Quasisymmetric Schubert calculus

Abstract

The ring of symmetric functions occupies a central place in algebraic combinatorics, with a particularly notable role in Schubert calculus, where the standard cell decompositions of Grassmannians yield the celebrated family of Schur functions and the cohomology ring is governed by Littlewood-Richardson rules. The past 50 years have seen an analogous development of quasisymmetric function theory, with applications to enumerative combinatorics, Hopf algebras, graph theory, representation theory, and other areas. Despite such successes, this theory has lacked a quasisymmetric analogue of Schubert calculus. In particular, there has been much interest, since work of Lam and Pylyavskyy (2007), in developing "$K$-theoretic" analogues of quasisymmetric function theory, for which a major obstacle has been the lack of topological interpretations. Here, building on work of Baker and Richter (2008), we apply the philosophy of Schubert calculus to the loop space $\Omega(\Sigma(\mathbb{C}\mathbb{P}^\infty))$ through the homotopy model given by James reduced product $J(\mathbb{C}\mathbb{P}^\infty)$. We describe a canonical Schubert cell decomposition of $J(\mathbb{C}\mathbb{P}^\infty)$, yielding a canonical basis of its cohomology, which we explicitly identify with monomial quasisymmetric functions. Our constructions apply equally to James reduced products of generalized flag varieties $G/P$, and we show how Littlewood-Richardson rules for any $G/P$ lift to $H^*(J(G/P))$. If $J(\mathbb{C}\mathbb{P}^\infty)$ carried the structure of a normal projective algebraic variety, the structure sheaves of the cell closures would yield a "cellular $K$-theory" Schubert basis. We show this is impossible. Nonetheless, we introduce and study a more subtle $K$-theory Schubert basis. We characterize this $K$-theory ring and develop quasisymmetric representatives with an explicit combinatorial description.