%0 Journal Article
%T Gelfand-Zetlin basis, Whittaker vectors and a bosonic formula for the $\sln$ principal subspace
%A B. Feigin
%A M. Jimbo
%A T. Miwa
%J Mathematics
%D 2009
%I arXiv
%X 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 $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$. 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.
%U http://arxiv.org/abs/0907.2045v1