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In this paper, we study the generalized (co)homology Hopf algebras of theloop spaces on the infinite classical groups, generalizing the work due toKono-Kozima and Clarke. We shall give a description of these Hopf algebras interms of symmetric functions. Based on topological considerations in the firsthalf of this paper, we then introduce a universal analogue of the factorialSchur $P$- and $Q$-functions due to Ivanov and Ikeda-Naruse. We investigatevarious properties of these functions such as the cancellation property, whichwe 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 functions with the$\mathbb{L}$-supersymmetric property. By using the universal analogue of theCauchy identity, we then define the dual universal Schur $P$- and$Q$-functions. We describe the duality of these functions in terms of Hopfalgebras.

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