## Advanced Studies in Pure Mathematics

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

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

Masaki Nakagawa and Hiroshi Naruse

#### 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.

#### Article information

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

**Dates**

Received: 20 October 2013

Revised: 6 May 2014

First available in Project Euclid:
4 October 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1538623005

**Digital Object Identifier**

doi:10.2969/aspm/07110337

**Mathematical Reviews number (MathSciNet)**

MR3644829

**Zentralblatt MATH identifier**

1378.57045

**Subjects**

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

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aspm/1538623005