Séminaires : Groupes, Représentations et Géométrie

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Following some work of Aluffi-Mihalcea-Schürmann-Su for the CSM classes of Schubert cells and some elaborate computer calculations by R. Rimanyi and L. Mihalcea, I conjecture that the CSM classes of the Richardson cells expressed in the Schubert basis have nonnegative coefficients. This conjecture was principally motivated by a new product $\square$ coming from the Segre classes in the cohomology of flag varieties (such that the associated Gr of this product is the standard cup product) and the conjecture at that time that the structure constants of this new product $\square$ in the standard Schubert basis have alternating sign behavior. I prove that this sign alternating behavior of the structure constants of $\square$ follows from my above positivity conjecture about the CSM classes of Richardson cells. In a recent work, Schürmann-Simpson-Wang have proved the above mentioned alternating sign behavior for the product $\square$.