The Kostant cone and the combinatorial structure of the root lattice

Abstract

Let R be an irreducible root system with base B, weight lattice ,4, fundamental dominant weights D with respect to B, dominant weights /i + with respect to D, root lattice A,, and A,? the nonnegative integral sums of members of B. Then /i,’ C$ /i +, but A, c A. Let Z, be the nonnegative integers. Let x E AT. Then x has coordinates in Z; as a root sum with respect to B, and coordinates in Z” as a weight with respect to D. The difference of these coordinates y, in Z”, plays a key role in the study of Kostant’s partition function, the deeper combinatorial study, now required in various areas of mathematics and physics, of the representation theory of complex semisimple Lie algebras, and the modular version of this representation theory. If YEZ”,, i.e., with all nonnegative entries, then x is a sum of positive nonsimple roots, a member of the Kostant cone of A,?, so called because all the calculations of Kostant’s partition function, P, in A,? can be done in the Kostant cone, by means of an easy operator, s, on A,?, defined in terms of y. If y E z; , but with all strictly positive entries, then x is in the interior of the Kostant cone; otherwise it is on the boundary. For xE,4,f, let E+(x) be a realization of the partition function, i.e., the set of all expressions of x as a sum of positive roots. The coordinates of any such expression would be in Zy , where m is the number of positive roots of R. This E+(x) is rich in combinatorial structure, most notably i-blocks and zero blocks. The jth i-block of E+(x) is the set of expressions in E+(x) whose coefficient of the simple root ai is exactly j. The relations between the i-blocks inside of E+(x), using y, correspond to the relations between certain weight spaces in irreducible modules. Also, y provides the entering wedge into the crucial lowest i-blocks. The zero block of E+(x) is the intersection, over i, of all the zeroth i-blocks of E+(x), i.e., the expressions in 81 0021-8693/89 $3.00