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Kirillov's conjecture on Hecke-Grothendieck polynomials

Ben Brubaker, A. Suki Dasher, Michael Hu, Nupur Jain, Yifan Li, Yi Lin, Maria Mihaila, Van Tran, I. Deniz Ünel

Abstract

We use algebraic methods in statistical mechanics to represent a multi-parameter class of polynomials in severable 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.

Kirillov's conjecture on Hecke-Grothendieck polynomials

Abstract

We use algebraic methods in statistical mechanics to represent a multi-parameter class of polynomials in severable 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 . 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.

Paper Structure

This paper contains 9 sections, 15 theorems, 43 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The lattice model
  3. Basic construction and Boltzmann weights
  4. Boundary conditions and partition functions
  5. Solvability of the lattice model
  6. Solution to the Yang--Baxter equation
  7. Connections to known $R$-matrices
  8. Proof of main result
  9. Applications and Kirillov's conjectures

Key Result

Theorem 1.2

For each positive integer $n$, there exists a solvable lattice model depending upon parameters $\alpha, \beta, \gamma$ and variables $\boldsymbol{x} = (x_1, \ldots, x_n)$ such that for each permutation $w \in S_n$ and integer partition $\lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n)

Figures (14)

  • Figure 2.1: Boltzmann weights in the style of borodin-wheeler. In place of their multiset $\mathbf{I}$ of colors, we use a subset $\Sigma$ of the colors $\{ 1, 2, \ldots, n \}$. We assume $c < d$. The function $h_k(\alpha,\beta)$ denotes the $k$-th complete homogeneous symmetric function of degree $k$, where we interpret $h_{-1} = 0$ and $h_{-2} = \frac{-1}{\alpha \beta}$ according to the usual recursion $h_k = \alpha h_{k-1} + \beta^k$. We adopt the notation $\Sigma_{[c+1, n]} \coloneqq \{s \in \Sigma : s \geq c + 1\}$, as well as $\Sigma_c^+ \coloneqq \Sigma \cup \{ c \}$, $\Sigma_c^- \coloneqq \Sigma \setminus \{ c \}$, and $\tensor*{\Sigma}{*^+_c^-_d} = (\Sigma \cup \{ c \}) \setminus \{ d \}$ when $c \notin \Sigma$ and $d \in \Sigma$.
  • Figure 2.2: Boundary condition for the lattice model determined by $\mu = (3, 1, 1)$ and the permutations $w_1 = s_1 s_2$ and $w_2 = s_2$ in $S_3$, with $N = 6$.
  • Figure 2.3: Admissible states of the system $\mathfrak{S}_w^\lambda$ for $\lambda = (1, 1, 0)$ and $w = s_2 \in S_3$. Here $\star = 1 + (\alpha + \gamma)x_2$ and $\ast = \gamma + (\alpha + \gamma)(\beta + \gamma)x_2$, and Boltzmann weights are overlaid on their respective vertices.
  • Figure 3.1: The Yang--Baxter equation as an equality of partition functions. The Boltzmann weights of the vertices labeled $T_{ik}$ and $T_{jk}$ take values in Figure \ref{['coloredalabw']} if they are non-zero, and they are functions of the spectral parameters $x_i$ and $x_j$, respectively. The Boltzmann weights of the vertices labeled $R_{ij}$ are functions of the spectral parameters $x_i$ and $x_j$.
  • Figure 3.2: Solution to the Yang--Baxter equation (\ref{['YBEeq']}). Here $a < b$ and $c$ is any color.
  • ...and 9 more figures

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3: kirillov2016notes
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Theorem 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • ...and 18 more