A combinatorial rule for (co)minuscule Schubert calculus

Details A combinatorial rule for (co)minuscule Schubert calculus A combinatorial rule for (co)minuscule Schubert calculus

2006-08-11

arxiv.org/abs/math/0608276v2

Adv. Math. 222 (2009), no. 2, 596--620.

Mathematics

Algebraic Geometry

23 pages. Companion software available at the authors' websites. v2 is the submitted version, with typo-corrections

Scientific paper

We prove a root system uniform, concise combinatorial rule for Schubert calculus of_minuscule_ and_cominuscule_ flag manifolds G/P (the latter are also known as "compact Hermitian symmetric spaces"). We connect this geometry to the poset combinatorics of [Proctor '04], thereby giving a generalization of the [Sch\"{u}tzenberger `77]_jeu de taquin_ formulation of the Littlewood-Richardson rule that computes the intersection numbers of Grassmannian Schubert varieties. Our proof introduces_cominuscule recursions_, a general technique to relate the numbers for different Lie types. A discussion about connections of our rule to (geometric) representation theory is also briefly entertained.

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