244 research outputs found

    Enumeration of Standard Young Tableaux

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    A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction

    Crystal approach to affine Schubert calculus

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    We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the type-AA affine Weyl group and produce a crystal reflecting the internal structure of the generalized Young modules whose Frobenius image is represented by stable Schubert polynomials. We apply the crystal framework to products of a Schur function with a kk-Schur function, consequently proving that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for Cn\mathbb C^n enumerate the highest weight elements under these operators. Included in this class are the Schubert structure constants in the (quantum) product of a Schubert polynomial with a Schur function sλs_\lambda for all ∣λ∨∣<n|\lambda^\vee|< n. Another by-product gives a highest weight formulation for various fusion coefficients of the Verlinde algebra and for the Schubert decomposition of certain positroid classes.Comment: 42 pages; version to appear in IMR

    Signed Young Modules and Simple Specht Modules

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    By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are parametrized by certain pairs of partitions. The main result of this article establishes the signed Young module labels of simple Specht modules. Along the way we prove a number of results concerning indecomposable signed Young modules that are of independent interest. In particular, we determine the label of the indecomposable signed Young module obtained by tensoring a given indecomposable signed Young module with the sign representation. As consequences, we obtain the Green vertices, Green correspondents, cohomological varieties, and complexities of all simple Specht modules and a class of simple modules of symmetric groups, and extend the results of Gill on periodic Young modules to periodic indecomposable signed Young modules.Comment: To appear in Adv. Math. 307 (2017) 369--416. Proposition 4.3 (F4), (F5) corrected, Lemma 4.9 adjusted accordingl
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