The actual question I want to ask is whether there is a geometric proof of this famous identity
$$\sum\_{\sigma \in S\_n} q^{\operatorname{inv} \sigma}=\sum\_{\sigma\in S\_n}q^{\operatorname{maj}\sigma},$$
along the lines of interpreting both sides as the Poincare polynomial of some nice variety computed in two different ways.


Here $\operatorname{inv}$ and $\operatorname{maj}$ stand for [number of inversions](http://en.wikipedia.org/wiki/Permutation#Inversions) and [Major index](http://en.wikipedia.org/wiki/Major_index), respectively.


One possible candidate variety would be the full flag variety, where LHS can be easily interpreted as the generating function for the dimensions of the Schubert cells. Is there a way to see the major index in that context?



Is there a geometric meaning of the Major index?
On @Gjergji Zaimi's request, I turn my comment into an answer (and extend it a bit):


There were many (as far as I know unsuccessful) attempts to understand the major index in a way that makes it possible to uniformly generalize it to other finite Weyl/Coxeter groups, see e.g. <http://arxiv.org/abs/math/0002245> and <http://www.emis.de/journals/SLC/wpapers/s61Abiazen.pdf> and the references therein.


Observe that any geometric approach that does generalize in this sense would provide such a generalized definition. I would therefore rather hope for a geometric interpretation that is strongly attached to the combinatorics/geometry of the symmetric group.


Finally: when not looking at the complete symmetric group $\mathcal{S}\_n$ but only on **noncrossing permutations**, or, equivalently, on **Dyck paths**, the major index is related to **MacMahon's q-Catalan numbers**, see <http://arxiv.org/abs/0808.2822>.


There is a uniform generalization thereof as a bigraded Hilbert series of certain graded modules over the **rational Cherednik algebra** for any **well-generated complex reflection group**. Those can be found in <http://arxiv.org/abs/math/0208138> and in <http://www.maths.ed.ac.uk/~igordon/pubs/diag3.pdf>.


But it is not at all clear if


* these notions can be used to define a statistic on noncrossing partitions (as a subset of all elements in a finite Weyl group),
* can be extended to notions of statistics on the complete finite Weyl group, or
* have anything to do with the weak length function on finite Weyl groups.


