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VOL. 71 | 2016 Generalized (co)homology of the loop spaces of classical groups and the universal factorial Schur $P$- and $Q$-functions
Masaki Nakagawa, Hiroshi Naruse

Editor(s) Hiroshi Naruse, Takeshi Ikeda, Mikiya Masuda, Toshiyuki Tanisaki


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.


Published: 1 January 2016
First available in Project Euclid: 4 October 2018

zbMATH: 1378.57045
MathSciNet: MR3644829

Digital Object Identifier: 10.2969/aspm/07110337

Primary: 05E05 , 55N20 , 57T25

Keywords: Generalized (co)homology theory , Hopf algebras , Lazard ring , loop spaces , Schur $P$- and $Q$-functions

Rights: Copyright © 2016 Mathematical Society of Japan


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