Refined canonical stable Grothendieck polynomials and their duals, Part 1

Abstract

In this paper we introduce refined canonical stable Grothendieck polynomials and their duals with two infinite sequences of parameters. These polynomials unify several generalizations of Grothendieck polynomials including canonical stable Grothendieck polynomials due to Yeliussizov, refined Grothendieck polynomials due to Chan and Pflueger, and refined dual Grothendieck polynomials due to Galashin, Liu, and Grinberg. We give Jacobi--Trudi-like formulas, combinatorial models, Schur expansions, Schur positivity, and dualities of these polynomials.

Figures (4)

• Figure 1: The Young diagram of $\lambda=(4,3,1)$ on the left and that of its transpose $\lambda'=(3,2,2,1)$ on the right. The integer in each cell indicates the content of the cell.
• Figure 2: A semistandard Young tableau of shape $(5,4,2)/(2,1)$.
• Figure 3: A $\mathbb{Z}$-elegant tableau of shape $(4,4,4,2)/(3,2)$ on the left and a $\mathbb{Z}$-inelegant tableau of shape $(6,6,4,2)/(3,3,1)$ on the right.
• Figure 4: A nonintersecting $n$-path $\overline{\bm{p}}=(\overline{p}_1,\dots,\overline{p}_n)$ with $\overline{p}_i\in P_{\Gamma}(\overline{u}_i,\overline{v}_i)$, where $u_i=(\mu_i-i,\min(\mu_i-i+1,1))$, $v_i=(\lambda_i-i,\lambda_i)$, $\mu=(4,4,4,3,3,3,2)$, $\lambda=(3,2,2,1)$ and $n=7$, and the corresponding tableau $T$. Since each $\overline{p}_i$ passes through $u_i=(\mu_i-i,\min(\mu_i-i+1,1))$ and $v_i=(\lambda_i-i,\lambda_i)$, we have $T\in{\operatorname{IET}}_\mathbb{Z}({\mu/\lambda})$.

Theorems & Definitions (52)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6: Hwang-preprint
• Theorem 1.7
• Proposition 2.1
• Lemma 2.2: Kim_JT22
• Definition 2.3