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We derive a bosonic formula for the character of the principal space in thelevel $k$ vacuum module for $\widehat{\mathfrak{sl}}_{n+1}$, starting from aknown fermionic formula for it. In our previous work, the latter was written asa sum consisting of Shapovalov scalar products of the Whittaker vectors for$U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$. In this paper we compute these scalarproducts in the bosonic form, using the decomposition of the Whittaker vectorsin the Gelfand-Zetlin basis. We show further that the bosonic formula obtainedin this way is the quasi-classical decomposition of the fermionic formula.

Gelfand-Zetlin basis, Whittaker vectors and a bosonic formula for the $\sln$ principal subspace