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

摘要

Denoting these bounds by Nd(n), 0 < or = d < or = n, the authors prove thatNd(n)/(n + 1) is a polynomial Pd(n) of degree d with rational coefficients. The authors give explicitly the polynomials for d < or = 5. The proof uses the fact that these bounds Nd(n) are also the number of d-faces of the Voronoi cell of the weight lattice of the Lie algebra An (it is also the Cayley diagram of the symmetric group S(sub n + 1) which is isomorphic to the Weyl group of An). Each d-face of this cell is a zonotope which can be defined by a symmetry group approximately = Gd(alpha), d-dimensional reflection subgroup of the An Weyl group. The authors show that for a given d and n large enough, all such subgroups of An are represented and the authors compute explicitly N(Gd(alpha),n), the number of d-faces of type Gd(alpha) in the Voronoi cell of L = A(sup omega, sub n). The final result is obtained by summoning over alpha. That also yields the