## Pechenik, Oliver and Yong, Alexander - Genomic Tableaux and Combinatorial $K$-Theory

dmtcs:2482 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Genomic Tableaux and Combinatorial $K$-Theory

Authors: Pechenik, Oliver and Yong, Alexander

We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant $K$-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of [Anderson-Griffeth-Miller ’11]. We thereby deduce an earlier conjecture of [Thomas-Yong ’13] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) $K$-theory of Grassmannians. From this perspective, we also obtain a new rule for $K$-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians.

Source : oai:HAL:hal-01337842v1
Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Submitted on: November 21, 2016
Keywords: Schubert calculus,equivariant $K$-theory,Grassmannians,genomic tableaux,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]