The Path Model for Representations of Symmetrizable Kac-Moody Algebras

Abstract

In the theory of finite-dimensional representations of complex reductive algebraic groups, the group GL n (C) is singled out by the fact that besides the usual language of weight lattices, roots, and characters, there exists an additional important com-binatorial tool: the Young tableaux. For example, the sum over the weights of all tableaux of a fixed shape is the character of the corresponding representation, and the Littlewood-Richardson rule describes the decomposition of tensor products of c7L n (C)-modules purely in terms of the combinatorie of these Young tableaux. The advantage of this type of formula is (for example compared to Steinberg's formula to decompose tensor products) that there is no cancellation of terms. This makes it much easier (and sometimes even possible) to prove for example that certain representations occur in a given tensor product. To construct objects like the tableaux in a more general setting, consider the weight lattice X of a complex symmetrizable Kac-Moody algebra g, and denote by n the set of all piecewise linear paths IT : [0, 1]Q —> XQ starting in 0 and ending in an integral weight. We associate to a simple root a linear operators e a and f a on the free Z-module Zn spanned by n. Let A C End^ Zn be the subalgebra generated by these operators. Fix 7T G n such that the image is completely contained in the dominant Weyl chamber. The .A-module Air C Zn generated by 7T is a "model" for the irreducible, integrable representation VA of Q of highest weight À = TT(1): for example, the sum over the endpoints of all paths in AIT is the character of V\, and the Littlewood-Richardson rule can be generalized in a straightforward way. As an application, one gets a purely combinatorial proof of the P-R-V conjecture. So the paths can be viewed in a natural way as a generalization of Young tableaux to the setting of symmetrizable Kac-Moody algebras. Though the theory of the path modules is completely independent of the theory of quantum groups, the path modules are strongly related to the crystal graph of representations of the q-analog of the enveloping algebra of g. In fact, they can be viewed as a geometric realization of the theory of crystals of representations. The path model for representations of symmetrizable Kac-Moody algebras 299 1 Review on the GL n (C)-case Let T be the maximal torus of GL n (C) of diagonal matrices, and denote by e^ : T —» C* the projection of a diagonal matrix onto its zth entry. The irreducible polynomial representations of GL n (C) are in bijection with the dominant weights X+ of the form A = piei + • • • + p n e m where pi > P2 > • • • > p n > 0-A Young diagram of shape A G X+ is a left-justified sequence of rows of boxes such that there are pi boxes in the first row, j J 2 boxes in the second row, etc. By a Young tableau T of shape A we mean a filling of the boxes of the diagram with numbers {1,... ,?i}. T is called semi-standard if the entries are strictly increasing in the columns and nondecreasing in the rows. A word w in the alphabet {1,... , n} is a finite sequence w = ii%2 ... i s with 1 < ii,... ,i s <8> V^ ~ ® Vx +; ,(r) ; where the sum runs over all semi-standard Young tableaux of shape p that are X-dominant.