Invariant algebras and major indices for classical Weyl groups.

Given a classical Weyl group $W$, that is, a Weyl group of type $A$</formtex>, $B$</formtex> or $D$</formtex>, one can associate with it a polynomial with integral coefficients $Z_W$</formtex> given by the ratio of the Hilbert series of the invariant algebras of the natural action of $W$</formtex> and $W^t$</formtex> on the ring of polynomials ${\bf C}[x_1, \ldots , x_n]^{\otimes t}$</formtex>. We introduce and study several statistics on the classical Weyl groups of type $B$</formtex> and $D$</formtex> and show that they can be used to give an explicit formula for $Z_{D_n}$</formtex>. More precisely, we define two Mahonian statistics, that is, statistics having the same distribution as the length function, $Dmaj$</formtex> and $ned$</formtex> on $D_n$</formtex>. The statistic $Dmaj$</formtex>, defined in a combinatorial way, has an analogous algebraic meaning to the major index for the symmetric group and the flag-major index of Adin and Roichman for $B_n$</formtex>; namely, it allows us to find an explicit formula for $Z_{D_n}$</formtex>. Our proof is based on the theory of $t$</formtex>-partite partitions introduced by Gordon and further studied by Garsia and Gessel. Using similar ideas, we define the Mahonian statistic $ned$</formtex> also on $B_n$</formtex> and we find a new and simpler proof of the Adin–Roichman formula for $Z_{B_n}$</formtex>. Finally, we define a new descent number $Ddes$</formtex> on $D_n$</formtex> so that the pair $(Ddes,Dmaj)$</formtex> gives a generalization to $D_n$</formtex> of the Carlitz identity on the Eulerian–Mahonian distribution of descent number and major index on the symmetric group. [ABSTRACT FROM AUTHOR]