Shifted Hecke insertion and the K-theory of OG(n,2n + 1)

Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We study shifted Hecke insertion, showing it preserves descent sets and relating it the K-theoretic jeu de taquin of Buch-Samuel and Clifford-Thomas-Yong. As a consequence, we construct symmetric functions that are closely related to Ikeda-Naruse's representatives for the K-theory of the orthogonal Grassmannian. Exploiting this relationship and introducing a shifted K-theoretic Poirier-Reutenauer algebra, we derive a Littlewood-Richardson rule for the K-theory of the orthogonal Grassmannian equivalent to the rules of Clifford-Thomas-Yong and Buch-Samuel. Our methods are independent of the Buch-Ravikumar Pieri rule.