Grassmannian

In mathematics,Grassmannian, often denoted G_k,n isspaceall k-dimensional subspacesan n-dimensional vector space. That is, forfixed field K, we can consideran n-dimensional vector space V,setsuch subspaces,appropriate extra structure (of topological space, homogeneous space, differential manifold or algebraic variety),notice that up to appropriate isomorphisms, we havewell-defined geometric object forgiven pair (n,k).

The simplest exampleG_1,n,setall lines (through zero), also known asprojective space. In euclidean 3-space,planecompletely characterized byoneonly line perpendicularit (and vice-versa); hence G_2,3isomorphicG_1,3.

Supposing first that K isreal number or complex number field,easiest approachGrassmanniansprobablyconsider them as homogeneous spaces. That is,group actionGL(V) onk-dimensional subspaces hassingle orbit, asshownlinear algebra. The stabilizer HK^kKⁿ, embedded usingfirst k co-ordinates, can be identified quickly asblock matrices defined bycondition a_ij = 0i = 1kj > k. We can therefore identify G_k,n ascoset space H/GL(Kⁿ). This then providestopology onGrassmannian, andsmooth structure.

There can be other approaches:example orthogonal groups also act transitively, so thatGrassmannians also appear as coset spacesthose groups. This shows directly thatreal Grassmannianscompact (forsame resultcomplex Grassmannians one appliesunitary group). This representation might also be preferredhomotopy theory.

Incase ofgeneral field K, something similar can be donealgebraic groupstheir cosets. Then Grassmannians can be shownbe projective varieties. Explicit homogeneous coordinatesknown,come fromk-th exterior power: applywedge product tobasis ofk-dimensional subspace andresulting k-vectorwell-defined, up toscalar multiple. It follows thatequations definingGrassmannian can be regarded asidentities satisfied by kxk minors.

For an exampleuseGrassmanniansdifferential geometry, see Gauss map.

Schubert cells

The detailed study ofGrassmannians usesdecomposition into subsets called Schubert cells, which were first appliedenumerative geometry. The Schubert cellsG_k,ndefinedtermsan auxiliary flag: take subspaces V₁, V₂, ..., V_k,V_i containedV_i+1. Then we considercorresponding subsetG_k,n, consisting ofW having intersectionV_idimension at least i,i = 1k.