Combinatorial constructions motivated by K-theory of Grassmannians

Abstract

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought to as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map in these cases. Next, using the Hecke insertion of Buch-Kresch-Shimozono-Tamvakis-Yong and the K-Knuth equivalence of Buch-Samuel in place of the Robinson-Schensted and Knuth equivalence, we introduce a K-theoretic analogue of the Poirier-Reutenauer Hopf algebra of standard Young tableaux. As an application, we rederive the K-theoretic Littlewood-Richardson rules of Thomas-Yong and of Buch-Samuel. Lastly, we define a K-theoretic analogue of Fomin's dual graded graphs, which we call dual filtered graphs. They key formula in this definition is DU-UD=D+I. Our major examples are K-theoretic analogues of Young's lattice, of shifted Young's lattice, and of the Young-Fibonacci lattice. We suggest notions of tableaux, insertion algorithms, and growth rules whenever such objects are not already present in the literature. We also provide a large number of other examples. Most of our examples arise via two constructions, which we call the Pieri construction and the Mobius construction. The Pieri construction is closely related to the construction of dual graded graphs from a graded Hopf algebra as described by Bergeron-Lam-Li, Lam-Shimozono, and Nzeutchap. The Mobius construction is more mysterious but also potentially more important as it corresponds to natural insertion algorithms.