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- W4306292802 abstract "The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of standard monomials for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of GL(n). This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases." @default.
- W4306292802 created "2022-10-15" @default.
- W4306292802 creator A5062755935 @default.
- W4306292802 date "2015-08-11" @default.
- W4306292802 modified "2023-09-27" @default.
- W4306292802 title "Accessible Proof of Standard Monomial Basis for Coordinatization of Schubert Sets of Flags" @default.
- W4306292802 doi "https://doi.org/10.48550/arxiv.1508.02744" @default.
- W4306292802 hasPublicationYear "2015" @default.
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