On the number and sizes of double cosets of Sylow subgroups of the symmetric group

TL;DR

Let P_n be a Sylow p-subgroup of the symmetric group S_n. The paper analyzes the P_nackslash S_n / P_n double cosets, proving that for odd p and large n, most double cosets have maximal size |P_n|^2, equivalently P_n ∩ P_n^x = 1 for most x; it also shows that, when n is large, every possible double coset size between |P_n| and |P_n|^2 occurs (with a few small exceptions). It develops exact and asymptotic formulas in the abelian Sylow case (n=kp, 1≤k≤p−1) and provides closed-form counts for the size |P_n|p^k double cosets in the general setting, including a detailed treatment of the case |P_n|p for p=2. The approach combines orbit-stabiliser and character-theoretic methods with recent random-generation results (EG) to handle odd primes, and yields implications for the associated Hecke algebras L_k(P_n ackslash S_n / P_n) and their representation theory. The work advances understanding of double coset structures in S_n and informs related questions in modular representation theory and combinatorial enumeration.

Abstract

Let $P_n$ be a Sylow $p$-subgroup of the symmetric group $S_n$. We investigate the number and sizes of the $P_n\setminus S_n\ /\ P_n$ double cosets, showing that most double cosets have maximal size when $p$ is odd, or equivalently, that $P_n\cap P_n^x=1$ for most $x\in S_n$ when $n$ is large. We also find that all possible sizes of such double cosets occur, modulo a list of small exceptions.

Theorems & Definitions (38)

• Theorem 1.1
• Example 1.2
• Remark 2.1
• Lemma 2.2
• Remark 2.3
• Theorem 2.4
• Remark 2.5
• Lemma 2.6
• proof
• Theorem 2.7