Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Corner Polynomials: A Generalization of Super Macdonald Polynomials and Their VOA Correspondence

Panupong Cheewaphutthisakun, Jun'ichi Shiraishi, Keng Wiboonton

Abstract

In this paper, we introduce a family of partially symmetric polynomials, which we call quantum corner polynomials, as a generalization of the Sergeev-Veselov super Macdonald polynomials. We show that these quantum corner polynomials are precisely the partially symmetric polynomials corresponding to the quantum corner VOAs. Furthermore, we provide a detailed proof of the partial symmetricity of these polynomials.

Quantum Corner Polynomials: A Generalization of Super Macdonald Polynomials and Their VOA Correspondence

Abstract

In this paper, we introduce a family of partially symmetric polynomials, which we call quantum corner polynomials, as a generalization of the Sergeev-Veselov super Macdonald polynomials. We show that these quantum corner polynomials are precisely the partially symmetric polynomials corresponding to the quantum corner VOAs. Furthermore, we provide a detailed proof of the partial symmetricity of these polynomials.

Paper Structure

This paper contains 17 sections, 38 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Quantum Corner VOA
  3. Quantum Toroidal $\mathfrak{gl}_1$ Algebra
  4. Quantum Corner VOA
  5. Quantum Corner Polynomial
  6. Reverse Semi-standard Young Tritableau and Quantum Corner Polynomial
  7. Quantum Corner VOA/Quantum Corner Polynomial Correspondence
  8. Analysis on the Quantity in \ref{['eqn48-1512-25jul']}
  9. Analysis on the Quantity in \ref{['eqn49-1512-25jul']}
  10. Analysis on the Quantities in \ref{['eqn310-1137-27jul']} and \ref{['eqn311-1137-27jul']}
  11. Proof of Lemma \ref{['lemm45-1458-25jul']}
  12. Partially Symmetricity of the Quantum Corner Polynomial
  13. Proof of Lemma \ref{['prp317-2237-16aug']}
  14. Proof of Lemma \ref{['lemm42-1252-25jul']}
  15. Triangle-form Cancellation
  16. ...and 2 more sections

Key Result

Proposition 2.2

The map $\Delta : U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1) \rightarrow U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1) \otimes U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$ defined by the formula below is an algebra homomorphism: where in the formulas above, $C_1 := C \otimes 1, C_2 := 1 \otimes C$. This map is called the coproduct of $U_{q,t}(\widehat{\widehat{\mathfrak{gl}}}_1)$.

Theorems & Definitions (95)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Definition 2.8: HMNW
  • Proposition 2.9: CSw2024
  • Definition 3.1: Partition
  • ...and 85 more