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Determinantal formulas for Sklyanin-Whittaker integrals

Taro Kimura

Abstract

We study multi-variable integrals, that we name Sklyanin-Whittaker integrals, and prove their determinantal formulas. We also discuss a $q$-deformation, a determinantal point process, and associated Mellin--Barnes integrals.

Determinantal formulas for Sklyanin-Whittaker integrals

Abstract

We study multi-variable integrals, that we name Sklyanin-Whittaker integrals, and prove their determinantal formulas. We also discuss a -deformation, a determinantal point process, and associated Mellin--Barnes integrals.
Paper Structure (13 sections, 24 theorems, 94 equations, 1 table)

This paper contains 13 sections, 24 theorems, 94 equations, 1 table.

Table of Contents

  1. Introduction and summary
  2. Sklyanin--Whittaker integrals
  3. Gaussian integral
  4. Matrix integral with an external source
  5. Determinantal point process
  6. $q$-Sklyanin--Whittaker integral
  7. Mellin--Barnes integral
  8. Type $A_{n-1}$
  9. Type $B_n$, $C_n$, $D_n$
  10. $q$-Mellin--Barnes integral
  11. Type $A_{n-1}$
  12. Type $B_n$, $C_n$, $D_n$
  13. Special functions

Key Result

Theorem 1.1

For a classical root system, the SW integral is given by a determinant.

Theorems & Definitions (45)

  • Theorem 1.1: Proposition \ref{['prop:SW_det']}
  • Proposition 1.2: Proposision \ref{['prop:SW_DDP']}
  • Theorem 1.3: Proposition \ref{['prop:q-SW_det']}
  • Theorem 1.4: Theorems \ref{['thm:MBSW_A']}, \ref{['thm:MBSW_BCD']}
  • Lemma 2.1: Andréief's formula
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 35 more