arXiv subjects
Publications and source records attributed to Hery Randriamaro.
In this work, we generalize the integer enumeration basis. We also construct bijections between the elements of special sets and the elements of some groups, and treat the special case of the hyperoctohedral groups. Then, we find a code analogous to the Lehmer code for the hyperoctahedral groups.
The descent algebra of finite Coxeter groups is studied by many famous mathematicians like Bergeron, Brown, Howlett, or Reutenauer. Blessenohl, Hohlweg, and Schocker, for example, proved a symmetry property of the descent algebra, when it is linked to the representation theory of its Coxeter group. The interest is particularly showed for the descent algebra of symmetric group. Thibon determined the eigenvalues and their multiplicities of the action on the group algebra of symmetric group of the descent algebra element, which is the sum over all permutations weighted by q^maj. And even the author diagonalized the matrix of the action of the descent algebra element, which is the sum over all permutations weighted by the new introduced statistic desX. In this article, we give a more general result by determining the eigenvalues and their multiplicities of the action on the group algebra of finite Coxeter group of an element of its descent algebra.
In the work of Varchenko, Zagier, Thibon, and Reiner, Saliola, Welker, linear algebraic properties of the multiplication map on the group algebra of the group algebra element are studied, which is the sum over all permutations weighted by q^{inv}, q^{maj}, inv. Here q is a variable, and inv and maj are the classical statistics inversion and major index. We define a multinomial descent statistic des_X and a multinomial inversion statistic inv_X. These new defined statistics are the multinomial expressions of the classical statistics descent des and inversion. We determine the spectrum and the multiplicity of each element of the spectrum of the analogously defined multiplication map on the group algebra for both des_X and inv_X. As corollaries we deduce the spectrum and the multiplicity of each element of the spectrum of the defined multiplication map on the group algebra for the statistics des, maj and inv.