Cominuscule tableau combinatorics

Details Cominuscule tableau combinatorics Cominuscule tableau combinatorics

2007-01-08

arxiv.org/abs/math/0701215v1

Mathematics

Combinatorics

10 pages

Scientific paper

We continue our study of "cominuscule tableau combinatorics" by generalizing constructions of M. Haiman, S. Fomin and M.-P. Sch\"{u}tzenberger. In particular, we extend the "dual equivalence" ideas of [Haiman, 1992] to reformulate the generalized Littlewood-Richardson rule for cominuscule G/P Schubert calculus from [Thomas-Yong, 2006]; this reformulation avoids certain arbitrary choices demanded by the original version. Also, we apply dual equivalence to give an alternative and independent proof of the jeu de taquin results of [Proctor, 2004] needed in our earlier work. Further, we extend Fomin's "growth diagram" description of jeu de taquin; the inherent symmetry of these diagrams leads to a generalization of Sch\"{u}tzenberger's "evacuation involution". We think of these results as characteristic of our viewpoint that the tableau combinatorics making the cohomology of Grassmannians attractive should have natural cominuscule generalizations. This principle incorporates the standard maxim that the Young tableau combinatorics of Schur polynomials should have "shifted analogues" for Schur $P-$ and $Q-$ polynomials (although this is often used with representation theory of the symmetric group, rather than geometry of flag manifolds, in mind).

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