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Character Formulas for Kirillov-Reshetikhin Modules via Folding of Supercharacters of $\mathfrak{gl}(M|N)$

Zengo Tsuboi

TL;DR

The paper proves that the characters of a broad class of Kirillov-Reshetikhin modules for quantum affine (and twisted) superalgebras can be realized as folded supercharacters of the finite-dimensional superalgebra $\mathfrak{gl}(M|N)$. By leveraging Cauchy-type identities for supersymmetric Schur functions and a systematic folding/reduction procedure, explicit character formulas are derived for KR modules across untwisted and twisted families, including $U_q(\mathfrak{osp}(2r+1|2s)^{(1)})$, $U_q(\mathfrak{sl}(2r|2s+1)^{(2)})$, $U_q(\mathfrak{sl}(2r+1|2s)^{(2)})$, $U_q(\mathfrak{sl}(2r|2s)^{(2)})$, $U_q(\mathfrak{osp}(2r|2s)^{(1)})$, and $U_q(\mathfrak{osp}(2r|2s)^{(2)})$. The results, which align with Bethe ansatz-derived expectations, unify KR-character formulas under a common folding framework and clarify when folded supercharacters correspond to irreducible KR modules. This approach not only proves a prior conjecture but also offers a versatile method to obtain determinant-type character expressions across a broad range of quantum-affine (super)algebras.

Abstract

We derive decomposition formulas for supercharacters of quantum affine ortho-symplectic superalgebras and twisted quantum affine superalgebras into supercharacters of their finite-type quantum sub-superalgebras, by employing Cauchy-type identities for supersymmetric Schur functions. These formulas are obtained via a folding (reduction) procedure applied to the supercharacters of the finite-dimensional general linear Lie superalgebra $\mathfrak{gl}(M|N)$. As a special case, our results provide explicit character formulas for a class of Kirillov--Reshetikhin modules of quantum affine algebras (and their Yangian counterparts), thereby proving a previously proposed conjecture derived from Bethe ansatz analysis (arXiv:2309.16660).

Character Formulas for Kirillov-Reshetikhin Modules via Folding of Supercharacters of $\mathfrak{gl}(M|N)$

TL;DR

The paper proves that the characters of a broad class of Kirillov-Reshetikhin modules for quantum affine (and twisted) superalgebras can be realized as folded supercharacters of the finite-dimensional superalgebra . By leveraging Cauchy-type identities for supersymmetric Schur functions and a systematic folding/reduction procedure, explicit character formulas are derived for KR modules across untwisted and twisted families, including , , , , , and . The results, which align with Bethe ansatz-derived expectations, unify KR-character formulas under a common folding framework and clarify when folded supercharacters correspond to irreducible KR modules. This approach not only proves a prior conjecture but also offers a versatile method to obtain determinant-type character expressions across a broad range of quantum-affine (super)algebras.

Abstract

We derive decomposition formulas for supercharacters of quantum affine ortho-symplectic superalgebras and twisted quantum affine superalgebras into supercharacters of their finite-type quantum sub-superalgebras, by employing Cauchy-type identities for supersymmetric Schur functions. These formulas are obtained via a folding (reduction) procedure applied to the supercharacters of the finite-dimensional general linear Lie superalgebra . As a special case, our results provide explicit character formulas for a class of Kirillov--Reshetikhin modules of quantum affine algebras (and their Yangian counterparts), thereby proving a previously proposed conjecture derived from Bethe ansatz analysis (arXiv:2309.16660).

Paper Structure

This paper contains 16 sections, 2 theorems, 94 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Lie superalgebras
  3. Type A
  4. Type B
  5. The case $r>0$:
  6. The case $r=0$:
  7. Type C and D
  8. Type C for $r=1$, $s \ge 1$:
  9. Decomposition of the supersymmetric Schur function
  10. Specialization
  11. $U_{q}(\mathfrak{osp}(2r+1|2s)^{(1)})$ case:
  12. $U_{q}(\mathfrak{sl}(2r|2s+1)^{(2)})$ case:
  13. $U_{q}(\mathfrak{sl}(2r+1|2s)^{(2)})$ case:
  14. $U_{q}(\mathfrak{sl}(2r|2s)^{(2)})$ case:
  15. $U_{q}(\mathfrak{osp}(2r|2s)^{(1)})$ case:
  16. ...and 1 more sections

Key Result

Proposition 3.1

Let $\xi = \pm 1$ and $\lambda \in {\mathcal{P}}$. The following relations are valid.

Figures (5)

  • Figure 1: $[M,N]$-hook: the Young diagram $\lambda$ is related to the highest weight \ref{['HW-A']} by \ref{['YW-A']}.
  • Figure 2: $[r,s]$-hook in $[2r,2s+1]$-hook: the Young diagram $\lambda$ is related to the highest weight \ref{['HW-B']} by \ref{['YW-B']}.
  • Figure 3: $[0,s]$-hook in $[0,2s+1]$-hook: the Young diagram $\lambda$ is related to the highest weight \ref{['HW-B0']} by \ref{['YW-B0']}.
  • Figure 4: $[r,s]$-hook in $[2r,2s+2]$-hook: the Young diagram $\lambda$ is related to the highest weight \ref{['HW-D']} by \ref{['YW-Dp']} or \ref{['YW-Dm']}.
  • Figure 5: $[1,s]$-hook in $[2,2s+2]$-hook: the Young diagram $\lambda$ is related to the highest weight \ref{['HW-Cp']} by \ref{['YW-Cp']}.

Theorems & Definitions (2)

  • Proposition 3.1
  • Theorem 4.1