Generalisation of Scott permanent identity

TL;DR

This paper extends Scott's classical identity by deriving a new determinant specialization involving an arbitrary set of indeterminates, broadening the understanding of such identities in algebraic combinatorics.

Contribution

It generalizes Scott's permanent identity to a broader class of determinants with arbitrary indeterminates, including special cases relevant to integrable models.

Findings

Derived a new determinant specialization for arbitrary indeterminate sets.

Extended Scott's identity to the case involving the product over x.

Connected the results to the Gaudin-Izergin-Korepin model.

Abstract

Scott considered the determinant of 1/(y-z)^2, with y,z running over two sets X,Y of size n, and determined its specialisation when Y and Z are the roots of y^n-a and z^n-b. We give the same specialisation for the determinant 1/\prod_x(xy-z), where {x} is an arbitrary set of indeterminates. The case of the Gaudin-Izergin-Korepin is for {x}={q,1/q}.