An alternative proof of $\widehat{\mathfrak{sl}}_2'$ standard module semi-infinite structure

Abstract

B. Feigin and A. Stoyanovsky found the basis of semi-infinite monomials in standard $\widehat{\mathfrak{sl}}_2'$-module $L_{(0, 1)}$ with Lefschetz formula for the corresponding flag variety. These semi-infinite monomials are constructed by modes of the current $e(z) = \sum\limits_{n\in\mathbb{Z}} e_n\,z^{- n - 1}$. We give an alternative proof of this fact using explicit fermionic construction of this module. Namely, we realize $L_{(0, 1)}$ inside of the zero-charge subspace of Fermionic Fock space and show linear independence of vectors corresponding to semi-infinite monomials.

Figures (2)

• Figure 1: Weight diagram for $L_{(0, 1)}$$\ch L_{(0, 1)} = \sum\limits_{m\in\mathbb{Z}}\,\frac{z^m\,q^{m^2}}{(q)_{\infty}}$
• Figure A2: Illustration of \ref{['qBinom2']}: any Young diagram corresponding to partition into ${m \: \text{distinct}\: \leq N - 1}$ parts with adjacent differing $\geq 2$ might be obtained by ascribing Young diagram contained in ${m\times (N - 2m)}$ rectangle to $(1, 3, \ldots, 2m - 1)$ shape from the right.

Theorems & Definitions (13)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Lemma 1.1
• Lemma 1.2
• Definition 2.1
• Definition 2.2
• Remark 4.1
• Lemma 4.1
• proof