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

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

2009-07-12

arxiv.org/abs/0907.2045v1

Mathematics

Quantum Algebra

Scientific paper

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.

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