JournalsprimsVol. 47, No. 2pp. 535–551

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

  • Boris Feigin

    Independent University of Moscow, Russian Federation
  • Michio Jimbo

    Rikkyo University, Tokyo, Japan
  • Tetsuji Miwa

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

We derive a bosonic formula for the character of the principal space in the level kk vacuum module for sl^n+1\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 Uv±1(gln+1)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 Uv±1(gln+1)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.

Cite this article

Boris Feigin, Michio Jimbo, Tetsuji Miwa, Gelfand-Zetlin basis, Whittaker Vectors and a Bosonic Formula for the sln+1{\mathfrak{sl}}_{n+1} Principal Subspace. Publ. Res. Inst. Math. Sci. 47 (2011), no. 2, pp. 535–551

DOI 10.2977/PRIMS/42