Geodesic
Classification of Ding's Schubert Varieties: Finer Rook Equivalence
Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 36-62

Voir la notice de l'article provenant de la source Cambridge University Press

K. Ding studied a class of Schubert varieties ${{X}_{\lambda }}$ in type A partial flag manifolds, indexed by integer partitions $\text{ }\!\!\lambda\!\!\text{ }$ and in bijection with dominant permutations. He observed that the Schubert cell structure of ${{X}_{\lambda }}$ is indexed by maximal rook placements on the Ferrers board ${{B}_{\lambda \text{ }}}$ , and that the integral cohomology groups ${{H}^{*}}\left( {{X}_{\lambda }};\,\mathbb{Z} \right),\,{{H}^{*}}\left( {{X}_{\mu }};\,\mathbb{Z} \right)$ are additively isomorphic exactly when the Ferrers boards ${{B}_{\lambda \text{ }}}$ , ${{B}_{\mu }}$ satisfy the combinatorial condition of rook-equivalence.We classify the varieties ${{X}_{\lambda }}$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.
DOI : 10.4153/CJM-2007-002-9
Mots-clés : 14M15, 05E05, Schubert variety, rook placement, Ferrers board, flag manifold, cohomology ring, nilpotence 
Develin, Mike; Martin, Jeremy L.; Reiner, Victor. Classification of Ding's Schubert Varieties: Finer Rook Equivalence. Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 36-62. doi: 10.4153/CJM-2007-002-9
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