2019/05/16 by Randriamaro, Hery
#05E15 #81R10 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1905.06813
The quon algebra is an approach to particle statistics introduced by Greenberg in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. We generalize these models by introducing a deformation of the quon algebra generated by a collection of operators \mathttai, i ∈ ℕ^* the set of positive integers, on an infinite dimensional module satisfying the qi,j-mutator relations \mathttai \mathttaj^† - qi,j \mathttaj^† \mathttai = δi,j. The realizability of our model is proved by means of the Aguiar-Mahajan bilinear form on the chambers of hyperplane arrangements. We show that, for suitable values of qi,j, the module generated by the particle states obtained by applying combinations of \mathttai's and \mathttai^†'s to a vacuum state |0⟩ is an indefinite Hilbert module. Furthermore, we refind the extended Zagier's conjecture established independently by Meljanac et al. and by Duchamp et al.