Grassmannian

By giving a collection of subspaces of a vector space a topological structure, it is possible to talk about a continuous choice of subspaces or open and closed collections of subspaces. Giving them the further structure of a differential manifold, one can talk about smooth choices of subspace.

A natural example comes from tangent bundles of smooth manifolds embedded in a Euclidean space. Suppose we have a manifold ${\displaystyle M}$ of dimension ${\displaystyle k}$ embedded in ${\displaystyle \mathbf {R} ^{n}}$. At each point ${\displaystyle x\in M}$, the tangent space to ${\displaystyle M}$ can be considered as a subspace of the tangent space of ${\displaystyle \mathbf {R} ^{n}}$, which is also just ${\displaystyle \mathbf {R} ^{n}}$. The map assigning to x its tangent space defines a map from M to ${\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}$. (In order to do this, we have to translate the tangent space at each ${\displaystyle x\in M}$ so that it passes through the origin rather than ${\displaystyle x}$, and hence defines a ${\displaystyle k}$-dimensional vector subspace. This idea is very similar to the Gauss map for surfaces in a 3-dimensional space.)

This can with some effort be extended to all vector bundles over a manifold ${\displaystyle M}$, so that every vector bundle generates a continuous map from ${\displaystyle M}$ to a suitably generalised Grassmannian—although various embedding theorems must be proved to show this. We then find that the properties of our vector bundles are related to the properties of the corresponding maps. In particular we find that vector bundles inducing homotopic maps to the Grassmannian are isomorphic. Here the definition of homotopy relies on a notion of continuity, and hence a topology.

For k = 1, the Grassmannian Gr(1, n) is the space of lines through the origin in n-space, so it is the same as the projective space ${\displaystyle \mathbf {P} ^{n-1}}$of n − 1 dimensions.

For k = 2, the Grassmannian is the space of all 2-dimensional planes containing the origin. In Euclidean 3-space, a plane containing the origin is completely characterized by the one and only line through the origin that is perpendicular to that plane (and vice versa); hence the spaces Gr(2, 3), Gr(1, 3), and P² (the projective plane) may all be identified with each other.

The simplest Grassmannian that is not a projective space is Gr(2, 4).

To endow ${\displaystyle \mathbf {Gr} _{k}(V)}$ with the structure of a differentiable manifold, choose a basis for ${\displaystyle V}$. This is equivalent to identifying ${\displaystyle V}$ with ${\displaystyle K^{n}}$, with the standard basis denoted ${\displaystyle (e_{1},\dots ,e_{n})}$, viewed as column vectors. Then for any ${\displaystyle k}$-dimensional subspace ${\displaystyle w\subset V}$, viewed as an element of ${\displaystyle \mathbf {Gr} _{k}(V)}$, we may choose a basis consisting of ${\displaystyle k}$ linearly independent column vectors ${\displaystyle (W_{1},\dots ,W_{k})}$. The homogeneous coordinates of the element ${\displaystyle w\in \mathbf {Gr} _{k}(V)}$ consist of the elements of the ${\displaystyle n\times k}$ maximal rank rectangular matrix ${\displaystyle W}$ whose ${\displaystyle i}$-th column vector is ${\displaystyle W_{i}}$, ${\displaystyle i=1,\dots ,k}$. Since the choice of basis is arbitrary, two such maximal rank rectangular matrices ${\displaystyle W}$ and ${\displaystyle {\tilde {W}}}$ represent the same element ${\displaystyle w\in \mathbf {Gr} _{k}(V)}$ if and only if

${\displaystyle {\tilde {W}}=Wg}$

for some element ${\displaystyle g\in GL(k,K)}$ of the general linear group of invertible ${\displaystyle k\times k}$ matrices with entries in ${\displaystyle K}$. This defines an equivalence relation between ${\displaystyle n\times k}$ matrices ${\displaystyle W}$ of rank ${\displaystyle k}$, for which the equivalence classes are denoted ${\displaystyle [W]}$.

We now define a coordinate atlas. For any ${\displaystyle n\times k}$ homogeneous coordinate matrix ${\displaystyle W}$, we can apply elementary column operations (which amounts to multiplying ${\displaystyle W}$ by a sequence of elements ${\displaystyle g\in GL(k,K)}$) to obtain its reduced column echelon form. If the first ${\displaystyle k}$ rows of ${\displaystyle W}$ are linearly independent, the result will have the form

${\displaystyle {\begin{bmatrix}1\\&1\\&&\ddots \\&&&1\\a_{1,1}&\cdots &\cdots &a_{1,k}\\\vdots &&&\vdots \\a_{n-k,1}&\cdots &\cdots &a_{n-k,k}\end{bmatrix}}}$

and the ${\displaystyle (n-k)\times k}$ affine coordinate matrix ${\displaystyle A}$ with entries ${\displaystyle (a_{ij})}$ determines ${\displaystyle w}$. In general, the first ${\displaystyle k}$ rows need not be independent, but since ${\displaystyle W}$ has maximal rank ${\displaystyle k}$, there exists an ordered set of integers ${\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}$ such that the ${\displaystyle k\times k}$ submatrix ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ whose rows are the ${\displaystyle (i_{1},\ldots ,i_{k})}$-th rows of ${\displaystyle W}$ is nonsingular. We may apply column operations to reduce this submatrix to the identity matrix, and the remaining entries uniquely determine ${\displaystyle w}$. Hence we have the following definition:

For each ordered set of integers ${\displaystyle 1\leq i_{1}<\cdots <i_{k}\leq n}$, let ${\displaystyle U_{i_{1},\dots ,i_{k}}}$ be the set of elements ${\displaystyle w\in \mathbf {Gr} _{k}(V)}$ for which, for any choice of homogeneous coordinate matrix ${\displaystyle W}$, the ${\displaystyle k\times k}$ submatrix ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ whose ${\displaystyle j}$-th row is the ${\displaystyle i_{j}}$-th row of ${\displaystyle W}$ is nonsingular. The affine coordinate functions on ${\displaystyle U_{i_{1},\dots ,i_{k}}}$ are then defined as the entries of the ${\displaystyle (n-k)\times k}$ matrix ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ whose rows are those of the matrix ${\displaystyle WW_{i_{1},\dots ,i_{k}}^{-1}}$ complementary to ${\displaystyle (i_{1},\dots ,i_{k})}$, written in the same order. The choice of homogeneous ${\displaystyle n\times k}$ coordinate matrix ${\displaystyle W}$ in ${\displaystyle [W]}$ representing the element ${\displaystyle w\in \mathbf {Gr} _{k}(V)}$ does not affect the values of the affine coordinate matrix ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ representing w on the coordinate neighbourhood ${\displaystyle U_{i_{1},\dots ,i_{k}}}$. Moreover, the coordinate matrices ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ may take arbitrary values, and they define a diffeomorphism from ${\displaystyle U_{i_{1},\dots ,i_{k}}}$ to the space of ${\displaystyle K}$-valued ${\displaystyle (n-k)\times k}$ matrices. Denote by

${\displaystyle {\hat {A}}^{i_{1},\dots ,i_{k}}:=W(W_{i_{1},\dots ,i_{k}})^{-1}}$

the homogeneous coordinate matrix having the identity matrix as the ${\displaystyle k\times k}$ submatrix with rows ${\displaystyle (i_{1},\dots ,i_{k})}$ and the affine coordinate matrix ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ in the consecutive complementary rows. On the overlap ${\displaystyle U_{i_{1},\dots ,i_{k}}\cap U_{j_{1},\dots ,j_{k}}}$ between any two such coordinate neighborhoods, the affine coordinate matrix values ${\displaystyle A^{i_{1},\dots ,i_{k}}}$ and ${\displaystyle A^{j_{1},\dots ,j_{k}}}$ are related by the transition relations

${\displaystyle {\hat {A}}^{i_{1},\dots ,i_{k}}W_{i_{1},\dots ,i_{k}}={\hat {A}}^{j_{1},\dots ,j_{k}}W_{j_{1},\dots ,j_{k}},}$

where both ${\displaystyle W_{i_{1},\dots ,i_{k}}}$ and ${\displaystyle W_{j_{1},\dots ,j_{k}}}$ are invertible. This may equivalently be written as

${\displaystyle {\hat {A}}^{j_{1},\dots ,j_{k}}={\hat {A}}^{i_{1},\dots ,i_{k}}({\hat {A}}_{j_{1},\dots ,j_{k}}^{i_{1},\dots ,i_{k}})^{-1},}$

where ${\displaystyle {\hat {A}}_{j_{1},\dots ,j_{k}}^{i_{1},\dots ,i_{k}}}$ is the invertible ${\displaystyle k\times k}$ matrix whose ${\displaystyle l}$th row is the ${\displaystyle j_{l}}$th row of ${\displaystyle {\hat {A}}^{i_{1},\dots ,i_{k}}}$. The transition functions are therefore rational in the matrix elements of ${\displaystyle A^{i_{1},\dots ,i_{k}}}$, and ${\displaystyle \{U_{i_{1},\dots ,i_{k}},A^{i_{1},\dots ,i_{k}}\}}$ gives an atlas for ${\displaystyle \mathbf {Gr} _{k}(V)}$ as a differentiable manifold and also as an algebraic variety.

An alternative way to define a real or complex Grassmannian as a manifold is to view it as a set of orthogonal projection operators (Milnor & Stasheff (1974) problem 5-C). For this, choose a positive definite real or Hermitian inner product ${\displaystyle \langle \cdot ,\cdot \rangle }$ on ${\displaystyle V}$, depending on whether ${\displaystyle V}$ is real or complex. A ${\displaystyle k}$-dimensional subspace ${\displaystyle w}$ determines a unique orthogonal projection operator ${\displaystyle P_{w}:V\rightarrow V}$ whose image is ${\displaystyle w\subset V}$ by splitting ${\displaystyle V}$ into the orthogonal direct sum

${\displaystyle V=w\oplus w^{\perp }}$

of ${\displaystyle w}$ and its orthogonal complement ${\displaystyle w^{\perp }}$ and defining

${\displaystyle P_{w}(v)={\begin{cases}v\quad {\text{ if }}v\in w\\0\quad {\text{ if }}v\in w^{\perp }.\end{cases}}}$

Conversely, every projection operator ${\displaystyle P}$ of rank ${\displaystyle k}$ defines a subspace ${\displaystyle w_{P}:=\mathrm {Im} (P)}$ as its image. Since the rank of an orthogonal projection operator equals its trace, we can identify the Grassmann manifold ${\displaystyle \mathbf {Gr} (k,V)}$ with the set of rank ${\displaystyle k}$ orthogonal projection operators ${\displaystyle P}$:

${\displaystyle \mathbf {Gr} (k,V)\sim \left\{P\in \mathrm {End} (V)\mid P=P^{2}=P^{\dagger },\,\mathrm {tr} (P)=k\right\}.}$

In particular, taking ${\displaystyle V=\mathbf {R} ^{n}}$ or ${\displaystyle V=\mathbf {C} ^{n}}$ this gives completely explicit equations for embedding the Grassmannians ${\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{N})}$, ${\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{N})}$ in the space of real or complex ${\displaystyle n\times n}$ matrices ${\displaystyle \mathbf {R} ^{n\times n}}$, ${\displaystyle \mathbf {C} ^{n\times n}}$, respectively.

Since this defines the Grassmannian as a closed subset of the sphere ${\displaystyle \{X\in \mathrm {End} (V)\mid \mathrm {tr} (XX^{\dagger })=k\}}$ this is one way to see that the Grassmannian is a compact Hausdorff space. This construction also turns the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$ into a metric space with metric

${\displaystyle d(w,w'):=\lVert P_{w}-P_{w'}\rVert ,}$

for any pair ${\displaystyle w,w'\subset V}$ of ${\displaystyle k}$-dimensional subspaces, where ‖⋅‖ denotes the operator norm. The exact inner product used does not matter, because a different inner product will give an equivalent norm on ${\displaystyle V}$, and hence an equivalent metric.

For the case of real or complex Grassmannians, the following is an equivalent way to express the above construction in terms of matrices.

Let ${\displaystyle M(n,\mathbf {R} )}$ denote the space of real ${\displaystyle n\times n}$ matrices and the subset ${\displaystyle P(k,n,\mathbf {R} )\subset M(n,\mathbf {R} )}$ of matrices ${\displaystyle P\in M(n,\mathbf {R} )}$ that satisfy the three conditions:

• ${\displaystyle P}$ is a projection operator: ${\displaystyle P^{2}=P}$.
• ${\displaystyle P}$ is symmetric: ${\displaystyle P^{T}=P}$.
• ${\displaystyle P}$ has trace ${\displaystyle \operatorname {tr} (P)=k}$.

There is a bijective correspondence between ${\displaystyle P(k,n,\mathbf {R} )}$ and the Grassmannian ${\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{n})}$ of ${\displaystyle k}$-dimensional subspaces of ${\displaystyle \mathbf {R} ^{n}}$ given by sending ${\displaystyle P\in P(k,n,\mathbf {R} )}$ to the ${\displaystyle k}$-dimensional subspace of ${\displaystyle \mathbf {R} ^{n}}$ spanned by its columns and, conversely, sending any element ${\displaystyle w\in \mathbf {Gr} (k,\mathbf {R} ^{n})}$ to the projection matrix

${\displaystyle P_{w}:=\sum _{i=1}^{k}w_{i}w_{i}^{T},}$

where ${\displaystyle (w_{1},\cdots ,w_{k})}$ is any orthonormal basis for ${\displaystyle w\subset \mathbf {R} ^{n}}$, viewed as real ${\displaystyle n}$ component column vectors.

An analogous construction applies to the complex Grassmannian ${\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{n})}$, identifying it bijectively with the subset ${\displaystyle P(k,n,\mathbf {C} )\subset M(n,\mathbf {C} )}$ of complex ${\displaystyle n\times n}$ matrices ${\displaystyle P\in M(n,\mathbf {C} )}$ satisfying

• ${\displaystyle P}$ is a projection operator: ${\displaystyle P^{2}=P}$.
• ${\displaystyle P}$ is self-adjoint (Hermitian): ${\displaystyle P^{\dagger }=P}$.
• ${\displaystyle P}$ has trace ${\displaystyle {\rm {{tr}(P)=k}}}$,

where the self-adjointness is with respect to the Hermitian inner product ${\displaystyle \langle \,\cdot ,\cdot \,\rangle }$ in which the standard basis vectors ${\displaystyle (e_{1},\cdots ,e_{n})}$ are orthonomal. The formula for the orthogonal projection matrix ${\displaystyle P_{w}}$ onto the complex ${\displaystyle k}$-dimensional subspace ${\displaystyle w\subset \mathbf {C} ^{n}}$ spanned by the orthonormal (unitary) basis vectors ${\displaystyle (w_{1},\cdots ,w_{k})}$ is

${\displaystyle P_{w}:=\sum _{i=1}^{k}w_{i}w_{i}^{\dagger }.}$

The quickest way of giving the Grassmannian a geometric structure is to express it as a homogeneous space. First, recall that the general linear group ${\displaystyle \mathrm {GL} (V)}$ acts transitively on the ${\displaystyle k}$-dimensional subspaces of ${\displaystyle V}$. Therefore, if we choose a subspace ${\displaystyle w_{0}\subset V}$ of dimension ${\displaystyle k}$, any element ${\displaystyle w\in \mathbf {Gr} (k,V)}$ can be expressed as

${\displaystyle w=g(w_{0})}$

for some group element ${\displaystyle g\in \mathrm {GL} (V)}$, where ${\displaystyle g}$ is determined only up to right multiplication by elements ${\displaystyle \{h\in H\}}$ of the stabilizer of ${\displaystyle w_{0}}$:

${\displaystyle H:=\mathrm {stab} (w_{0}):=\{h\in \mathrm {GL} (V)\,|\,h(w_{0})=w_{0}\}\subset \mathrm {GL} (V)}$

under the ${\displaystyle \mathrm {GL} (V)}$-action.

We may therefore identify ${\displaystyle \mathbf {Gr} (k,V)}$ with the quotient space

${\displaystyle \mathbf {Gr} (k,V)=\mathrm {GL} (V)/H}$

of left cosets of ${\displaystyle H}$.

If the underlying field is ${\displaystyle \mathbf {R} }$ or ${\displaystyle \mathbf {C} }$ and ${\displaystyle \mathrm {GL} (V)}$ is considered as a Lie group, this construction makes the Grassmannian a smooth manifold under the quotient structure. More generally, over a ground field ${\displaystyle K}$, the group ${\displaystyle \mathrm {GL} (V)}$ is an algebraic group, and this construction shows that the Grassmannian is a non-singular algebraic variety. It follows from the existence of the Plücker embedding that the Grassmannian is complete as an algebraic variety. In particular, ${\displaystyle H}$ is a parabolic subgroup of ${\displaystyle \mathrm {GL} (V)}$.

Over ${\displaystyle \mathbf {R} }$ or ${\displaystyle \mathbf {C} }$ it also becomes possible to use smaller groups in this construction. To do this over ${\displaystyle \mathbf {R} }$, fix a Euclidean inner product ${\displaystyle q}$ on ${\displaystyle V}$. The real orthogonal group ${\displaystyle O(V,q)}$ acts transitively on the set of ${\displaystyle k}$-dimensional subspaces ${\displaystyle \mathbf {Gr} (k,V)}$ and the stabiliser of a ${\displaystyle k}$-space ${\displaystyle w_{0}\subset V}$ is

${\displaystyle O(w_{0},q|_{w_{0}})\times O(w_{0}^{\perp },q|_{w_{0}^{\perp }})}$,

where ${\displaystyle w_{0}^{\perp }}$ is the orthogonal complement of ${\displaystyle w_{0}}$ in ${\displaystyle V}$. This gives an identification as the homogeneous space

${\displaystyle \mathbf {Gr} (k,V)=O(V,q)/\left(O(w,q|_{w})\times O(w^{\perp },q|_{w^{\perp }})\right)}$.

If we take ${\displaystyle V=\mathbf {R} ^{n}}$ and ${\displaystyle w_{0}=\mathbf {R} ^{k}\subset \mathbf {R} ^{n}}$ (the first ${\displaystyle k}$ components) we get the isomorphism

${\displaystyle \mathbf {Gr} (k,\mathbf {R} ^{n})=O(n)/\left(O(k)\times O(n-k)\right).}$

Over C, if we choose an Hermitian inner product ${\displaystyle h}$, the unitary group ${\displaystyle U(V,h)}$ acts transitively, and we find analogously

${\displaystyle \mathbf {Gr} (k,V)=U(V,h)/\left(U(w_{0},h|_{w_{0}})\times U(w_{0}^{\perp }|,h_{w_{0}^{\perp }})\right),}$

or, for ${\displaystyle V=\mathbf {C} ^{n}}$ and ${\displaystyle w_{0}=\mathbf {C} ^{k}\subset \mathbf {C} ^{n}}$,

${\displaystyle \mathbf {Gr} (k,\mathbf {C} ^{n})=U(n)/\left(U(k)\times U(n-k)\right).}$

In particular, this shows that the Grassmannian is compact, and of (real or complex) dimension k(n − k).

In the realm of algebraic geometry, the Grassmannian can be constructed as a scheme by expressing it as a representable functor.[4]

The Plücker embedding[6] is a natural embedding of the Grassmannian ${\displaystyle \mathbf {Gr} (k,V)}$ into the projectivization of the ${\displaystyle k}$th exterior power ${\displaystyle \Lambda ^{k}V}$ of ${\displaystyle V}$.

$\iota$ :\mathbf {Gr} (k,V)\to \mathbf {P} \left(\Lambda ^{k}V\right).

Suppose that ${\displaystyle w\subset V}$ is a ${\displaystyle k}$-dimensional subspace of the ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$. To define ${\displaystyle \iota (w)}$, choose a basis ${\displaystyle (w_{1},\cdots ,w_{k})}$ for ${\displaystyle w}$, and let ${\displaystyle \iota (w)}$ be the projectivization of the wedge product of these basis elements:

$${\displaystyle \iota (w)=[w_{1}\wedge \cdots \wedge w_{k}],}$$

where ${\displaystyle [\,\cdot \,]}$ denotes the projective equivalence class.

A different basis for ${\displaystyle w}$ will give a different wedge product, but the two will differ only by a non-zero scalar multiple (the determinant of the change of basis matrix). Since the right-hand side takes values in the projectivized space, ${\displaystyle \iota }$ is well-defined. To see that it is an embedding, notice that it is possible to recover ${\displaystyle w}$ from ${\displaystyle \iota (w)}$ as the span of the set of all vectors ${\displaystyle v\in V}$ such that

${\displaystyle v\wedge \iota (w)=0}$.

Every ${\displaystyle k}$-dimensional subspace ${\displaystyle W\subset V}$ determines an ${\displaystyle (n-k)}$-dimensional quotient space ${\displaystyle V/W}$ of ${\displaystyle V}$. This gives the natural short exact sequence:

${\displaystyle 0\rightarrow W\rightarrow V\rightarrow V/W\rightarrow 0.}$

Taking the dual to each of these three spaces and the dual linear transformations yields an inclusion of ${\displaystyle (V/W)^{*}}$ in ${\displaystyle V^{*}}$ with quotient ${\displaystyle W^{*}}$

${\displaystyle 0\rightarrow (V/W)^{*}\rightarrow V^{*}\rightarrow W^{*}\rightarrow 0.}$

Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between ${\displaystyle k}$-dimensional subspaces of ${\displaystyle V}$ and ${\displaystyle (n-k)}$-dimensional subspaces of ${\displaystyle V^{*}}$. In terms of the Grassmannian, this gives a canonical isomorphism

${\displaystyle \mathbf {Gr} _{k}(V)\leftrightarrow \mathbf {Gr} {(n-k},V^{*})}$

that associates to each subspace ${\displaystyle W\subset V}$ its annihilator ${\displaystyle W^{0}\subset V^{*}}$. Choosing an isomorphism of ${\displaystyle V}$ with ${\displaystyle V^{*}}$ therefore determines a (non-canonical) isomorphism between ${\displaystyle \mathbf {Gr} _{k}(V)}$ and ${\displaystyle \mathbf {Gr} _{n-k}(V)}$. An isomorphism of ${\displaystyle V}$ with ${\displaystyle V^{*}}$ is equivalent to the choice of an inner product, so with respect to the chosen inner product, this isomorphism of Grassmannians sends any ${\displaystyle k}$-dimensional subspace into its ${\displaystyle (n-k)}$}-dimensional orthogonal complement.

The detailed study of Grassmannians makes use of a decomposition into affine subpaces called Schubert cells, which were first applied in enumerative geometry. The Schubert cells for ${\displaystyle \mathbf {Gr} _{k}(V)}$ are defined in terms of a specified complete flag of subspaces ${\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V}$ of dimension ${\displaystyle \mathrm {dim} (V_{i})=i}$. For any partition

${\displaystyle \lambda =(\lambda _{1},\cdots ,\lambda _{k})}$

of weight

${\displaystyle |\lambda |=\sum _{i=1}^{k}\lambda _{i}}$

consisting of weakly decreasing non-negative integers

${\displaystyle \lambda _{1}\geq \cdots \geq \lambda _{k}\geq 0,}$

whose Young diagram fits within the rectangular one ${\displaystyle (n-k)^{k}}$, the Schubert cell ${\displaystyle X_{\lambda }(k,n)\subset \mathbf {Gr} _{k}(V)}$ consists of those elements ${\displaystyle W\in \mathbf {Gr} _{k}(V)}$ whose intersections with the subspaces ${\displaystyle \{V_{i}\}}$ have the following dimensions

${\displaystyle X_{\lambda }(k,n)=\{W\in \mathbf {Gr} _{k}(V)\,|\,\dim(W\cap V_{n-k+j-\lambda _{j}})=j\}.}$

These are affine spaces, and their closures (within the Zariski topology) are known as Schubert varieties.

As an example of the technique, consider the problem of determining the Euler characteristic ${\displaystyle \chi _{k,n}}$ of the Grassmannian ${\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}$ of k-dimensional subspaces of Rⁿ. Fix a ${\displaystyle 1}$-dimensional subspace ${\displaystyle \mathbf {R} \subset \mathbf {R} ^{n}}$ and consider the partition of ${\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}$ into those k-dimensional subspaces of Rⁿ that contain R and those that do not. The former is ${\displaystyle \mathbf {Gr} _{k-1}(\mathbf {R} ^{n-1})}$ and the latter is a rank ${\displaystyle k}$ vector bundle over ${\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n-1})}$. This gives recursive formulae:

${\displaystyle \chi _{k,n}=\chi _{k-1,n-1}+(-1)^{k}\chi _{k,n-1},\qquad \chi _{0,n}=\chi _{n,n}=1.}$

Solving these recursion relations gives the formula: ${\displaystyle \chi _{k,n}=0}$ if ${\displaystyle n}$ is even and ${\displaystyle k}$ is odd and

${\displaystyle \chi _{k,n}={\begin{pmatrix}\left\lfloor {\frac {n}{2}}\right\rfloor \\\left\lfloor {\frac {k}{2}}\right\rfloor \end{pmatrix}}}$

otherwise.

When ${\displaystyle V}$ is an ${\displaystyle n}$-dimensional Euclidean space, we may define a uniform measure on ${\displaystyle \mathbf {Gr} _{k}(V)}$ in the following way. Let ${\displaystyle \theta _{n}}$ be the unit Haar measure on the orthogonal group ${\displaystyle O(n)}$ and fix ${\displaystyle w\in \mathbf {Gr} _{k}(V)}$. Then for a set ${\displaystyle A\subset \mathbf {Gr} _{k}(V)}$ , define

${\displaystyle \gamma _{k,n}(A)=\theta _{n}\{g\in \operatorname {O} (n):gW\in A\}.}$

This measure is invariant under the action of the group ${\displaystyle O(n)}$; that is,

${\displaystyle \gamma _{k,n}(gA)=\gamma _{k,n}(A)}$

for all ${\displaystyle g\in O(n)}$. Since ${\displaystyle \theta _{n}(O(n))=1}$, we have ${\displaystyle \gamma _{k,n}(\mathbf {Gr} _{k}(V))=1}$. Moreover, ${\displaystyle \gamma _{k,n}}$ is a Radon measure with respect to the metric space topology and is uniform in the sense that every ball of the same radius (with respect to this metric) is of the same measure.

This is the manifold consisting of all oriented ${\displaystyle k}$-dimensional subspaces of ${\displaystyle \mathbf {R} ^{n}}$. It is a double cover of ${\displaystyle \mathbf {Gr} _{k}(\mathbf {R} ^{n})}$ and is denoted by ${\displaystyle {\widetilde {\mathbf {Gr} }}_{k}(\mathbf {R} ^{n})}$.