  The paper is devoted to the study of some well-knonw combinatorial functions
on the symmetric group $\sn$ --- the major index $\maj$, the descent number
$\des$, and the inversion number $\inv$ --- from the representation-theoretic
point of view. We show that each of these functions generates in the group
algebra the same ideal, and the restriction of the left regular representation
to this ideal is isomorphic to the representation of $\sn$ in the space of
$n\times n$ skew-symmetric matrices. This allows us to obtain formulas for the
functions $\maj,\des,\inv$ in terms of matrices of an exceptionally simple
form. These formulas are applied to find the spectra of the elements under
study in the regular representation, as well as to deduce a series of
identities relating these functions to one another and to the number of fixed
points
  $\fix$.
  \smallskip\noindent
  {\it Keywords:} major index, descent number, inversion number,
representations of the symmetric group, skew-symmetric matrices, dual
complexity.
