The recursive nature of cominuscule Schubert calculus

• Autores: Kevin Purbhoo, Frank Sottile
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 217, Nº 5, 2008, págs. 1962-2004
• Idioma: inglés
• DOI: 10.1016/j.aim.2007.09.010
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• Resumen

  • The necessary and sufficient Horn inequalities which determine the non-vanishing Littlewood�Richardson coefficients in the cohomology of a Grassmannian are recursive in that they are naturally indexed by non-vanishing Littlewood�Richardson coefficients on smaller Grassmannians. We show how non-vanishing in the Schubert calculus for cominuscule flag varieties is similarly recursive. For these varieties, the non-vanishing of products of Schubert classes is controlled by the non-vanishing products on smaller cominuscule flag varieties. In particular, we show that the lists of Schubert classes whose product is non-zero naturally correspond to the integer points in the feasibility polytope, which is defined by inequalities coming from non-vanishing products of Schubert classes on smaller cominuscule flag varieties. While the Grassmannian is cominuscule, our necessary and sufficient inequalities are different than the classical Horn inequalities.