Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 1 standard modules for $\tilde{\mathfrak sl}(\ell+1,\C)$

TL;DR

This paper constructs explicit combinatorial bases for Feigin-Stoyanovsky's type subspaces of level 1 standard modules of affine Lie algebras of type A, using difference and initial conditions and intertwining operators.

Contribution

It provides a new combinatorial basis for these subspaces, extending understanding of their structure and basis construction methods.

Findings

Explicit combinatorial bases for subspaces are obtained.

Linear independence is proved using intertwining operators.

A basis for the entire module is constructed as an inductive limit.

Abstract

Let $\tilde{\mathfrak g}$ be an affine Lie algebra of type $A_\ell^{(1)}$. Suppose we're given a $\mathbb Z$-gradation of the corresponding simple finite-dimensional Lie algebra ${\mathfrak g}={\mathfrak g}_{-1}\oplus{\mathfrak g}_0 \oplus {\mathfrak g}_1$; then we also have the induced $\mathbb Z$-gradation of the affine Lie algebra $$\tilde{\mathfrak g}=\tilde{\mathfrak g}_{-1} \oplus \tilde{\mathfrak g}_0 \oplus \tilde{\mathfrak g}_1.$$ Let $L(\Lambda)$ be a standard module of level 1. Feigin-Stoyanovsky's type subspace $W(\Lambda)$ is the $\tilde{\mathfrak g}_1$-submodule of $L(\Lambda)$ generated by the highest-weight vector $v_\Lambda$, $$W(\Lambda)=U(\tilde{\mathfrak g}_1)\cdot v_\Lambda\subset L(\Lambda).$$ We find a combinatorial basis of $W(\Lambda)$ given in terms of difference and initial conditions. Linear independence of the generating set is proved inductively by using coefficients of intertwining operators. A basis of $L(\Lambda)$ is obtained as an ``inductive limit'' of the basis of $W(\Lambda)$.