Séminaire Lotharingien de Combinatoire, 95B.16 (2026), 12 pp.
Eric Marberg, Kam Hung Tong and Tianyi Yu
Grothendieck Positivity for Square Root Crystals
Abstract.
Normal crystals (also called Stembridge crystals) are a standard tool for proving Schur positivity, since their characters are sums of Schur polynomials.
In this extended abstract, we develop an analogous combinatorial framework for symmetric Grothendieck polynomials, the K-theoretic counterparts of Schur polynomials.
Specifically, we study certain directed graphs called square root crystals, whose properties mirror classical normal crystals.
Our main result is that the character of any normal square root crystal is a sum of symmetric Grothendieck polynomials.
This provides a systematic method for establishing positive symmetric Grothendieck expansions, including a new proof of Buch's K-theoretic Littlewood-Richardson rule.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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