Schubert unions in Grassmann varieties

descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Research 01 Nov 2007Embargo end date: 01 Jan 2005 English Publisher:Elsevier BVJournal:Finite Fields and Their Applications, volume 13, pages 738-750 (issn: 1071-5797,

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Authors: Johan P. Hansen; Trygve Johnsen; Kristian Ranestad;

doi: 10.1016/j.ffa.2007.06.003 , 10.48550/arxiv.math/0503121

arXiv: math/0503121

Schubert unions in Grassmann varieties

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Abstract

We study subsets of Grassmann varieties G(l,m) over a field F, such that these subsets are unions of Schubert cycles, with respect to a fixed flag. We study the linear spans of, and in case of positive characteristic, the number of points on such unions over finite fields. Moreover we study a geometric duality of such unions, and give a combinatorial interpretation of this duality. We discuss the maximum number of points over a finite field for the Schubert unions of a given spanning dimension, and we give some applications to coding theory. We define Schubert union codes, and study the parameters and support weights of these codes and of the well-known Grassmann codes.

37 pages, 2 figures

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Keywords

Schubert cycles;, Applications to coding theory and cryptography of arithmetic geometry, Schubert varieties, Grassmannians, Schubert varieties, flag manifolds, hypersurfaces, Theoretical Computer Science, Finite ground fields in algebraic geometry, Mathematics - Algebraic Geometry, FOS: Mathematics, Schubert cycles, Mathematics - Combinatorics, Algebraic Geometry (math.AG), Engineering(all), flag manifolds, Algebra and Number Theory, finite ground fields, Applied Mathematics, Grassmann codes, 14M15 (05E15, 94B27), Combinatorics (math.CO), Grassmannians, applications to coding theory and cryptography

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