BKP tau-functions as square roots of KP tau-functions

Abstract

It is well-known that a BKP tau-function is the square root of a certain KP tau-function, provided one puts the even KP times equal to zero. In this paper we compute for all polynomial BKP tau-function its corresponding KP ”square”. We also give, in the polynomial case, a representation theoretical proof of a recent result by Alexandov, viz. that a KdV tau-function becomes a BKP tau-function when one divides all KdV times by 2.

The generalized Giambelli formula and polynomial KP and CKP tau-functions

“…In the conclusion of the paper we compare our results on polynomial tau-functions of the CKP hierarchy with that of the BKP hierarchy, found in [13,16,20].…”

“…. , a k ), and are of the form (20), where d = ι C (c), c consists of the first b 1 rows of c, and ι C stands for changing the sign of even numbered rows of the matrix c; in addition, the matrix c must satisfy the constraint (75) in section 5 (which holds for c = 0).…”

Extended Schur's $Q$-functions and the full Kostant--Toda hierarchy on the Lie algebra of type $D$

“…The Schur-type functions also appear in the τ -functions of the KP-type hierarchies [24,23,13,9,19]. In particular, You [24] gives the polynomial τ -functions of the BKP hierarchy in terms of Schur's Q functions, which were first introduced by Schur in the study of projective representations of symmetric group.…”