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I don't know much about Schubert calculus, but I would like to know all of the intersection numbers $$ \#(X_{w_1} \cap X_{w_2} \cap X_{w_3}) $$ where $X_w$ indicates a Schubert variety (maybe translated) in $\mathrm{GL}_5/B$, $w_1, w_2, w_3$ are in $S_5$ and $w_1 w_2 w_3$ is the long element of $S_5$.

There are a priori too many such numbers to fit in a table (medium-sized fraction of 120^3), but maybe there are some symmetries and redundancies that cut it down?

Failing that, is there some computer code that will tell me all of these numbers?

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A package to compute intersection numbers in Macaulay2 is Schubert2 (which has its full documentation here). Sage has a similar package, and a forthcoming implementation called Schubert3.

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  • $\begingroup$ Carlo, "Computations in intersection theory" goes to what looks like a short abstract for a talk Grayson gave. Is that what you meant to link to? $\endgroup$
    – ya-tayr
    Sep 5 '13 at 14:57
  • $\begingroup$ apologies, I removed that link, which was not as informative as I had hoped. $\endgroup$ Sep 5 '13 at 15:21
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I haven't used it, but the documentation for the Sage package lrcalc suggests it can do this, and even gives an $Fl(5)$ example.

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