The Grassmannian Gr(n,k) is the set of k-dimensional subspaces in an n-dimensional vector space. For example, the set of lines Gr(n+1,1) is projective space. The real Grassmannian (as well as the complex Grassmannian) are examples of manifolds. For example, the subspace <(1,0,0,0,0),(0,1,0,0,0)> subset R^5 has a neighborhood U subset Gr(5,2). A subspace W=<w_1,w_2> is in U if w_1=(w_(11),w_(12),w_(13),w_(14),w_(15)) and w_2=(w_(21),w_(22),w_(23),w_(24),w_(25)) and w_(11)w_(22)-w_(12)w_(21)!=0. Then for any W in U, the vectors w_1 and w_2 are uniquely determined by requiring w_(11)=1=w_(22) and w_(12)=0=w_(21). The other six entries provide coordinates for U.

In general, the Grassmannian can be given coordinates in a similar way at a point V subset R^n. Let U be the open set of k-dimensional subspaces which project onto V. First one picks an orthonormal basis b_1,...,b_n for R^n such that b_1,...,b_k span V. Using this basis, it is possible to take any k vectors and make a k×n matrix. Doing this for the basis of W, another k-dimensional subspace in U, gives a k×n-matrix, which is well-defined up to linear combinations of the rows. The final step is to row-reduce so that the first k×k block is the identity matrix. Then the last k×(n-k) block is uniquely determined by W. The entries in this block give coordinates for the open set U.

If e_1,...,e_n is the standard basis of V, a basis of  ^ ^(r)V is given by the (n; m) vectors e_(i_1) ^ ... ^ e_(i_m), 1<=i_1<...<i_m<=n. If v_1,...,v_m is a basis of a subspace W of dimension m of V, W corresponds to a point (x_1,...,x_((n; m))) of P_(K^((n; m)-1)), whose coordinates are the components of v_1 ^ ... ^ v_m with respect to the basis of  ^ ^(r)V given above. These coordinates are the so-called Grassmann coordinates of W.

A different choice of the basis of W yields a different m-tuple of coordinates, which differs from the original m-tuple by a nonzero multiplicative constant, hence it corresponds to the same point.

The Grassmannian is also a homogeneous space. A subspace is determined by its basis vectors. The group that permutes basis vectors is Gl(n). The matrix that fixes <e_1,...e_k> is a diagonal block matrix, with a k×k nonsingular matrix in the top left, and a n-k×n-k invertible matrix in the lower right. GL(n) acts transitively on the Grassmannian G(n,k). Let P subset GL(n) be the stabilizer (or isotropy) of span(e_1,...,e_k) in G(n,k). Then P is the subgroup of GL(n) consisting of matrices A=[a_(i,j)] such that a_(i,j)=0 for all i, j such that i>k and j<k+1. G(n,k) is isomorphic to GL(n)/P.

The tangent space to the Grassmannian is given by k×(n-k) matrices, i.e., linear maps from V to the quotient vector space R^n/V.

The elements x_1,...,x_((n; m)) are the m-minors of the m×n matrix whose ith row contains the components of v_i with respect to the basis e_1,...,e_n. It corresponds to a linear transformation T:K^m->K^n whose range is W. In general, the range of such a linear transformation has dimension m iff the corresponding m×n matrix has rank m.

Let U be the subset of P_(K^(mn-1)) defined by the condition that all the m+1-minors of the matrix (x_(ij))_(i=1,...,m,j=1,...,n) (which can be viewed as a sequence of mn coordinates) be equal to zero, and one m-minor be nonzero. The Grassmannian G(n,m,K) can be viewed as the image of the map U->P_(K^((n; m)-1)) which maps each matrix of U to the sequence of its m-minors.

It as an algebraic projective algebraic variety defined by equations called Plücker's equations. It is a nonsingular variety of dimension m(n-m).

See also

Grassmann Manifold, Indecomposable, Manifold, Plücker Embedding, Plücker's Equations, Schubert Variety, Variety

Portions of this entry contributed by Todd Rowland

Portions of this entry contributed by Margherita Barile

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Fulton, W. Schubert Varieties and Degeneracy Loci. New York: Springer-Verlag, 1998.Harris, J. "Grassmannians and Related Varieties." Lecture 6 in Algebraic Geometry: A First Course. New York: Springer-Verlag, pp. 63-71, 1992.Kleiman, S. and Laksov, D. "Schubert Calculus." Amer. Math. Monthly 79, 1061-1082, 1972.Shafarevich, I. R. Basic Algebraic Geometry, Vol. 1, 2nd ed. Berlin: Springer-Verlag, pp. 42-44, 1994.

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Cite this as:

Barile, Margherita; Rowland, Todd; and Weisstein, Eric W. "Grassmannian." From MathWorld--A Wolfram Web Resource.

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