1: (A) Partition $\lambda = (5,4,2^2,1)$ with rim hook $R_{1,3}(\lambda)$ highlighted. (B) Removal of $R_{1,3}(\lambda)$ from $\lambda$, resulting in partition $\mu:= (3,2^3,1)$. (C) Partition $\mu=(3,2^3,1)$ with rim hook $R_{3,1}(\mu)$ highlighted. (D) Removal of $R_{3,1}(\mu)$ from $\mu$ resulting in partition $\nu:= (3,2,1)$. In particular, we have that $C_4(\lambda)=\nu$.
  • Figure 2: The partitions in \ref{['fig: partition hook removal']} as seen on a $4$-abacus with $\beta$-sets of size 5. The highlighted bead $b$ in (A) (and (C) respectively) corresponds to the highlighted rim hooks in \ref{['fig: partition hook removal']} (A) (and (C) respectively). Moving $b$ up one step on its runner yields the abacus (B) (and (D) respectively), which corresponds to removing the highlighted rim hooks in \ref{['fig: partition hook removal']} (A) (and (C) respectively).
  • Figure 3: $2$-abacus configuration of various $\beta$-sets. (A) $\beta$-set $Y=\left\{9,7,5,3,1\right\}$ with associated partition $P(Y) = (5,4,3,2,1)$ of size $15$. (B) $\beta$-set $\left\{65,7,5,3,1\right\}$ corresponding to a partition of size $71$. (C) $\beta$-set $\left\{65,9,5,3,1\right\}$ corresponding to a partition of size $73$. (D) $\beta$-set $X=\left\{65,9,7,2,1\right\}$ with associated partition $P(X)=(61,6,5,2,1)$ of size $75$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Theorem 3.1
  • Example 3.2
  • proof : Proof of \ref{['thm: main part A']}
  • ...and 6 more