@article{626c910a1f1e42ffbd4b5bac4577d705,

title = "A combinatorial rule for (co)minuscule Schubert calculus",

abstract = "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, thereby giving a generalization of Sch{\"u}tzenberger's 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.",

keywords = "Algebraic combinatorics, Littlewood-Richardson rules, Minuscule Schubert varieties, Schubert calculus",

author = "Hugh Thomas and Alexander Yong",

note = "Funding Information: We thank Kevin Purbhoo and Frank Sottile for inspiring conversations about [23]. We are most grateful to Allen Knutson for bringing to our attention relations of our work to representation theory and for many other comments, corrections and insights on this text. We would also like to thank Prakash Belkale, Mark Haiman, Joseph Landsberg, Bernd Sturmfels, Terry Tao and Alexander Woo for helpful discussions, and John Stembridge for supplying code to compute Schur P -, Q-products as well as answering our questions about its theory. We would also like to thank the referees for insightful comments. H.T. was partially supported by an NSERC Discovery grant. A.Y. was partially supported by NSF grant DMS-0601010. This work was partially completed during a visit by H.T. to the University of Minnesota, supported by a McKnight grant awarded to Victor Reiner, and separately, while A.Y. was an NSERC sponsored visitor at the Fields Institute in Toronto. Finally, A.Y. would like to thank the Institute for Pure and Applied Mathematics at UCLA, where exceptional hospitality greatly facilitated the writing of this paper during an NSF supported visit of April–June, 2006.",

year = "2009",

month = oct,

day = "1",

doi = "10.1016/j.aim.2009.05.008",

language = "English (US)",

volume = "222",

pages = "596--620",

journal = "Advances in Mathematics",

issn = "0001-8708",

publisher = "Academic Press Inc.",

number = "2",

}