Proof of a conjecture of Green and Liebeck on codes in symmetric groups
Teng Fang, Jinbao Li
TL;DR
The paper resolves Green and Liebeck's conjecture on subgroup codes in the symmetric group $S_n$ by an explicit representation-theoretic criterion. It shows that, for $n>2k$, a conjugacy class $X=x^{S_n}$ and a Young subgroup $Y_k=S_k\times S_{n-k}$ satisfy $r S_n=X\boldsymbol{\cdot}Y_k$ for some $r>0$ if and only if the cycle structure of $x$ has a unique pattern of powers of two up to $2^j$ and all other cycles are long, precisely characterized via the Frobenius formula and the irreducible characters of $S_n$. The argument hinges on analyzing which Specht modules occur in the left ideal $(\mathbb{C}S_n)\overline{Y_k}$ and how conjugacy-sums act on irreducibles, culminating in an inductive proof on $k$. The results bridge combinatorial cycle structure with algebraic decompositions, contributing to the theory of tilings and codes in nonabelian groups and suggesting further questions about maximal and Young subgroups as codes.
Abstract
Let $A$ and $B$ be subsets of a finite group $G$ and $r$ a positive integer. If for every $g\in G$, there are precisely $r$ pairs $(a,b)\in A\times B$ such that $g=ab$, then $B$ is called a code in $G$ with respect to $A$ and we write $r G=A\boldsymbol{\cdot}B$. If in addition $B$ is a subgroup of $G$, then we say that $B$ is a subgroup code in $G$. In this paper we resolve a conjecture by Green and Liebeck \cite[Conjecture 2.3]{Green20} on certain subgroup codes in the symmetric group $S_n$. Let $n>2k$ and let $j$ be such that $2^j\leqslant k<2^{j+1}$. Suppose that $X$ is a conjugacy class in $S_n$ containing $x$, and $Y_k$ is the subgroup $S_k\times S_{n-k}$ of $S_n$, where the factor $S_k$ permutes the subset $\{1,\ldots,k\}$ and the factor $S_{n-k}$ permutes the subset $\{k+1,\ldots,n\}$. We prove that $r S_n=X\boldsymbol{\cdot}Y_k$ for some positive integer $r$ if and only if the cycle type of $x$ has exactly one cycle of length $2^i$ for $0\leqslant i\leqslant j$ and all other cycles have length at least $k+1$. We also propose several problems concerning the existence of certain subgroup codes in a finite group $G$ with respect to a conjugation-closed subset in $G$.