James reduced product schemes and double quasisymmetric functions

Abstract

Symmetric function theory is a key ingredient in the Schubert calculus of Grassmannians. Quasisymmetric functions are analogues that are similarly central to algebraic combinatorics, but for which the associated geometry is poorly developed. Baker and Richter (2008) showed that QSym manifests topologically as the cohomology ring of the loop suspension of infinite projective space or equivalently of its combinatorial homotopy model, the James reduced product JCP∞. In recent work, we used this viewpoint to develop topologically-motivated bases of QSym and initiate a Schubert calculus for JCP∞ in both cohomology and K-theory.Here, we study the torus-equivariant cohomology of JCP∞. We identify a cellular basis and introduce double monomial quasisymmetric functions as combinatorial representatives, analogous to the factorial Schur functions and double Schubert polynomials of classical Schubert calculus. We also provide a combinatorial Littlewood–Richardson-type rule for the structure coefficients of this basis.Furthermore, we introduce an algebro-geometric analogue of the James reduced product construction. In particular, we prove that the James reduced product of a complex (quasi-)projective variety also carries the structure of a (quasi-)projective variety.