Complete Description of the Voronoï Cell of the Lie Algebra A _n Weight Lattice. On the Bounds for the Number of d-Faces of the n-Dimensional Voronoï Cells

Abstract

Denoting these bounds by N_d(n), 0 ≤ d ≤ n we prove that Nd(n)/(n + 1)! is a polynomial Pd(n) of degree d with rational coefficients. We give the polynomials for d ≤ 5 explicitly. The proof uses the fact that these bounds Nd(n) are also the number of d-faces of the Voronoï cell of the weight lattice of the Lie algebra A_n (it is also the Cayley diagram of the symmetric group S_n+1, which is isomorphic to the Weyl group of A_n). Each d-face of this cell is a zonotope that can be defined by a symmetry group ~ G _d (α), (d-dimensional reflection subgroup of the A _n Weyl group. We show that for a given d and n large enough, all such subgroups of A _n are represented, and we compute explicitly N(G _d (α),n) the number of d-faces of type G _d (α) in the Voronoï cell of L = A ^w _n. The final result is obtained by summing over a. That also yields the simple expression $Nd(n) = (n + 1 - d)!S_{n + 1}^{(n + 1 - d)}$ where the last symbol is the Stirling number of second kind.

This paper was received in August 1997.