Multiplicative formulas in Cohomology of $G/P$ and in quiver representations

Nicolas Ressayre (1)

1 I3M - Institut de Mathématiques et de Modélisation de Montpellier

Consider a partial flag variety $X$ which is not a grassmaninan. Consider also its cohomology ring ${\rm H}^*(X,\ZZ)$ endowed with the base formed by the Poincaré dual classes of the Schubert varieties. In \cite{Richmond:recursion}, E.~Richmond showed that some coefficient structure of the product in ${\rm H}^*(X,\ZZ)$ are products of two such coefficients for smaller flag varieties. Consider now a quiver without oriented cycle. If $\alpha$ and $\beta$ denote two dimension-vectors, $\alpha\circ\beta$ denotes the number of $\alpha$-dimensional subrepresentations of a general $\alpha+\beta$-dimensional representation. In \cite{DW:comb}, H.~Derksen and J.~Weyman expressed some numbers $\alpha\circ\beta$ as products of two smaller such numbers. The aim of this work is to prove two generalisations of the two above results by the same way.

Mots clés: Cohomology; Homogeneous spaces; Quiver representations

Domaines: Géométrie algébrique [math.AG]

Nicolas Ressayre. Multiplicative formulas in Cohomology of $G/P$ and in quiver representations. 2008. ⟨hal-00346257⟩