Generalized (co)homology of the loop spaces of classical groups and the universal factorial Schur $P$- and $Q$-functions

Abstract

In this paper, we study the generalized (co)homology Hopf algebras of the loop spaces on the infinite classical groups, generalizing the work due to Kono-Kozima and Clarke. We shall give a description of these Hopf algebras in terms of symmetric functions. Based on topological considerations in the first half of this paper, we then introduce a universal analogue of the factorial Schur $P$- and $Q$-functions due to Ivanov and Ikeda-Naruse. We investigate various properties of these functions such as the cancellation property, which we call the $\mathbb{L}$-supersymmetric property, the factorization property, and the vanishing property. We prove that the universal analogue of the Schur $P$-functions form a formal basis for the ring of symmetric functions with the $\mathbb{L}$-supersymmetric property. By using the universal analogue of the Cauchy identity, we then define the dual universal Schur $P$- and $Q$-functions. We describe the duality of these functions in terms of Hopf algebras.