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

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 $W(\Lambda )$.

22 pages, 7 figures, AMS-LaTeX

References in corpus (1)

• Intertwining vertex operators and certain representations of sl(n)^

Cited by in corpus (1)

• Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 2 standard modules for $D_4^{(1)}$