Advanced Studies in Pure Mathematics

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

Masaki Nakagawa and Hiroshi Naruse

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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.

Article information

Schubert Calculus — Osaka 2012, H. Naruse, T. Ikeda, M. Masuda and T. Tanisaki, eds. (Tokyo: Mathematical Society of Japan, 2016), 337-417

Received: 20 October 2013
Revised: 6 May 2014
First available in Project Euclid: 4 October 2018

Permanent link to this document euclid.aspm/1538623005

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05E05: Symmetric functions and generalizations 55N20: Generalized (extraordinary) homology and cohomology theories 57T25: Homology and cohomology of H-spaces

Loop spaces Hopf algebras Schur $P$- and $Q$-functions Generalized (co)homology theory Lazard ring


Nakagawa, Masaki; Naruse, Hiroshi. Generalized (co)homology of the loop spaces of classical groups and the universal factorial Schur $P$- and $Q$-functions. Schubert Calculus — Osaka 2012, 337--417, Mathematical Society of Japan, Tokyo, Japan, 2016. doi:10.2969/aspm/07110337.

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