A. Suki Dasher | Searcharxiv

arXiv2026

Kirillov's conjecture on Hecke-Grothendieck polynomials

We use algebraic methods in statistical mechanics to represent a multi-parameter class of polynomials in several variables as partition functions of a new family of solvable lattice models. The class of polynomials, defined by A. N. Kirillov, is derived from the largest class of divided difference operators satisfying the braid relations of Cartan type $A$. It includes as specializations Schubert, Grothendieck, and dual-Grothendieck polynomials, among others. In particular, our results prove positivity conjectures of Kirillov for the subfamily of Hecke-Grothendieck polynomials, while the larger family is shown to exhibit rare instances of negative coefficients.

math.CO↗

arXiv2022

The Three Gap Theorem and Periodic Functions

The Three Gap Theorem, also known as the Steinhaus Conjecture, is a classical result on the combinatorics of the fractional part function, and has since been generalized in many ways. In this paper, we pose a new problem related to these results: for which other periodic functions does an analogue of the Three Gap Theorem hold? We prove analogous results for certain classes of piecewise-linear periodic functions and demonstrate the existence of functions for which no bound exists on the number of gap lengths.

math.CO↗