Publications of the Research Institute for Mathematical Sciences


Full-Text PDF (308 KB) | Metadata | Table of Contents | PRIMS summary
Volume 47, Issue 2, 2011, pp. 535–551
DOI: 10.2977/PRIMS/42

Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace

Boris Feigin[1], Michio Jimbo[2] and Tetsuji Miwa[3]

(1) Independent University of Moscow, Russian Federation
(2) Rikkyo University, Tokyo, Japan
(3) Kyoto University, Japan

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. %for $U_{v^{\pm1}}(\mathfrak{gl}_{n+1})$ % modules computing it %by using the decomposition %of the Whittaker vectors in the Gelfand-Zetlin basis. %We show that the bosonic formula obtained in this way %is the quasi-classical decomposition of the fermionic formula.

Keywords: difference Toda Hamiltonian, quantum groups, fermionic formulas, bosonic formulas

Feigin Boris, Jimbo Michio, Miwa Tetsuji: Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the ${\mathfrak{sl}}_{n+1}$ Principal Subspace. Publ. Res. Inst. Math. Sci. 47 (2011), 535-551. doi: 10.2977/PRIMS/42