More Like this

1. Frozen pipes: lattice models for Grothendieck polynomials

   https://doi.org/10.5802/alco.277  

   Brubaker, Ben; Frechette, Claire; Hardt, Andrew; Tibor, Emily; Weber, Katherine (June 2023, Algebraic combinatorics)

   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. 

   more » « less
2. Tilted biorthogonal ensembles, Grothendieck random partitions, and determinantal tests

   https://doi.org/10.1007/s00029-024-00945-3  

   Gavrilova, Svetlana; Petrov, Leonid (July 2024, Selecta Mathematica)

   We study probability measures on partitions based on symmetric Grothendieck polynomials. These deformations of Schur polynomials introduced in the K-theory of Grassmannians share many common properties. Our Grothendieck measures are analogs of the Schur measures on partitions introduced by Okounkov (Sel Math 7(1):57–81, 2001). Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry, and we employ a determinantal test first obtained by Nanson in 1897 for the 4 × 4 problem. We also propose a procedure for getting Nanson-like determinantal tests for matrices of any size n ≥ 4, which appear new for n ≥ 5. By placing the Grothendieck measures into a new framework of tilted biorthogonal ensembles generalizing a rich class of determinantal processes introduced by Borodin (Nucl Phys B 536:704–732, 1998), we identify Grothendieck random partitions as a cross-section of a Schur process, a determinantal process in two dimensions. This identification expresses the correlation functions of Grothendieck measures through sums of Fredholm determinants, which are not immediately suitable for asymptotic analysis. A more direct approach allows us to obtain a limit shape result for the Grothendieck random partitions. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. The gradient of this surface is expressed through the argument of a complex root of a cubic equation. 

   more » « less
3. Interpolation for Brill–Noether curves

   https://doi.org/10.1017/fmp.2023.22  

   Larson, Eric; Vogt, Isabel (October 2023, Forum of Mathematics, Pi)

   Abstract In this paper, we determine the number of general points through which a Brill–Noether curve of fixed degree and genus in any projective space can be passed. 

   more » « less
4. Hook-length Formulas for Skew Shapes via Contour Integrals and Vertex Models

   Panova, Greta; Petrov, Leonid (September 2024, arXiv)

   The number of standard Young tableaux of a skew shape $$\lambda/\mu$$ can be computed as a sum over excited diagrams inside $$\lambda$$. Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also nonintersecting lattice paths inside $$\lambda$$. We give two new proofs of a multivariate generalization of this formula, which allow us to extend the setup beyond standard Young tableaux and the underlying Schur symmetric polynomials. The first proof uses multiple contour integrals. The second one interprets excited diagrams as configurations of a six-vertex model at a free fermion point, and derives the formula for the number of standard Young tableaux of a skew shape from the Yang-Baxter equation. 

   more » « less
5. Brill–Noether Theory on ℙ2 for Bundles with Many Sections

   https://doi.org/10.1093/imrn/rnaf064  

   Coskun, Izzet; Huizenga, Jack; Raha, Neelarnab (March 2025, International Mathematics Research Notices)

   Abstract The Brill–Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill–Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill–Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let $$E$$ be a semistable sheaf on $${\mathbb{P}}^{2}$$. In this paper, we give an upper bound $$\beta _{r, \mu }$$ for $$h^{0}(E)$$ in terms of the rank $$r$$ and the slope $$\mu $$ of $$E$$. We show that the bound is achieved precisely when $$E$$ is a twist of a Steiner bundle. We classify the sheaves $$E$$ such that $$h^{0}(E)$$ is within $$\lfloor \mu (E) \rfloor + 1$$ of $$\beta _{r, \mu }$$. We determine the nonemptiness, irreducibility and dimension of the Brill–Noether loci in the moduli spaces of sheaves on $${\mathbb{P}}^{2}$$ with $$h^{0}(E)$$ in this range. When they are proper subvarieties, these Brill–Noether loci are irreducible though almost always of larger than the expected dimension. 

   more » « less