Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

WKB approximation, crystals and combinatorics of Young tableaux

Xiaomeng Xu

TL;DR

This work establishes a deep bridge between the analytic Stokes phenomena of the quantum confluent hypergeometric equation and the combinatorics of Young tableaux via WKB methods. By evaluating Stokes/connection data at caterpillar points and in GT bases, it derives a transcendental realization of the gl_n-crystal structure on semistandard Young tableaux and recovers the Robinson–Schensted correspondence, the Littlewood–Richardson rule, and Schützenberger involutions as WKB phenomena. It further connects these to quantum groups, providing explicit formulas for quantum Stokes/connection matrices and a coproduct formalism that mirrors crystal tensor products. The framework unifies analytic, algebraic, and combinatorial perspectives, offering new tools to study representation theory through isomonodromic and WKB techniques. It also illuminates how wall-crossing phenomena encode combinatorial symmetries in tableau theory and crystal operators.

Abstract

In this paper, we show how various combinatorial algorithms of Young tableaux naturally arise from the WKB approximation of the connection matrix of quantum confluent hypergeometric equation, including the Robinson-Schensted algorithm, the Littlewood-Richardson rule and the Schützenberger involution.

WKB approximation, crystals and combinatorics of Young tableaux

TL;DR

This work establishes a deep bridge between the analytic Stokes phenomena of the quantum confluent hypergeometric equation and the combinatorics of Young tableaux via WKB methods. By evaluating Stokes/connection data at caterpillar points and in GT bases, it derives a transcendental realization of the gl_n-crystal structure on semistandard Young tableaux and recovers the Robinson–Schensted correspondence, the Littlewood–Richardson rule, and Schützenberger involutions as WKB phenomena. It further connects these to quantum groups, providing explicit formulas for quantum Stokes/connection matrices and a coproduct formalism that mirrors crystal tensor products. The framework unifies analytic, algebraic, and combinatorial perspectives, offering new tools to study representation theory through isomonodromic and WKB techniques. It also illuminates how wall-crossing phenomena encode combinatorial symmetries in tableau theory and crystal operators.

Abstract

In this paper, we show how various combinatorial algorithms of Young tableaux naturally arise from the WKB approximation of the connection matrix of quantum confluent hypergeometric equation, including the Robinson-Schensted algorithm, the Littlewood-Richardson rule and the Schützenberger involution.

Paper Structure

This paper contains 26 sections, 29 theorems, 107 equations, 1 table.

Table of Contents

  1. Introduction
  2. ${\mathfrak {gl}}_n$-crystals arising from the WKB approximation of Stokes matrices
  3. Schensted deletion operator arising from the WKB approximation of connection matrices
  4. Robinson–Schensted correspondence
  5. Littlewood–Richardson rule and the WKB approximation of coproduct
  6. Schützenberger involution and Wall-crossing
  7. Crystals and Young tableaux
  8. Crystals and tensor products
  9. Crystal of words
  10. Crystal of Young tableaux
  11. The Robinson-Schensted correspondence
  12. The Littlewood–Richardson rule
  13. The Gelfand-Tsetlin patterns
  14. Quantum Stokes matrices and connection matrices
  15. Canonical solutions and monodromy
  16. ...and 11 more sections

Key Result

Theorem 1.1

Xu2 For any fixed positive real number $h$, the map (with $q=e^{h/2}$) defines a representation of the Drinfeld-Jimbo quantum group $U_q(\frak{gl}_n)$ on the vector space $W$. Here $U_q(\frak{gl}_n)$ is a unital associative algebra with generators $q^{\pm h_i}, e_j, f_j,$$1\le j\le n-1, 1\le i\le n$, see Definition defqg.

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • ...and 49 more