[Paper Review] On averages of randomized class functions on the symmetric groups and their asymptotics

Paul-Olivier Dehaye, Dirk Zeindler|ArXiv.org|Nov 20, 2009

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper generalizes the study of characteristic polynomials of permutation matrices on symmetric groups by introducing randomized eigenvalue shifts—either independently or per cycle—and extends the framework to arbitrary holomorphic functions and other groups like the alternating group and Weyl groups. The key contribution is explicit generating functions for moments and asymptotic results as n→∞, derived using the Feller coupling to analyze cycle length distributions.

ABSTRACT

The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices (n points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other groups than the symmetric group, for instance the alternating group and other Weyl groups. Finally, we compute some asymptotics results when n tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.

Motivation & Objective

• To generalize the moments of characteristic polynomials of permutation matrices beyond the standard case by introducing randomized eigenvalue shifts.
• To extend the analysis from polynomials to holomorphic functions of eigenvalues, preserving combinatorial structure.
• To compute explicit generating functions for moments over symmetric groups and their subgroups, including the alternating group and Weyl groups of classical Lie groups.
• To derive asymptotic behavior of these moments as n→∞, particularly for |x|<1.
• To unify and simplify earlier results using cycle-type decompositions and probabilistic tools like the Feller coupling.

Proposed method

• Reformulates the characteristic polynomial of a permutation matrix as a product over cycles: ∏(1−x^l) for each cycle of length l.
• Introduces two models of randomization: one where each eigenvalue is independently rotated (W¹), and one where eigenvalues within each cycle are rotated together (W²).
• Derives generating functions for moments of randomized class functions using combinatorial lemmas on conjugacy classes and cycle indices.
• Applies the Feller coupling to model the asymptotic distribution of cycle lengths in random permutations, enabling asymptotic analysis.
• Uses generating function identities involving ∏(1 + x^k t)^{-b_k} to express moment generating functions for holomorphic functions f.
• Reduces the asymptotic analysis to the case b₀=1 via normalization and applies convergence theorems to establish limit distributions.

Experimental results

Research questions

• RQ1 How do randomized eigenvalue shifts—per cycle or independently—affect the moment generating functions of class functions on symmetric groups?
• RQ2 What is the asymptotic behavior of these moments as n→∞, particularly for holomorphic functions of the eigenvalues?
• RQ3 Can the generating functions for moments be extended beyond polynomials to holomorphic functions, and what is their structure?
• RQ4 How do the results generalize to other groups such as the alternating group and Weyl groups of SO(2n), SO(2n+1), and SU(n)?
• RQ5 What functional equations or modular properties emerge in the asymptotic limit, especially when f(x)=1/(1−x)?

Key findings

• For polynomial functions f, the moment generating function over S_n is given by ∏_{k=0}^∞ (1 + x^k t)^{-b_k}, where b_k are coefficients of f.
• For holomorphic f with b₀=1, the limit of E[Wⁿ(f)(x)] as n→∞ is ∏_{k=1}^∞ (1 - x^k)^{-1}, which up to a constant is the Dedekind eta function when x=e^{2πiτ} with τ in the upper half-plane.
• The asymptotic distribution of W¹(f)(x) converges to a non-degenerate limit, with convergence proven via two distinct methods.
• For the Weyl group of SO(2n), the moment generating function is ∏_{k₁,k₂: 2|(k₁−k₂)} (1 - x₁^{k₁}x₂^{k₂}t)^{(-1)^{k₁+k₂+1} inom{s₁}{k₁}inom{s₂}{k₂}}.
• For SO(2n+1), the generating function includes both (1 - x₁^{k₁}x₂^{k₂}t) and (1 + x₁^{k₁}x₂^{k₂}t) terms due to the alternating group structure.
• For SU(n), the generating function is adjusted by a factor (1−x₁)^{-s₁}(1−x₂)^{-s₂} to account for the trace-zero condition on the Weyl group.