Colored Tournaments and Weyl's Denominator Formula

Abstract

The Weyl denominator formulae for the root systems B n , C n and D n are proved by associating to each term in the expansion of the product a colored tournament with handicaps at the vertices and then demonstrating by a direct involution on these handicapped colored tournaments that all terms cancel except those in the summation side of the denominator formulae. This generalizes I. Gessel's proof of the Vandermonde determinant formula.