On the irreducible character degrees of symmetric groups and their multiplicities
David A. Craven
TL;DR
This work probes the largest irreducible character degrees and their multiplicities in symmetric groups and the unipotent degrees in $GL_n(q)$. By combining computational data (notably via Magma) with theoretical bounds, it establishes explicit growth thresholds for the maximal multiplicity $m(n)$ in $S_n$ (e.g., $m(n)\ge 4$ for $n\ge 11$, $m(n)\ge 6$ for $n\ge 17$, and $m(n)\ge 8$ for $n\ge 21$) and constructs eight equal-degree characters for large $n$, while analyzing partition clusters and extending analogies to $GL_n(q)$ via hook-length patterns and Lusztig's $a$-function. The paper also develops conjectures bounding group order by multiplicity ratios and investigates the distribution of the largest degrees in $S_n$, including asymptotic behavior $b_m(n)/b_1(n)\to 1$ for fixed $m$, supported by extensive data up to $n\approx150$. A Magma-based computational framework underpins these results, and the work raises open questions about algorithmic approaches to largest-degree character computations and multiplicities in the unipotent setting.
Abstract
We consider problems concerning the largest degrees of irreducible characters of symmetric groups, and the multiplicities of character degrees of symmetric groups. Using evidence from computer experiments, we posit several new conjectures or extensions of previous conjectures, and prove a number of results. One of these is that, if $n\geq 21$, then there are at least eight irreducible characters of $S_n$, all of which have the same degree, and which have irreducible restriction to $A_n$. We explore similar questions about unipotent degrees of $\mathrm{GL}_n(q)$. We also make some remarks about how the experiments here shed light on posited algorithms for finding the largest irreducible character degree of $S_n$.