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Fixed-point statistics from spectral measures on tensor envelope categories

Arthur Forey, Javier Fresán, Emmanuel Kowalski

TL;DR

The paper develops spectral measures in tensor categories and uses Deligne–Knop tensor envelopes to study fixed-point statistics. It proves that the fixed-point distribution of random permutations converges to the Poisson law by identifying the Poisson measure as the spectral measure of the standard object in ˆRep(S_t), with moments matching Bell numbers. It extends these ideas to fixed-point statistics for vector spaces over finite and complex fields, interpreting limiting measures as spectral measures for corresponding objects (e.g., standard Gaussian in the GL_t setting). The work also connects these categorical insights to FI-modules and proposes arithmetic speculations, including Chebotarev-type explanations and potential S_t–GL_t dualities with broader implications for number theory and representation theory.

Abstract

We prove some old and new convergence statements for fixed-points statistics using tensor envelope categories, such as the Deligne--Knop category of representations of the "symmetric group" $S_t$ for an indeterminate~$t$. We also discuss some arithmetic speculations related to Chebotarev's density theorem.

Fixed-point statistics from spectral measures on tensor envelope categories

TL;DR

The paper develops spectral measures in tensor categories and uses Deligne–Knop tensor envelopes to study fixed-point statistics. It proves that the fixed-point distribution of random permutations converges to the Poisson law by identifying the Poisson measure as the spectral measure of the standard object in ˆRep(S_t), with moments matching Bell numbers. It extends these ideas to fixed-point statistics for vector spaces over finite and complex fields, interpreting limiting measures as spectral measures for corresponding objects (e.g., standard Gaussian in the GL_t setting). The work also connects these categorical insights to FI-modules and proposes arithmetic speculations, including Chebotarev-type explanations and potential S_t–GL_t dualities with broader implications for number theory and representation theory.

Abstract

We prove some old and new convergence statements for fixed-points statistics using tensor envelope categories, such as the Deligne--Knop category of representations of the "symmetric group" for an indeterminate~. We also discuss some arithmetic speculations related to Chebotarev's density theorem.
Paper Structure (9 sections, 15 theorems, 73 equations)

This paper contains 9 sections, 15 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Existence and uniqueness of spectral measures in tensor categories
  3. Tensor envelopes and fixed-point statistics
  4. Proof of Theorem
  5. Fixed-point statistics for vector and affine spaces over finite fields
  6. Fixed-point statistics for complex vector spaces
  7. Proof of Theorem
  8. Arithmetic speculations
  9. Knop's construction of the category $\mathop{\mathrm{\underline{\mathrm{Rep}}}}\nolimits(S_t)$

Key Result

Theorem 1.4

Let $(X_n)_{n\geqslant 1}$ be a sequence of random variables with $X_n$ a uniformly chosen random permutation in the symmetric group $S_n$. The sequence $(|\mathop{\mathrm{Fix}}\nolimits(X_n)|)_{n\geqslant 1}$, where $\mathop{\mathrm{Fix}}\nolimits(\sigma)$ denotes the set of fixed points of $\sigma

Theorems & Definitions (43)

  • Definition 1.1: Spectral measure
  • Example 1.2
  • Example 1.3
  • Theorem 1.4: "Problème des rencontres"; Montmort montmort; N. Bernoulli I; de Moivre de-moivre
  • Theorem 1.5
  • Corollary 1.6
  • proof
  • Proposition 2.1: Spectral measures for self-dual objects
  • proof
  • Proposition 2.2: Spectral measures for general objects
  • ...and 33 more