Covering numbers for characters of symmetric groups

TL;DR

The paper determines sharp covering thresholds for irreducible characters of the symmetric group $S_n$ (with $n>4$). By formulating covering numbers $d(\chi)$ and $e(\chi)$ via the sets of irreducible constituents of powers of characters, it proves that for all nonlinear $\chi$, the entire Irr$(S_n)$ is obtained from powers $\chi^k$ precisely when $k\ge n-1$. The proof combines semigroup-type lemmas for irreducible constituents, induced characters $\theta_{n,u}$, and a partition-theoretic analysis distinguishing rectangular and non-rectangular partitions. Consequently, the global maxima satisfy $d(S_n) \le e(S_n) = n-1$ with the lower bound $d(S_n) \ge n-1$, establishing a tight, explicit bound. This yields a clear, explicit criterion for covering Irr$(S_n)$ via a single nonlinear character’s powers, linking character theory to partition combinatorics and providing explicit thresholds for symmetric groups.

Abstract

If $n>4$ and $c(θ)$ denotes the set of irreducible constituents of a character $θ$, then $c(χ^k)={\rm Irr}(S_n)$ for all nonlinear $χ\in {\rm Irr}(S_n)$ if and only if $k\geq n-1$.

Theorems & Definitions (21)

• Theorem 1.1
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5