## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, J.-P. Bourguignon, M. Kotani, Y. Maeda and N. Tose, eds. (Tokyo: Mathematical Society of Japan, 2009), 309 - 319

### Generalized Q-functions and UC hierarchy of B-type

#### Abstract

We define a generalization of Schur's Q-function for an arbitrary pair of strict partitions, which is called the *generalized Q-function*. We prove that all the generalized Q-functions solve a series of non-linear differential equations called the *UC hierarchy of B-type* (*BUC hierarchy*). We furthermore investigate the BUC hierarchy from the viewpoint of representation theory. We consider the Fock representation of the algebra of neutral fermions and establish the boson-fermion correspondence. Using this, we discuss the relationship between the BUC hierarchy and a certain infinite dimensional Lie algebra.

#### Article information

**Dates**

Received: 8 February 2007

Revised: 14 July 2008

First available in Project Euclid:
28 November 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1543447918

**Digital Object Identifier**

doi:10.2969/aspm/05510309

**Mathematical Reviews number (MathSciNet)**

MR2463507

**Zentralblatt MATH identifier**

1196.17012

**Keywords**

Schur's Q-functions BKP hierarchy universal characters UC hierarchy

#### Citation

Ogawa, Yuji. Generalized Q-functions and UC hierarchy of B-type. Noncommutativity and Singularities: Proceedings of French–Japanese symposia held at IHÉS in 2006, 309--319, Mathematical Society of Japan, Tokyo, Japan, 2009. doi:10.2969/aspm/05510309. https://projecteuclid.org/euclid.aspm/1543447918