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Schubert Variety

A class of subvarieties of the Grassmannian G(n,m,K). Given m integers 1<=a_1<...<a_m<=n, the Schubert variety Omega(a_1,...,a_m) is the set of points of G(n,m,K) representing the m-dimensional subspaces W of K^n such that, for all i=1,...,m,

dim_K(W intersection <e_(n-a_i+1),...,e_n>)>=i.

It is a projective algebraic variety of dimension

sum_(i=1)^ma_i-1/2m(m+1).

See also

Algebraic Variety, Projective Algebraic Variety

This entry contributed by Margherita Barile

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References

Chipalkatti, J. V. "Notes on Grassmannians and Schubert Varieties." Queen's Papers in Pure and Applied Math. 13, No. 119, 2001. http://server.maths.umanitoba.ca/~jaydeep/Papers/grassmannians.pdf.Fulton, W. Schubert Varieties and Degeneracy Loci. New York: Springer-Verlag, 1998.Hodge, W. V. D. and Pedoe, D. Methods of Algebraic Geometry, Vol. 2. Cambridge, England: Cambridge University Press, 1952.Kleiman, S. and Laksov, D. "Schubert Calculus." Amer. Math. Monthly 79, 1061-1082, 1972.

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Schubert Variety

Cite this as:

Barile, Margherita. "Schubert Variety." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SchubertVariety.html

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