Complete description of the Voronoï cell of the Lie Algebras An weight lattice. On the Bounds for the number of d-faces of the n-dimensional Voronoï cells

https://repo-archives.ihes.fr/FONDS_IHES/I_Prepublications/MICHEL/1990-1999/P_97_53/P_97_53.pdf

Michel, Louis

MICHEL

07/1997

Version étendue d'une conférence faite le 9 janvier 1997 au Centre de Recherches Mathématiques de l'Université de Montréal au cours du colloque pour le 60e anniversaire de Jiri Patera et Pavel Winternitz.

Abstract :Denoting these bounds by Nd(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 An (it is also the Cayley diagram of the symmetric group Sn+1, which is isomorphic to the Weyl group of An). Each d-face of this cell is a zonotope that can be defined by a symmetry group ~ G d (?), (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 (?),n) the number of d-faces of type G d (?) 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?d)n+1Nd(n)=(n+1?d)!Sn+1(n+1?d) where the last symbol is the Stirling number of second kind.

ALGEBRES DE LIE

P/97/53

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MICHEL, “Complete description of the Voronoï cell of the Lie Algebras An weight lattice. On the Bounds for the number of d-faces of the n-dimensional Voronoï cells,” Archives de l'IHES, consulté le 20 juin 2024, https://omeka.ihes.fr/document/P_97_53.pdf.