×

Standard bases for affine \(\text{SL}(n)\)-modules. (English) Zbl 1083.17009

The canonical basis and crystal base approaches to understanding representations of simple and Kac-Moody Lie algebras have yielded important combinatorial information but are notorious for the difficulty and tediousness of explicit computations. Here, the authors offer a new tool, using what they call a roof operator to derive a standard basis for the basic representation of \(\hat {\mathfrak{sl}}_n\) and its Demazure modules. The advantage of the authors’ approach is the ease of explicit computation, and they offer an extended example to illustrate.
The idea is to associate semi-infinite sets to the vertices of the crystal graph and use the roof operator to move each vertex to an extremal weight vector. However, the action of the operator is not along the edges of the crystal graph, and therein lies the gain in computational simplicity.

MSC:

17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
PDFBibTeX XMLCite
Full Text: DOI arXiv