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

It is well-known that a B type Kadomtsev–Petviashvili (BKP) tau-function is the square root of a certain Kadomtsev–Petviashvili (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 Alexandrov, viz that a Korteweg–de Vries (KdV) tau-function becomes a BKP tau-function when one divides all KdV times by 2.