Séminaire Lotharingien de Combinatoire, 91B.5 (2024), 12 pp.
Jesse Campion Loth, Michael Levet, Kevin Liu, Sheila Sundaram and Mei Yin
Colored Permutation Statistics by Conjugacy Class
Abstract.
We consider the moments of statistics on conjugacy classes of colored
permutation groups Sn,r =
Zr ≀
Sn.
We first show that any fixed moment of a statistic
coincides on all conjugacy classes when all cycle lengths are
sufficiently long. For permutation statistics that can be realized via
a process called symmetric extension, we show that for fixed r, this
moment on these conjugacy classes is a polynomial in n. Hamaker and
Rhoades (arχiv, 2022) established analogous results for the symmetric
group as part of their far-reaching representation-theoretic
framework. Independently, Campion Loth, Levet, Liu, Stucky, Sundaram,
and Yin (arχiv, 2023) arrived at independence and polynomiality
results for the symmetric group using instead an elementary
combinatorial framework. Our techniques in this paper build on this
latter elementary approach. Finally, we extend the work of Fulman
(J. Comb. Theory Ser. A, 1998), to establish a central limit
theorem for descents in conjugacy classes of the hyperoctahedral group
with sufficiently long cycles.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
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