## Multiplication tables for H*(G/P)?

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Hi everyone. My recent work has me developing software to compute in $H^\ast(G/P)$, where $G$ is a complex connected semisimple algebraic group and $P$ is a standard parabolic subgroup (usually, $B$ or a maximal $P$). While my programs are built on sound theory, one can never be too sure. It's always good practice to check your work.

I'm looking for references of multiplication tables of these cohomology rings.

In particular, I'm interested in cases where $G$ is NOT simply-laced (that is, Lie type $B_n$, $C_n$, $F_4$, $G_2$), though simply-laced tables would be nice too. Any tables would depend on a choice of additive basis for $H^\ast(G/P)$. I typically use cohomology classes either Hom-dual or Poincare dual to the usual Schubert varieties living in $G/P$, and I like to parameterize my Schubert varieties with $W^P$, the minimal length coset representatives of $W/W_P$ where $W$ is the Weyl group and $W_P$ is the Weyl group of the Levi associated to $P$. Tables using this convention would be great. Of course, tables in any basis would be fine. :D

Thanks so much.

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## 2 Answers

You may consult with Duan Haibao's papers on arxiv.

Also, I have a Maple realization of my own, available at

http://www.staff.uni-mainz.de/semenov/software.html

Unfortunately, there is no documentation, and the coding style is ugly, but examples may help.

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Actually, I was using Haibao's result's for my program (which I've written in Mathematica). – brandyn Feb 11 2011 at 15:47

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Some people have written code to do this, but not necessarily the most efficient possible way. Alex Yong wrote one that works for general type, and it's linked in the references for the paper by H. Thomas and A. Yong, "A combinatorial rule for (co)minuscule Schubert calculus" (in Adv. Math., or on the arXiv). As for published tables, for small rank some are given in Griffeth-Ram, "Afﬁne Hecke algebras and the Schubert calculus". (I also wrote down the table for $G_2$ in my thesis.) There are probably many other sources, these are just the ones that come to mind.

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