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

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

2008-12-11

arxiv.org/abs/0812.2122v1

Mathematics

Algebraic Geometry

The main result of this note was obtained simultenously by E. Richmond (see arXiv:0812.1856)

Scientific paper

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 Poincar\'e 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.

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