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We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$$ -- {\it biaxial double} $$(\beta,q)$$-{\it Grothendieck polynomials} -- which specialize at $$q=0$ and $v=1$ to double $$\beta$$-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $$n$$ pairs of variables is a Drinfeld twist of the $$U_q(\widehat{\mathfrak{sl}}_{n+1})$$ $$R$$-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $$\beta$$-Grothendieck polynomials and dual double $$\beta$$-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for $$\beta$$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $$\beta$$-Grothendieck polynomials, and prove a new branching rule for double $$\beta$$-Grothendieck polynomials.
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This content will become publicly available on April 21, 2026
Zeros of Hook Polynomials and Related Questions
We study the zero set of polynomials built from partition statistics, complementing earlier work in this direction by Boyer, Goh, Parry, and others. In particular, addressing a question of Males with two of the authors, we prove asymptotics for the values of $$t$$-hook polynomials away from an annulus and isolated zeros of a theta function. We also discuss some open problems and present data on other polynomial families, including those associated to deformations of Rogers-Ramanujan functions.
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- Award ID(s):
- 2200728
- PAR ID:
- 10613100
- Publisher / Repository:
- SIGMA
- Date Published:
- Journal Name:
- Symmetry, Integrability and Geometry: Methods and Applications
- ISSN:
- 1815-0659
- Subject(s) / Keyword(s):
- integer partitions hook length zeros of polynomials zero attractor asymptotic behavior theta functions
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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