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Stable Invariants of Words from Random Matrices II: Formulas and Extensions

Doron Puder, Yotam Shomroni

TL;DR

The paper advances a unifying framework linking stable Fourier coefficients of w-random group elements to topological invariants, extending from scl to the stable primitivity rank and its variants. It provides explicit, computable formulas for stable Fourier coefficients in the symmetric groups $S_N$ and in wreath products $G\wr S_N$, via graph-theoretic objects (core graphs) and Möbius inversions, and develops new invariants such as $\mathrm{s\pi^{\phi}}$ that control decay rates. A key technical achievement is a generalized PP15 formula for averages of products of fixed-point statistics, together with a profinite-invariance result for $\mathrm{s\pi}$, showing these quantities depend only on finite-group measurements. The methods combine algebraic-morphism decompositions, Pieri-type rules, and induction on representation size, yielding stable-character formulas valid for arbitrary compact groups and enabling sharp bounds on Fourier coefficients. These results deepen the connection between random-matrix-inspired techniques and combinatorial-topological invariants, with potential implications for understanding word measures across broad families of groups.

Abstract

Let $w$ be a word in a free group. As was revealed by Magee and Puder in [arXiv:1802.04862], the stable commutator length (scl) of $w$, a well-known topological invariant, can also be defined in terms of certain stable Fourier coefficients of $w$-random unitary matrices. In the first part of the current work [arXiv:2311.17733], we demonstrated how this phenomenon is much broader: we proved more instances of such results and conjectured others. These new results and conjectures involved other topological invariants (relatives of scl) and different families of groups. In the current paper we further extend and support this theory. We provide another instance of the theory and prove that the stable primitivity rank, too, can be expressed in terms of stable Fourier coefficients of $w$-random elements of groups. We introduce concrete formulas for stable Fourier coefficients of $w$-random elements in the symmetric group $S_N$ and its generalizations in the form of the wreath products $G\wr S_N$ where $G$ is an arbitrary compact group. We also define new stable invariants related to these groups, and prove they give bounds to many of the stable Fourier coefficients. As an aside, we generalize to tuples of words a result of Puder and Parzanchevski [arXiv:1202.3269] about the expected number of fixed points of $w$-random permutations.

Stable Invariants of Words from Random Matrices II: Formulas and Extensions

TL;DR

The paper advances a unifying framework linking stable Fourier coefficients of w-random group elements to topological invariants, extending from scl to the stable primitivity rank and its variants. It provides explicit, computable formulas for stable Fourier coefficients in the symmetric groups and in wreath products , via graph-theoretic objects (core graphs) and Möbius inversions, and develops new invariants such as that control decay rates. A key technical achievement is a generalized PP15 formula for averages of products of fixed-point statistics, together with a profinite-invariance result for , showing these quantities depend only on finite-group measurements. The methods combine algebraic-morphism decompositions, Pieri-type rules, and induction on representation size, yielding stable-character formulas valid for arbitrary compact groups and enabling sharp bounds on Fourier coefficients. These results deepen the connection between random-matrix-inspired techniques and combinatorial-topological invariants, with potential implications for understanding word measures across broad families of groups.

Abstract

Let be a word in a free group. As was revealed by Magee and Puder in [arXiv:1802.04862], the stable commutator length (scl) of , a well-known topological invariant, can also be defined in terms of certain stable Fourier coefficients of -random unitary matrices. In the first part of the current work [arXiv:2311.17733], we demonstrated how this phenomenon is much broader: we proved more instances of such results and conjectured others. These new results and conjectures involved other topological invariants (relatives of scl) and different families of groups. In the current paper we further extend and support this theory. We provide another instance of the theory and prove that the stable primitivity rank, too, can be expressed in terms of stable Fourier coefficients of -random elements of groups. We introduce concrete formulas for stable Fourier coefficients of -random elements in the symmetric group and its generalizations in the form of the wreath products where is an arbitrary compact group. We also define new stable invariants related to these groups, and prove they give bounds to many of the stable Fourier coefficients. As an aside, we generalize to tuples of words a result of Puder and Parzanchevski [arXiv:1202.3269] about the expected number of fixed points of -random permutations.

Paper Structure

This paper contains 23 sections, 36 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Comparison with other formulas for stable Fourier coefficients of $S_{\bullet}$
  3. Outline and notation
  4. Preliminaries
  5. Serre graphs, core graphs and algebraic and free morphisms
  6. Previous analysis of word measures on $S_{\bullet}$
  7. A geometric interpretation of $\mathbb{E}_{S_{N}}[\mathrm{fix}(w)]$ and a generalization
  8. Möbius inversions
  9. Useful properties of the Möbius inversions
  10. The groups $G\wr S_{N}$ and their stable irreducible representations
  11. The expectations $\mathbb{E}_{w}[\chi]$ and the parameters $\beta(w,\chi)$
  12. The average value of $(\mathrm{fix}(w_{1})-1)\cdot\cdot\cdot(\mathrm{fix}(w_{k})-1)$
  13. The stable invariant $\mathrm{s\pi^{\phi}}$
  14. Upper bounds for stable characters
  15. A formula for induction of characters in $G\wr S_{N}$
  16. ...and 8 more sections

Key Result

Theorem 1.2

For $m\ge2$, let ${\cal J}_{m}$ denote the set of stable non-trivial irreducible characters of $S_{m}\wr S_{\bullet}$ corresponding to $\mathrm{\overrightarrow{\mu}}\colon\mathrm{Irr}(S_{m})\to{\cal P}$ with $\mathrm{triv}\notin\mathrm{Supp}(\mathrm{\overrightarrow{\mu}})$.When $m=1$ there are no su Moreover, the infimum on the right hand side is attained.

Theorems & Definitions (85)

  • Conjecture 1.1: PSh23
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: free morphisms
  • ...and 75 more