MarriageTheoremのこと

_ プレプリント確認状況：arXiv:math 8月10日分まで、arXiv:quant-ph 5月31日分まで、IACR ePrint：2012/056まで

_ ↑上のarXivチェックでThe Probability that a Pair of Elements of a Finite Group are Conjugate（arXiv:1108.1784）という論文を見つけた。曰く、

Let \(G\) be a finite group, and let \(\kappa(G)\) be the probability that elements \(g\), \(h\in G\) are conjugate, when \(g\) and \(h\) are chosen independently and uniformly at random. The paper classifies those groups \(G\) such that \(\kappa(G) \geq 1/4\), and shows that \(G\) is abelian whenever \(\kappa(G)|G| < 7/4\). It is also shown that \(\kappa(G)|G|\) depends only on the isoclinism class of \(G\). Specialising to the symmetric group \(S_n\), the paper shows that \(\kappa(S_n) \leq C/n^2\) for an explicitly determined constant \(C\). This bound leads to an elementary proof of a result of Flajolet \emph{et al}, that \(\kappa(S_n) \sim A/n^2\) as \(n\rightarrow \infty\) for some constant \(A\). The same techniques provide analogous results for \(\rho(S_n)\), the probability that two elements of the symmetric group have conjugates that commute.

対称群のネタを見るとCoxeter群に一般化できるか気になってしまう病なので今回も例に漏れずということなのだが、具体的に何か面白いことができるかどうかは知らない。