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Shifted Hecke insertion and the \(K\)-theory of \(\mathrm{OG}(n,2n+1)\). (English) Zbl 1366.05118
Summary: R. Patrias and P. Pylyavskyy [in: Proceedings of the 27th international conference on formal power series and algebraic combinatorics, FPSAC 2015, Daejeon, South Korea, July 6–10, 2015. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). 1–12 (2015; Zbl 1335.05201)] 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 A. Skovsted Buch and M. J. Samuel [J. Reine Angew. Math. 719, 133–171 (2016; )] and E. Clifford et al. [J. Reine Angew. Math. 690, 51–63 (2014; Zbl 1348.14127)]. As a consequence, we construct symmetric functions that are closely related to Ikeda-Naruse’s representatives [T. Ikeda and H. Naruse, Adv. Math. 243, 22–66 (2013; Zbl 1278.05240)] 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 [loc. cit.] and Buch-Samuel [loc. cit.]. Our methods are independent of the A. Skovsted Buch and R. Ravikumar [J. Reine Angew. Math. 668, 109–132 (2012; Zbl 1298.14059)] rule.

MSC:
05E10 Combinatorial aspects of representation theory
16T05 Hopf algebras and their applications
16W50 Graded rings and modules (associative rings and algebras)
57T15 Homology and cohomology of homogeneous spaces of Lie groups
14M15 Grassmannians, Schubert varieties, flag manifolds
14N15 Classical problems, Schubert calculus
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