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Generating functions for fixed points of the Mullineux map

David J. Hemmer

TL;DR

This work extends the enumeration of Mullineux-fixed points from the classical odd-prime setting to arbitrary $e$, providing explicit generating functions that count fixed points by size and by $e$-weight. It derives closed-form one-variable generating functions $MF_e(q)$ in terms of the Ramanujan chi-function and a two-variable generating function $MF_e(x,q)$ to capture weight structure, with parity-dependent forms for even and odd $e$. By decomposing fixed points into contributions from self-conjugate $e$-cores and weighted partitions, the paper expresses counts as products of known series $f_e$ and $g_e$ times $ ext{sc}_e(n-ew)$, and it formalizes these results via a bijective framework that leverages $t$-bar cores for even $e$. The methods connect deep combinatorial constructions (Mullineux symbol, $e$-cores/weights, and $t$-bar cores) to explicit generating functions, enabling precise fixed-point counts across all $e$ and linking to representations of symmetric and alternating groups in prime characteristics.

Abstract

Mullineux defined an involution on the set of $e$-regular partitions of $n$. When $e=p$ is prime, these partitions label irreducible symmetric group modules in characteristic $p$. Mullineux's conjecture, since proven, was that this ``Mullineux map" described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group $A_n$ in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux's map when $e=p$ is an odd prime (providing evidence in support of Mullineux's conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a $p$-block of weight $w$. We extend both results to arbitrary $e$, and determine the corresponding generating functions. When $e$ is odd but not prime the extension is immediate, while $e$ even requires additional work and the results, which are different, have not appeared in the literature.

Generating functions for fixed points of the Mullineux map

TL;DR

This work extends the enumeration of Mullineux-fixed points from the classical odd-prime setting to arbitrary , providing explicit generating functions that count fixed points by size and by -weight. It derives closed-form one-variable generating functions in terms of the Ramanujan chi-function and a two-variable generating function to capture weight structure, with parity-dependent forms for even and odd . By decomposing fixed points into contributions from self-conjugate -cores and weighted partitions, the paper expresses counts as products of known series and times , and it formalizes these results via a bijective framework that leverages -bar cores for even . The methods connect deep combinatorial constructions (Mullineux symbol, -cores/weights, and -bar cores) to explicit generating functions, enabling precise fixed-point counts across all and linking to representations of symmetric and alternating groups in prime characteristics.

Abstract

Mullineux defined an involution on the set of -regular partitions of . When is prime, these partitions label irreducible symmetric group modules in characteristic . Mullineux's conjecture, since proven, was that this ``Mullineux map" described the effect on the labels of taking the tensor product with the one-dimensional signature representation. Counting irreducible modules fixed by this tensor product is related to counting irreducible modules for the alternating group in prime characteristic. In 1991, Andrews and Olsson worked out the generating function counting fixed points of Mullineux's map when is an odd prime (providing evidence in support of Mullineux's conjecture). In 1998, Bessenrodt and Olsson counted the fixed points in a -block of weight . We extend both results to arbitrary , and determine the corresponding generating functions. When is odd but not prime the extension is immediate, while even requires additional work and the results, which are different, have not appeared in the literature.
Paper Structure (7 sections, 15 theorems, 24 equations, 4 figures, 1 table)

This paper contains 7 sections, 15 theorems, 24 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Mullineux Map, $e$-cores and $e$-weights
  3. $e$-cores and $e$-weights
  4. Main Results
  5. Mullineux fixed points of a given $e$-weight
  6. Proof of the main results
  7. t-bar cores

Key Result

Theorem 1.1

AndrewsOlssonPartitionIdentities Let $p>2$ be prime. The number of fixed points of $m_p$ is the number of partitions of $n$ with distinct odd parts, none of which are divisible by $p$.

Figures (4)

  • Figure 2.1: The $5$-rim of $\lambda=(7,7,7,4,4,1,1)$.
  • Figure 2.2: Calculating the Mullineux symbox $G_5(7,7,7,4,4,1,1)$
  • Figure 4.1: Abacus display for $\lambda=(23, 21, 17, 13, 11, 9, 7)$ and $t=6$.
  • Figure 4.2: Abacus display for $\tilde{\lambda}_{(6)}=(9,5,3)$

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Remark 2.6
  • Theorem 3.1
  • Corollary 3.2
  • ...and 15 more