From Characters to Quantum (Super)Spin Chains via Fusion

TL;DR

This paper provides an elementary proof of a determinant formula for transfer matrices in twisted quantum super-spin chains, advancing the analytical Bethe ansatz method for models based on gl(K|M) algebra.

Contribution

It introduces a systematic approach to generalize the Jacobi-Trudi formula to quantum super-spin chains, enabling broader fusion of transfer matrices.

Findings

Proves the Bazhanov-Reshetikhin determinant formula for rational transfer matrices.

Extends classical character formulas to quantum super-spin chains.

Facilitates analytical Bethe ansatz for complex algebraic structures.

Abstract

We give an elementary proof of the Bazhanov-Reshetikhin determinant formula for rational transfer matrices of the twisted quantum super-spin chains associated with the gl(K|M) algebra. This formula describes the most general fusion of transfer matrices in symmetric representations into arbitrary finite dimensional representations of the algebra and is at the heart of analytical Bethe ansatz approach. Our technique represents a systematic generalization of the usual Jacobi-Trudi formula for characters to its quantum analogue using certain group derivatives.