Invariant
algebras and major indices for classical Weyl groups
Given a classical Weyl group W,
that is, a Weyl group
of type A, B or D, one
can associate with it a polynomial
with integral coefficients ZW
given
by the ratio of the Hilbert
series of the invariant algebras of the natural
action of W and Wt on the ring of polynomials C[x1, ...,xn]\otimes t.
We introduce and study several statistics on the classical
Weyl groups of type
B and D and show that they can be used to
give an explicit formula for ZDn.
More precisely, we define
two Mahonian statistics, that is, statistics having
the same
distribution as the length function, Dmaj
and ned on Dn.
The statistic Dmaj, 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 Bn;
namely, it allows us to find an
explicit formula for ZDn.
Our proof is based on the theory of t-partite
partitions
introduced by Gordon and further studied by Garsia and Gessel.
Using similar ideas, we define the
Mahonian statistic ned also
on Bn and we
find a new
and simpler proof of the Adin-Roichman formula for ZBn.
Finally, we define a new descent number Ddes on Dn so that
the pair
(Ddes,Dmaj) gives a
generalization to Dn of the
Carlitz identity on the
Eulerian-Mahonian distribution of descent
number and major index
on the symmetric group.
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