Generalized \(Q\)-functions and \(UC\) hierarchy of \(B\)-type. (English) Zbl 1195.05080
A generalization of the Schur \(Q\)-function is introduced. This generalization, called generalized \(Q\)-function, is indexed by any pair of strict partitions, and can be expressed by the Pfaffian. A connection to the theory of integrable systems is clarified. Firstly, the bilinear identities satisfied by the generalized \(Q\)-function are given and proved to be equivalent to a system of partial differential equations of infinite order. The system is called the \(UC\) hierarchy of \(B\)-type(\(BUC\) hierarchy). Secondly, the aigebraic structure of the \(BUC\) hierarchy is investigated from the representation theoretic viewpoint. Some new kind of the boson-fermion correspondence is established, and a representation of an infinite dimensional Lie algebra, denoted by \({\frac go}_{2\infty}\), is obtained. The bilinear identities are translated to the language of neutral fermions, which turn out to characterize a \(G\)-orbit of the vacuum vector, where \(G\) is the group corresponding to \({\frac go}_{2\infty}\).
Reviewer: Victor Sharapov (Volgograd)
MSC:
| 05E05 | Symmetric functions and generalizations |
| 05E10 | Combinatorial aspects of representation theory |
| 17B80 | Applications of Lie algebras and superalgebras to integrable systems |
| 37Kxx | Dynamical system aspects of infinite-dimensional Hamiltonian and Lagrangian systems |
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