5 research outputs found
Integer Representations of the Generalized Symmetric Groups
In this paper, we construct a mixed-base number system over the generalized
symmetric group , which is a complex reflection group with a root
system of type . We also establish one-to-one correspondence between
all positive integers in the set and the elements of
by constructing the subexceedant function in relation to this group.
In addition, we provide a new enumeration system for by defining the
inversion statistic on . Finally, we prove that the
\textit{flag-major index} is equi-distributed with this inversion statistic on
. Therefore, the flag-major index is Mahonian on with
respect to the length function
An Inversion Statistic on the Hyperoctahedral Group
In this paper, we introduce an inversion statistic on the hyperoctahedral
group by using an decomposition of a positive root system of this
reflection group. Then we prove some combinatorial properties for the inversion
statistic. We establish an enumeration system on the group and give an
efficient method to uniquely derive any group element known its enumeration
order with the help of the inversion table. In addition, we prove that the
\textit{flag-major index} is equi-distributed with this inversion statistic on