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Concept
Schubert variety
Summary
In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain kinds of subspaces V, specified using linear algebra, inside a fixed vector subspace W. Here W may be a vector space over an arbitrary field, though most commonly over the complex numbers.
A typical example is the set X whose points correspond to those 2-dimensional subspaces V of a 4-dimensional vector space W, such that V non-trivially intersects a fixed (reference) 2-dimensional subspace W2:
Over the real number field, this can be pictured in usual xyz-space as follows. Replacing subspaces with their corresponding projective spaces, and intersecting with an affine coordinate patch of , we obtain an open subset X° ⊂ X. This is isomorphic to the set of all lines L (not necessarily through the origin) which meet the x-axis. Each such line L corresponds to a point of X°, and continuously moving L in space (while keeping contact with the x-axis) corresponds to a curve in X°. Since there are three degrees of freedom in moving L (moving the point on the x-axis, rotating, and tilting), X is a three-dimensional real algebraic variety. However, when L is equal to the x-axis, it can be rotated or tilted around any point on the axis, and this excess of possible motions makes L a singular point of X.
More generally, a Schubert variety is defined by specifying the minimal dimension of intersection between a k-dimensional V with each of the spaces in a fixed reference flag , where . (In the example above, this would mean requiring certain intersections of the line L with the x-axis and the xy-plane.)
In even greater generality, given a semisimple algebraic group G with a Borel subgroup B and a standard parabolic subgroup P, it is known that the homogeneous space X = G/P, which is an example of a flag variety, consists of finitely many B-orbits that may be parametrized by certain elements of the Weyl group W.
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Related courses (1)
MATH-314: Representation theory of finite groups
Group representation theory studies the actions of groups on vector spaces. This allows the use of linear algebra to study certain group theoretical questions. In this course the groups in question wi