Browsing by Author "Randriamaro, Hery"

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    A Deformed Quon Algebra
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Randriamaro, Hery
    The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators a_(i,k), (i,k)∈N^∗ × [m], on an infinite dimensional vector space satisfying the deformed q-mutator relations aj,a^(\dag)_(i,k) = q^(\dag)_(i,k)aj,l + q^(β−k,l)δ_(i,j). We prove the realizability of our model by showing that, for suitable values of q, the vector space generated by the particle states obtained by applying combinations of ai,k's and a^(\dag)_(i,k)'s to a vacuum state |0⟩ is a Hilbert space. The proof particularly needs the investigation of the new statistic cinv and representations of the colored permutation group.
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    The Tutte Polynomial of Ideal Arrangements
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Randriamaro, Hery
    The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.
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    The Varchenko determinant of a Coxeter arrangement
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2017) Pfeiffer, Götz; Randriamaro, Hery
    The Varchenko determinant is the determinant of a matrix defined from an arrangement of hyperplanes. Varchenko proved that this determinant has a beautiful factorization. It is, however, not possible to use this factorization to compute a Varchenko determinant from a certain level of complexity. Precisely at this point, we provide an explicit formula of this determinant for the hyperplane arrangements associated to the finite Coxeter groups. The intersections of hyperplanes with the chambers of such arrangements have nice properties which play a central role for the calculation of their relating determinants.