Refined canonical stable Grothendieck polynomials and their duals, Part 2

TL;DR

The paper develops combinatorial models for the two-parameter refined canonical stable Grothendieck polynomials $G_\lambda(\bm{x};\bm{\alpha},\bm{\beta})$ and their duals $g_\lambda(\bm{x};\bm{\alpha},\bm{\beta})$, extending to flagged and skew variants. It introduces marked multiset-valued tableaux and marked reverse plane partitions as the core tableaux models, and derives Jacobi–Trudi–type determinant formulas for row- and column-flagged cases, with the flagged duals obtained via MRPP. The results unify and extend prior frameworks (Buch’s set-valued tableaux, Lam–Pylyavskyy’s reverse plane partitions, Matsumura’s flagged Grothendieck polynomials, and Yeliussizov’s canonical cases) within the two-parameter refinement, and provide recurrence-based proofs that illuminate the combinatorial structure underpinning these polynomials. Collectively, the work enhances the combinatorial toolkit for K-theoretic Schubert calculus and related areas by delivering explicit, flag-respecting models for both the polynomials and their duals, including skew variants and independence from the ambient rank $n$ in appropriate limits.

Abstract

This paper is the sequel of the paper under the same title with part 1, where we introduced refined canonical stable Grothendieck polynomials and their duals with two families of infinite parameters. In this paper we give combinatorial interpretations for these polynomials using generalizations of set-valued tableaux and reverse plane partitions, respectively. Our results extend to their flagged and skew versions.

Figures (8)

• Figure 1: The Young diagram of $\lambda=(4,3,1)$ on the left and its transpose $\lambda'=(3,2,2,1)$ on the right.
• Figure 2: An example of $T\in\operatorname{MMSVT}((3,2)/(1))$, where the marked integers are indicated with $*$. The weight of $T$ is given by $\operatorname{wt}(T)=x_1^2x_2^4x_3^2x_4\alpha_2^2\alpha_3(-\beta_1)^2$.
• Figure 3: An example of $T\in{\operatorname{MRPP}}({\lambda/\mu})$, where $\lambda=(6,5,3,3)$ and $\mu=(2,1,1)$. We have the weight $\operatorname{wt}(T)=x_1x_2x_3^2x_4x_5 (-\alpha_1)(-\alpha_2)^2(-\alpha_5)\beta_1\beta_2^2$.
• Figure 4: The Young diagram of a dented partition $\lambda=(3,3,4,4,1)$. The cell $(3,4)$ is the minimal cell, which is colored gray.
• Figure 5: An example of $T\in{\operatorname{MRPP}}_I^{\operatorname{row}(\bm{r},\bm{s})}({\lambda/\mu})$, where $\lambda=(3,4,4,1),\mu=(1,1),I=\{ 1,3 \}, \bm{r}=(1,1,2,2),$ and $\bm{s}=(3,3,4,5)$. Note that $T(2,4)=3$ cannot be marked because $2\notin I$ and its contribution to the weight is $\beta_1$ since $1\in I$ and $T(2,4)=T(1,4)=3$. Thus we have $\operatorname{wt}(T)= x_1x_2^3x_4(-\alpha_3)(-\alpha_4)\beta_1^2\beta_2$.

Theorems & Definitions (99)

• Definition 1.1
• Theorem 1.2: HJKSS2024
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Lemma 2.2: Kim_JT22
• Lemma 2.3: Kim_JT22
• Lemma 2.4: HJKSS2024
• Lemma 2.5: HJKSS2024
• Definition 2.6