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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 and 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?

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You can prove similar symmetric distribution statements using $h$-vectors of polytopes (see e.g. Corollary 1.4 of arxiv.org/abs/1104.5292). –  Sam Hopkins Jul 30 at 3:18
    
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. Since any geometric approach that does generalize in this sense would provide such a genealization, I would rather hope for a geometric interpretation that is strongly attached to the combinatorics/geometry of the symmetric group. –  Christian Stump Jul 30 at 18:10
    
@ChristianStump, If you want you can upgrade your comment to an answer (I would accept it :-)). What I had in mind was to compute cohomology in different ways (perhaps localize with respect to different group actions or something along those lines), but since this wouldn't distinguish the symmetric group among other Weyl/Coxeter groups, I am less hopeful because of the reasons you mention. –  Gjergji Zaimi Oct 24 at 4:06

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up vote 1 down vote accepted

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.
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