Grothendieck positivity for normal square root crystals

TL;DR

This work develops a combinatorial framework of normal square root crystals to establish Grothendieck positivity for their characters. By modeling normal $\sqrt{\mathfrak{gl}_n}$-crystals via set-valued words and a rectification operator, the authors prove that ch$(\mathcal{B}) = \sum_{b \in \mathsf{HW}(\mathcal{B})} G_{\mathrm{wt}(b)}(x_1,\dots,x_n)$, connecting highest-weight structure to symmetric Grothendieck polynomials through a novel use of Hecke insertion. This bridge yields new proofs of Buch’s combinatorial rule for multiplying symmetric Grothendieck polynomials, a positive expansion for skew Grothendieck functions, and Grothendieck-positivity results for permutation-indexed Grothendieck functions. The framework thus links crystal theory with $K$-theoretic Schubert calculus, offering avenues to shifted/queer analogues and deeper representation-theoretic interpretations with potential broad impact on positivity phenomena in algebraic combinatorics.

Abstract

Normal crystals (also known as Stembridge crystals) are commonly used to establish the Schur positivity of symmetric functions, as their characters are sums of Schur polynomials. In this paper, we develop a combinatorial framework for a novel family of objects called normal square root crystals, which are closely related to symmetric Grothendieck functions, the $K$-theoretic analogue of Schur functions. Among other applications, this tool leads to a new proof of Buch's combinatorial rule for the multiplication of symmetric Grothendieck functions. The definition of a normal square root crystal, originally formulated by the first two authors, largely mirrors that of normal crystals. Our main result is to show that the character of such a crystal is always a sum of symmetric Grothendieck polynomials. The proof relies on an unexpected connection between the raising operators for our crystals and the Hecke insertion algorithm developed by Buch, Kresch, Shimozono, Tamvakis, and Yong.

Figures (3)

• Figure 1: The standard $\mathfrak{gl}_n$-crystal, for which the weight map is $\mathrm{wt}(\boxed{i})=\mathbf{e}_i\in\mathbb{Z}^n$.
• Figure 2: The standard $\sqrt{\mathfrak{gl}_3}$-crystal, for which the weight map is $\mathrm{wt}(S) = \sum_{i \in S} \mathbf{e}_i \in \mathbb{Z}^3$.
• Figure 3: The polynomial $G\space P_{(m)}(x_1,\dots,x_n)$ is the weight-generating function for all fillings $T$ of a one-row Young diagram with $m$ boxes by nonempty subsets of $\{1'<1<2'<2<\dots<n'<n\}$ such that $\max(T_{1j}) \leq \min(T_{1,j+1})$ for all $j \in[m-1]$, with strict inequality if $\max(T_{1j})$ is primed IkedaNaruse. One can inductively construct a weight-preserving bijection between such tableaux and the union $\bigsqcup_{i=k}^m \mathsf{SetTab}_{(i, 1^{m-i})}\sqcup \bigsqcup_{i=k+1}^m \mathsf{SetTab}_{(i, 1^{m+1-i})}$ for $k=\max\{1,m-n-1\}$. Rather than presenting a formal definition, we show an example of this bijection to illustrate the main idea.

Theorems & Definitions (61)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3
• proof
• Corollary 1.4: Buch2002
• proof
• Remark 1.5
• Corollary 1.6: Buch2002
• proof
• Definition 1.7