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

TL;DR

This paper presents a new bosonic formula for the character of the principal space in the level k vacuum module of fsl_{n+1}, derived from fermionic formulas and Gelfand-Zetlin basis decompositions, revealing a quasi-classical structure.

Contribution

It introduces a bosonic formula for the principal space character of fsl_{n+1} modules, connecting fermionic formulas with Gelfand-Zetlin basis computations.

Findings

Derived a bosonic formula from fermionic formulas.

Expressed scalar products of Whittaker vectors in bosonic form.

Showed the bosonic formula as a quasi-classical limit of the fermionic formula.

Abstract

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.