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Symmetric function theory is a key ingredient in the Schubert calculus ofGrassmannians. Quasisymmetric functions are analogues that are similarlycentral to algebraic combinatorics, but for which the associated geometry ispoorly developed. Baker and Richter (2008) showed that $\textrm{QSym}$manifests topologically as the cohomology ring of the loop suspension ofinfinite projective space or equivalently of its combinatorial homotopy model,the James reduced product $J\mathbb{C}\mathbb{P}^\infty$. In recent work, weused this viewpoint to develop topologically-motivated bases of $\textrm{QSym}$and initiate a Schubert calculus for $J\mathbb{C}\mathbb{P}^\infty$ in bothcohomology and $K$-theory.  Here, we study the torus-equivariant cohomology of$J\mathbb{C}\mathbb{P}^\infty$. We identify a cellular basis and introducedouble monomial quasisymmetric functions as combinatorial representatives,analogous to the factorial Schur functions and double Schubert polynomials ofclassical Schubert calculus. We also provide a combinatorialLittlewood--Richardson rule for the structure coefficients of this basis.  Furthermore, we introduce an algebro-geometric analogue of the James reducedproduct construction. In particular, we prove that the James reduced product ofa complex projective variety also carries the structure of a projectivevariety.

James reduced product schemes and double quasisymmetric functions