Question

Is there a good software package for doing computations in the cohomology ring of Grassmannians? Things like, I can write down a polynomial in, in fact, special Schubert classes, but it's one where doing the multiplication out is too tedious for me to have any chance at accuracy in the final answer, and want an efficient way to tell a computer to do it (things that will just multiply pairs, and then you input the next set of pairs don't count).

Answer 1

There's a Littlewood-Richardson calculator here: http://math.rutgers.edu/~asbuch/lrcalc/

I usually use the "SchurRings" package in Macaulay 2 ( http://www.math.uiuc.edu/Macaulay2/ ) though. No particular reason why, just that Macaulay 2 is something I am used to using. It's very easy to use, here's an example (it doesn't print correctly on this page) where the 4 in the first command means use 4 variables (so we're working in Gr(4, infinity)).

i1 : S = schurRing(s,4)

o1 = S

o1 : SchurRing

i2 : s_{2,2} * s_{3,1}

o2 = s + s + s + s + s + s + s

  5,3    5,2,1    4,3,1    4,2,2    4,2,1,1    3,3,2    3,2,2,1

o2 : S

Answer 2

The package Schubsingular.m2 by Alexander Woo, Alexander Yong (available from the authors website) won't do the Schubert calculus for you but it can determine which Schubert varieties are smooth, factorial and Gorenstein by examining pattern avoidance of the defining permutations.

Answer 3

There is something called Symmetrica, which is now bundled with Sage.