Twisted Baker-Akhiezer function from determinants

TL;DR

The paper investigates Chalykh Baker-Akhiezer functions, which describe eigenfunctions of integrable Hamiltonians associated with integer DIM rays, by recasting the problem into a linear system solved via Cramer's rule. It shows that determinant factorization occurs at the special roots of unity $\epsilon = e^{2\pi i / a}$, yielding comparatively simple expressions that generalize Macdonald polynomials for $a=1$ and extend to higher $a$ with intricate phase-structured behavior. Despite the linear nature of the equations, the resulting determinants exhibit rich combinatorial and product-sum structures, revealing why Noumi-Shiraishi non-polynomial extensions and related NS constructions arise in this context. The work clarifies the algebraic underpinnings of BAF on the DIM rays, discusses phase transitions at $k=na$, and points to elliptic deformations and Shiraishi-type liftings as natural avenues for further exploration within the Macdonald-Noumi-Shiraishi framework.

Abstract

General description of eigenfunctions of integrable Hamiltonians associated with the integer rays of Ding-Iohara-Miki (DIM) algebra, is provided by the theory of Chalykh Baker-Akhiezer functions (BAF) defined as solutions to a simply looking linear system. Solutions themselves are somewhat complicated, but much simpler than they could. It is because of simultaneous partial factorization of all the determinants, entering Cramer's rule. This is a conspiracy responsible for a relative simplicity of the Macdonald polynomials and of the Noumi-Shirashi functions, and it is further continued to all integer DIM rays. Still, factorization is only partial, moreover, there are different branches and abrupt jumps between them. We explain this feature of Cramer's rule in an example of a matrix that defines BAF and exhibits a non-analytical dependence on parameters. Moreover, the matrix is such that there is no natural expansion around non-degenerate approximations, which causes an unexpected complexity of formulas.