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Mullineux map: $d$-balanced partitions and $d$-runner matrices

Pavel Turek

TL;DR

This work develops a purely combinatorial, abacus-based framework to study the Mullineux map $m_e$ on structured partition families indexed by a coprime pair $(d,e)$. By introducing the $d$-runner matrix and the Abacus Mullineux Algorithm, it shows that for $e$-regular partitions with suitable $d$-balancedness, the image $m_e(\\lambda)'$ is uniquely determined as a partition that preserves the same $e$-core and $d$-runner data while switching to a $d$-shift balanced form; this yields a concrete description of $m_e(\\lambda)'$ in these cases. The paper also develops $d$-combined pairs and two algorithms, A1 and A2, to analyze extremal elements of sets $E_{\\mathcal{R}}(\\gamma)$ and proves that maximal/minimal elements correspond to $d$-shift skewed and $d$-skewed partitions, respectively, with $d=2$ cases enjoying stronger equivalences. Collectively, these results provide a robust combinatorial handle on the Mullineux map’s behavior in RoCK-like blocks and illuminate connections to decomposition matrices in symmetric group representation theory.

Abstract

Let $1<d<e$ be two coprime integers and let $m_e$ denote the Mullineux map, which for $e$ prime describes tensor products of the irreducible modules of symmetric groups with the sign in characteristic $e$. We prove that if $λ$ is an $e$-regular partition such that $d$ divides the arm length of any rim hook of $λ$ of size divisible by $e$, then $m_e(λ)'$ is a partition such that the arm length of any of its rim hooks of size divisible by $e$ is congruent to $-1$ modulo $d$. We introduce a new parameter for partitions called the $d$-runner matrix and show that if $λ$ is as above, then the $d$-runner matrices of $λ$ and $m_e(λ)'$ agree. This determines $m_e(λ)'$ uniquely. We approach the whole problem combinatorially and take advantage of a new Abacus Mullineux Algorithm introduced in this paper. We also establish equivalent descriptions of the above partitions which provide an alternative version of the main result about the Mullineux map, which becomes particularly strong when $d=2$.

Mullineux map: $d$-balanced partitions and $d$-runner matrices

TL;DR

This work develops a purely combinatorial, abacus-based framework to study the Mullineux map on structured partition families indexed by a coprime pair . By introducing the -runner matrix and the Abacus Mullineux Algorithm, it shows that for -regular partitions with suitable -balancedness, the image is uniquely determined as a partition that preserves the same -core and -runner data while switching to a -shift balanced form; this yields a concrete description of in these cases. The paper also develops -combined pairs and two algorithms, A1 and A2, to analyze extremal elements of sets and proves that maximal/minimal elements correspond to -shift skewed and -skewed partitions, respectively, with cases enjoying stronger equivalences. Collectively, these results provide a robust combinatorial handle on the Mullineux map’s behavior in RoCK-like blocks and illuminate connections to decomposition matrices in symmetric group representation theory.

Abstract

Let be two coprime integers and let denote the Mullineux map, which for prime describes tensor products of the irreducible modules of symmetric groups with the sign in characteristic . We prove that if is an -regular partition such that divides the arm length of any rim hook of of size divisible by , then is a partition such that the arm length of any of its rim hooks of size divisible by is congruent to modulo . We introduce a new parameter for partitions called the -runner matrix and show that if is as above, then the -runner matrices of and agree. This determines uniquely. We approach the whole problem combinatorially and take advantage of a new Abacus Mullineux Algorithm introduced in this paper. We also establish equivalent descriptions of the above partitions which provide an alternative version of the main result about the Mullineux map, which becomes particularly strong when .

Paper Structure

This paper contains 6 sections, 40 theorems, 30 equations, 28 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abacus Mullineux Algorithm
  4. Balanced partitions
  5. $d$-combined pairs
  6. Skewed partitions

Key Result

Theorem 1.3

Let $1<d<e$ be coprime integers and $\lambda$ be a $d$-balanced partition. If its $d$-runner matrix $\mathcal{R}_d(\lambda)$ contains at least one $0$ in its first row, then $m_e(\lambda)'$ is a $d$-shift balanced partition with the same $d$-runner matrix and $e$-core as $\lambda$. Moreover, this de

Figures (28)

  • Figure 1: The hierarchy of the introduced partitions, where $m_e'$ is the composition of the Mullineux map $m_e$ followed by the conjugation, that is $m_e'(\lambda) = m_e(\lambda)'$. The dashed arrows show the effect of $m_e'$ if we apply it to a $d$-balanced partition. If $d=2$, then both implications become equivalences, the dashed arrows become solid and in turn we obtain \ref{['co:d=2']}.
  • Figure 2: The Hasse diagram of partitions in $E_{\mathcal{R}}(\gamma)$ with $\gamma=(1^2)$ and $\mathcal{R}=(0010000001)$ from \ref{['ex:summary']}. Their $e$-divisible hooks are denoted with dashed lines. For each row, we wrote numbers $x,y$ in front of it if the row contributes by $1$ to the $(x,y)$-entry of the $d$-runner matrix of the corresponding partition. Thus we see that all displayed partitions have $d$-runner matrix equal to $\mathcal{R}$. One easily checks that all displayed partitions have $e$-core equal to $\gamma$.
  • Figure 3: These are two copies of the Young diagram of partition $\lambda = (6,4,2)$. The highlighted boxes in the left diagram form the rim of $\lambda$, while in the right diagram they form the $5$-hook $R_{2,1}$ with top $2$, bottom $3$, arm length $3$ and leg length $1$. Note that $\lambda$ has one more $5$-hook, namely, $R_{1,3}$. It has top $1$, bottom $2$, arm length $3$ and leg length $1$.
  • Figure 4: Ignoring the dashed lines, the diagram in the middle is $\lambda=(6,4,2)$ displayed on the James abacus. It is created from $\lambda$ by extending the left and upper borderlines of $Y(\lambda)$, placing beads on every vertical unit interval of its inner border and unbending the inner border into a straight line. We obtain a $\beta$-set of $\lambda$ by identifying the James abacus with the real line such that its positions lie at integers. If we identify the rightmost bead with $5$ as in the figure, we get the canonical $\beta$-set $\left\lbrace 5,2,-1,-4,-5,-6, \dots \right\rbrace$. The bottom left picture is a James abacus display of $\lambda$ with $5$ runners, obtained by dividing the middle picture into intervals of length $5$ (separated by the dashed lines) and placing them on top of each other. We obtain the canonical $\beta$-set by labelling the last displayed row as row $1$, so the bead on this row, which lies on runner $0$, is identified with $1*e + 0 =5$. The identifications of all shown positions in the canonical $\beta$-set with integers are displayed in the bottom right diagram. In the rest of the figures we will omit the dots indicating that all runners are infinite.
  • Figure 5: The $5$-core of partition $(6,4,2)$ is $(1^2)$. On the left-hand side, this is found by removing two $5$-hooks, while on the right-hand side this is done by sliding the corresponding beads up by one place on their runners. The $e$-weight of $(6,4,2)$ is $2$.
  • ...and 23 more figures

Theorems & Definitions (96)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Proposition 1.7
  • proof : Proof of the 'moreover' part of \ref{['th:max to min']}
  • Corollary 1.8
  • Example 1.9
  • ...and 86 more