  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.
