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Group character averages via a single Laguerre

Alexei Morozov, Kazumi Okuyama

Abstract

Average of exponential ${\rm Tr}_R e^X$, i.e. of a group rather than an algebra character, in Gaussian matrix model is known to be an amusing generalization of Schur polynomial, where time variables are substituted by traces of products of non-commuting matrices ${\rm Tr} \left(\prod_i A_{k_i}\right)$ and are thus labeled by weak compositions. The entries of matrices $A_k$ are made from extended Laguerre polynomials, what introduces additional difficulties. We describe the generic sum rules, which express arbitrary traces through convolutions of a single Laguerre polynomial $L_{N-1}^1(z_{k_i})$, what is a considerable simplification.

Group character averages via a single Laguerre

Abstract

Average of exponential , i.e. of a group rather than an algebra character, in Gaussian matrix model is known to be an amusing generalization of Schur polynomial, where time variables are substituted by traces of products of non-commuting matrices and are thus labeled by weak compositions. The entries of matrices are made from extended Laguerre polynomials, what introduces additional difficulties. We describe the generic sum rules, which express arbitrary traces through convolutions of a single Laguerre polynomial , what is a considerable simplification.
Paper Structure (10 sections, 50 equations)

This paper contains 10 sections, 50 equations.

Table of Contents

  1. Introduction
  2. ${\rm Tr}\, (\mathcal{A}_1)^m$ via Laguerres
  3. ${\bf m=1}$:
  4. ${\bf m=2}$:
  5. ${\bf m=3}$:
  6. Generic traces of the matrices $\mathcal{A}(s)$
  7. Connected correlators
  8. Comparison to Morozov:2009uy
  9. Conclusion and outlook
  10. Alternative derivation of \ref{['eq:tr-conv']}