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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

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