Integer representations of classical Weyl groups

In this paper, we define a mixed-base number system over a Weyl group Dn , the group of even-signed permutations. We introduce one-to-one correspondence between the positive integers of the set {1, · · · , 2n−1n!} and elements of this group, after constructing the subexceedant function associated with the group. Thus, the integer representations of all the classical Weyl groups are now completed. Furthermore, we present an inversion statistic on the group Dn by using a decomposition of a positive root system of this reflection group. This inversion statistic is compatible with the length function on the group Dn . Then we derive some combinatorial properties for the inversion statistic. In addition, we prove that the D-major index is equi-distributed with this inversion statistic on Dn . Finally, we propose a public-key cryptosystem based on both the generalized hidden discrete logarithm problem and the integer representation over the group Dn .