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On matrix Kloosterman sums and Hall-Littlewood polynomials

Elad Zelingher

TL;DR

The paper proves a generalization of Erd\'elyi–T\'oth for matrix Kloosterman sums by establishing a clean link between twisted matrix sums on regular elliptic elements and scalar twisted sums at the eigenvalues, with an explicit factor depending on $n$ and $k$. It then extends the framework to generalized Jordan matrices, showing that twisted matrix Kloosterman sums can be expressed as modified Hall–Littlewood polynomials evaluated at the roots of the twisted Kloosterman sheaf, thereby connecting arithmetic sums to symmetric function theory. The approach combines non-abelian Gauss sums (Kondo), Green’s basic characters, and Green polynomials, with Hall–Littlewood polynomials and Katz’s purity results for Kloosterman sheaves to obtain explicit formulas and bounds. The results provide concrete formulas and structural insight, unifying representation-theoretic techniques with geometric objects (twisted Kloosterman sheaves) and yielding applications such as exact identities and decay bounds for twisted matrix Kloosterman sums.

Abstract

We prove an identity relating twisted matrix Kloosterman sums to modified Hall-Littlewood polynomials evaluated at the roots of the characteristic polynomial associated to a twisted Kloosterman sheaf. This solves a conjecture of Erdélyi and Tóth.

On matrix Kloosterman sums and Hall-Littlewood polynomials

TL;DR

The paper proves a generalization of Erd\'elyi–T\'oth for matrix Kloosterman sums by establishing a clean link between twisted matrix sums on regular elliptic elements and scalar twisted sums at the eigenvalues, with an explicit factor depending on and . It then extends the framework to generalized Jordan matrices, showing that twisted matrix Kloosterman sums can be expressed as modified Hall–Littlewood polynomials evaluated at the roots of the twisted Kloosterman sheaf, thereby connecting arithmetic sums to symmetric function theory. The approach combines non-abelian Gauss sums (Kondo), Green’s basic characters, and Green polynomials, with Hall–Littlewood polynomials and Katz’s purity results for Kloosterman sheaves to obtain explicit formulas and bounds. The results provide concrete formulas and structural insight, unifying representation-theoretic techniques with geometric objects (twisted Kloosterman sheaves) and yielding applications such as exact identities and decay bounds for twisted matrix Kloosterman sums.

Abstract

We prove an identity relating twisted matrix Kloosterman sums to modified Hall-Littlewood polynomials evaluated at the roots of the characteristic polynomial associated to a twisted Kloosterman sheaf. This solves a conjecture of Erdélyi and Tóth.
Paper Structure (22 sections, 20 theorems, 120 equations)

This paper contains 22 sections, 20 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. Twisted Kloosterman sums
  5. Conjugacy classes of $\mathrm{GL}_n\left(\mathbb{F}\right)$
  6. Representation theory of $\mathrm{GL}_n\left(\mathbb{F}\right)$
  7. Non-abelian Gauss sums
  8. Formula for regular elliptic elements
  9. A character identity for matrix Kloosterman sums
  10. Representations supported on regular elliptic elements
  11. Proof of the theorem for regular elliptic elements
  12. Formula for general elements
  13. Green's basic characters
  14. Reduction of twisted matrix Kloosterman sums to twisted Kloosterman sums and Green polynomials
  15. Expansion of twisted matrix Kloosterman sums in terms of basic characters
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any finite field $\mathbb{F}$ and any regular elliptic $x \in \mathrm{GL}_n\left(\mathbb{F}\right)$ with eigenvalues $\{\xi, \xi^q, \dots, \xi^{q^{n-1}}\}$, and for any $k > 1$, we have the identity More generally, for any character $\alpha \colon \left(\mathbb{F}^{\times}\right)^k \to \mathbb{C}^{\times}$, we have where $\mathrm{Kl}\left(\alpha, \psi, x\right)$ and $\mathrm{Kl}_n\left(\alph

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3
  • ...and 14 more