A Multiparametric Quon Algebra

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 $\mathtt{a}_i$, $i \in \mathbb{N}^*$ the set of positive integers, on an infinite dimensional module satisfying the $q_{i,j}$-mutator relations $\mathtt{a}_i \mathtt{a}_j^{\dag} - q_{i,j}\, \mathtt{a}_j^{\dag} \mathtt{a}_i = \delta_{i,j}$. 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 $q_{i,j}$, the module generated by the particle states obtained by applying combinations of $\mathtt{a}_i$'s and $\mathtt{a}_i^{\dag}$'s to a vacuum state $|0\rangle$ is an indefinite Hilbert module. Furthermore, we refind the extended Zagier's conjecture established independently by Meljanac et al. and by Duchamp et al.