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Key expansion of the flagged refined skew stable Grothendieck polynomial

Siddheswar Kundu

TL;DR

This work develops a Demazure-crystal framework for flagged refined skew stable Grothendieck polynomials by endowing the set of flagged semi-standard set-valued tableaux with a Demazure crystal structure. The main theorem proves that $\mathrm{SVT}_{\mathbf{e}}(\lambda/\mu,\Phi)$ decomposes into Demazure crystals, yielding a Demazure expansion of $G_{\lambda/\mu}(X_{\Phi};\mathbf{t})$ in terms of key polynomials $\kappa_{\widehat{\beta(R')}}$. The paper then derives multiple expansions: refined dual stable Grothendieck polynomials and row-refined skew stable Grothendieck polynomials in terms of stable and dual stable bases, as well as Schur $P$-functions, using tableau- and crystal-theoretic methods. Overall, this links flagged, set-valued combinatorics to Demazure characters and K-theoretic Schubert calculus, broadening tableau-based expansions and enabling explicit computations across several Grothendieck-type families.

Abstract

The flagged refined stable Grothendieck polynomials of skew shapes generalize several polynomials like stable Grothendieck polynomials, flagged skew Schur polynomials. In this paper, we provide a combinatorial expansion of the flagged refined skew stable Grothendieck polynomial in terms of key polynomials. We present this expansion by imposing a Demazure crystal structure on the set of flagged semi-standard set-valued tableaux of a given skew shape and a flag. We also provide expansions of the row-refined stable Grothendieck polynomials and refined dual stable Grothendieck polynomials and the Schur P-functions in terms of stable Grothendieck polynomials $G_λ$ and in terms of dual stable Grothendieck polynomials $g_λ$.

Key expansion of the flagged refined skew stable Grothendieck polynomial

TL;DR

This work develops a Demazure-crystal framework for flagged refined skew stable Grothendieck polynomials by endowing the set of flagged semi-standard set-valued tableaux with a Demazure crystal structure. The main theorem proves that decomposes into Demazure crystals, yielding a Demazure expansion of in terms of key polynomials . The paper then derives multiple expansions: refined dual stable Grothendieck polynomials and row-refined skew stable Grothendieck polynomials in terms of stable and dual stable bases, as well as Schur -functions, using tableau- and crystal-theoretic methods. Overall, this links flagged, set-valued combinatorics to Demazure characters and K-theoretic Schubert calculus, broadening tableau-based expansions and enabling explicit computations across several Grothendieck-type families.

Abstract

The flagged refined stable Grothendieck polynomials of skew shapes generalize several polynomials like stable Grothendieck polynomials, flagged skew Schur polynomials. In this paper, we provide a combinatorial expansion of the flagged refined skew stable Grothendieck polynomial in terms of key polynomials. We present this expansion by imposing a Demazure crystal structure on the set of flagged semi-standard set-valued tableaux of a given skew shape and a flag. We also provide expansions of the row-refined stable Grothendieck polynomials and refined dual stable Grothendieck polynomials and the Schur P-functions in terms of stable Grothendieck polynomials and in terms of dual stable Grothendieck polynomials .
Paper Structure (18 sections, 18 theorems, 99 equations, 7 figures)

This paper contains 18 sections, 18 theorems, 99 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partitions and skew Shapes
  4. Semi-standard set-valued tableaux
  5. The flagged refined skew stable Grothendieck polynomials
  6. Key polynomials
  7. The Burge correspondence
  8. Crystal structure on set-valued tableaux
  9. Crystals
  10. Tensor products of crystals
  11. Crystal morphism
  12. Crystal structure on set-valued tableaux of skew shape
  13. Demazure crystals
  14. proof of the main theorem
  15. various expansions
  16. ...and 3 more sections

Key Result

Theorem 1

The Burge correspondence gives a bijection between $\mathop{\mathrm{Mat}}\nolimits_{m \times n}(\mathbb{Z}_{+})$ and the set of pairs $(P, Q)$, where $P, Q$ are both semi-standard Young tableaux of same shape and entries of $P, Q$ are in $[n],[m]$ respectively. We write $(w_A \rightarrow \emptyset)=

Figures (7)

  • Figure 1: Skew shape $(3,3,2)/(2,1)$
  • Figure 2: A semi-standard set-valued tableau of shape $(4,3,2,1)/(2,1)$, weight $(3,3,4,2)$, excess $(2,1,2,0)$.
  • Figure 3: Crystal structure on the set of all elements in $\mathop{\mathrm{SVT}}\nolimits_{3}((2,2)/(1)),$ whose excess is $(0,1)$.
  • Figure 4: The Demazure crystal $\mathcal{B}_{s_1s_2} (2,1,0)$
  • Figure 5: $\mathop{\mathrm{SVT}}\nolimits_{\mathbf{e}}(\lambda/\mu, \Phi;P)$ is isomorphic to $\mathcal{B}_{s_2} (2,2,0)$
  • ...and 2 more figures

Theorems & Definitions (49)

  • Example 1
  • Theorem 1
  • Example 2
  • Proposition 1
  • Lemma 1
  • Remark 1
  • Proposition 2
  • Definition 1
  • Example 3
  • Example 4
  • ...and 39 more