Grothendieck positivity for normal square root crystals

TL;DR

This paper introduces normal square root crystals, a new combinatorial framework related to symmetric Grothendieck functions, providing a novel proof of Buch's rule for their multiplication.

Contribution

It develops the theory of normal square root crystals and proves their characters are sums of symmetric Grothendieck polynomials, linking crystal theory with K-theoretic symmetric functions.

Findings

Normal square root crystals' characters are sums of symmetric Grothendieck polynomials.

Established a connection between raising operators and Hecke insertion algorithm.

Provided a new combinatorial proof of Buch's multiplication rule for symmetric Grothendieck functions.

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.