Communications in Algebra
We give an explicit (new) morphism of modules between H∗T (G/P)⊗H∗T (P/B) and H∗T (G/B) and prove (the known result) that the two modules are isomorphic. Our map identifies submodules of the cohomology of the flag variety that are isomorphic to each of H∗T (G/P) and H∗ T (P/B). With this identification, the map is simply the product within the ring H∗T (G/B). We use this map in two ways. First we describe module bases for H∗T (G/B) that are different from traditional Schubert classes and from each other. Second we analyze a W-representation on H∗T (G/B) via restriction to subgroups WP. In particular we show that the character of the Springer representation on H∗T (G/B) is a multiple of the restricted representation of WP on H∗T (P/B).
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Drellich, Elizabeth and Tymoczko, Julianna, "A Module Isomorphism Between H∗T (G/P)⊗H∗T (P/B) and H∗T (G/B)" (2014). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.