Gelfand-Zetlin basis, whittaker vectors and a bosonic formula for the sl_n+1 principal subspace

We derive a bosonic formula for the character of the principal space in the level k vacuum module for 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 Whittaker vectors for Uv ±1 (gl_n+1). In this paper we compute these scalar products in bosonic form, using the decomposition of 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.