Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 1 standard modules for $\tilde{\mathfrak sl}(\ell+1,\C)$
Abstract
Description
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 $L(Λ)$ be a standard module of level 1. Feigin-Stoyanovsky's type subspace $W(Λ)$ is the $\tilde{\mathfrak g}_1$-submodule of $L(Λ)$ generated by the highest-weight vector $v_Λ$, $$W(Λ)=U(\tilde{\mathfrak g}_1)\cdot v_Λ\subset L(Λ).$$ We find a combinatorial basis of $W(Λ)$ 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 $L(Λ)$ is obtained as an ``inductive limit'' of the basis of $W(Λ)$.
22 pages, 7 figures, AMS-LaTeX
22 pages, 7 figures, AMS-LaTeX