Gelfand–Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace

Abstract

We derive a bosonic formula for the character of the principal space in the level $k$ vacuum module for ${\widehat{\mathfrak{sl}}}_{n+1}$, starting from a known fermionic formula for it. In our previous work, the latter was written as a sum consisting of Shapovalov scalar products of the Whittaker vectors for $U_{v^{\pm 1}}({\mathfrak{gl}}_{n+1})$. In this paper we compute these scalar products in the bosonic form, using the decomposition of the Whittaker vectors in the Gelfand–Zetlin basis. We show further that the bosonic formula obtained in this way is the quasi-classical decomposition of the fermionic formula.