Grassmann manifold

Grassmannian

The set $G _ {n, m } ( k)$, $m \leq n$, of all $m$- dimensional subspaces in an $n$- dimensional vector space $V$ over a skew-field $k$. If $k$ is a field, then $G _ {n, m } ( k)$ can be imbedded in a $( _ { m } ^ {mn} ) - 1$- dimensional projective space over $k$ as a compact algebraic variety with the aid of Grassmann coordinates (cf. Exterior algebra). In the study of the geometrical properties of a Grassmann manifold an important role is played by the so-called Schubert varieties $S _ {a _ {0} \dots a _ {m} }$, $0 \leq a _ {0} < \dots < a _ {m} \leq n$, which are defined as follows. If $0 = V _ {0} \subset V _ {1} \subset \dots \subset V _ {n} = V$ is a flag of subspaces, i.e. a chain of subspaces such that also $\mathop{\rm dim} V _ {k} = k$, then

$$S _ {a _ {0} \dots a _ {m} } = \ \{ {W \in G _ {n, m } ( k) } : { \mathop{\rm dim} ( W \cap V _ {a _ {r} } ) \geq r,\ 0 \leq r \leq m } \} .$$

Any $\rho$- dimensional algebraic subvariety in a Grassmann manifold $G _ {n, m } ( k)$ is equivalent to a unique integer combination of the varieties $S _ {a _ {0} \dots a _ {m} }$, where $\sum _ {i = 0 } ^ {m} a _ {i} - m ( m + 1)/2 = \rho$( see [1]).

If $k$ is the field $\mathbf R$ of real numbers, the field $\mathbf C$ of complex numbers or the skew-field $\mathbf H$ of quaternions, a Grassmann manifold over $k$ can be regarded as a compact analytic manifold (which is real if $k = \mathbf R$ or $\mathbf H$ and complex if $k = \mathbf C$). These manifolds are distinguished by the fact that they are the classifying spaces for the classical groups (cf. Classical group) $O ( m)$, $U ( m)$ and $\mathop{\rm Sp} ( m)$, respectively. More exactly, for any CW-complex $X$ of dimension $\leq c( n + 1) - 2$, where $c =$ 1, 2 and 4, respectively, the set of isomorphism classes of $m$- dimensional vector bundles over $k$ with base $X$ is in a natural one-to-one correspondence with the set of homotopy classes of continuous mappings $X \rightarrow G _ {n + m, m } ( k)$[2]. A similar theory concerning the groups $\mathop{\rm SO} ( m)$ and $\mathop{\rm SU} ( m)$ leads to the study of the Grassmann manifold $G _ {n, m } ^ {0} ( k)$( $k = \mathbf R$ or $\mathbf C$) of oriented $m$- dimensional spaces in $k ^ {n}$. The manifolds listed above are closely connected, in particular, with the theory of characteristic classes (cf. Characteristic class).

The role played by Grassmann manifolds in topology necessitated a detailed study of their topological invariants. The oldest method of this study was based on Schubert varieties, with the aid of which a cell decomposition for $G _ {n, m } ( k)$( $k = \mathbf R , \mathbf C , \mathbf H$) is readily constructed. It is found, in particular, that the cycles $S _ {a _ {0} \dots a _ {m} }$ form a basis of the homology groups $H _ {*} ( G _ {n, m } ( \mathbf C ), \mathbf Z )$, $H _ {*} ( G _ {n, m } ( \mathbf R ), \mathbf Z _ {2} )$, $H _ {*} ( G _ {n, m } ( \mathbf H ), \mathbf Z )$. Cohomology algebras of Grassmann manifolds and the effect of Steenrod powers on them have also been thoroughly studied .

Another aspect of the theory of Grassmann manifolds is that they are homogeneous spaces of linear groups over the corresponding skew-field, and represent basic examples of irreducible symmetric spaces (cf. Symmetric space).

Manifolds which are analogous to Grassmann manifolds can also be constructed from subspaces of infinite-dimensional vector spaces. In particular, an important role in the theory of deformation of analytic structures is played by a Banach analytic manifold $G _ {B}$, the elements of which are the closed subspaces of a Banach space $B$ over $\mathbf C$ with a closed direct complement.

References

 [1] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 2 , Cambridge Univ. Press (1952) MR0048065 Zbl 0048.14502 [2] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804 [3a] A. Borel, "Sur la cohomogie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts" Ann of Math. , 57 (1953) pp. 115–207 [3b] A. Borel, "La cohomologie mod 2 de certains espaces homogènes" Comm. Math. Helv. , 27 (1953) pp. 165–197 MR0057541 Zbl 0052.40301 [3c] A. Borel, J.-P. Serre, "Groupes de Lie et puissances réduites de Steenrod" Amer. J. Math. , 75 (1953) pp. 409–448 MR0058213 [4] S.S. Chern, "Complex manifolds without potential theory" , Springer (1979) MR0533884 Zbl 0444.32004

Choose a basis in $k ^ {n}$. For each $x \in G _ {n,m} ( k)$ choose $m$ vectors generating $x$. These vectors generate an $( n \times m )$ matrix $A$. Now assign to $x$ the point in the projective space $\mathbf P ^ {N-} 1 ( k)$, $N = ( _ {m} ^ {n} )$, whose homogeneous coordinates are the determinants of all $( m \times m )$ submatrices of $A$. This point does not depend on the choices made. This defines an imbedding $G _ {n,m} ( k) \rightarrow \mathbf P ^ {N-} 1 ( k)$, called the Plücker imbedding. Correspondingly, these coordinates are called Plücker coordinates; they are also called Grassmann coordinates (cf. Exterior algebra and above). As a subvariety of $\mathbf P ^ {N-} 1 ( k)$ the Grassmann manifold $G _ {n,m} ( k)$ is given by a number of quadratic relations, called the Plücker relations, cf. [a1], Sect. 1.5.
There are a number of different notations in use; thus, the Grassmann manifold of $m$- planes in $k ^ {n}$ is variously denoted $G _ {n,m} ( k)$( as here), $G _ {m,n} ( k)$, $G ( m , n ; k )$, and $G _ {m} ( k ^ {n} )$, the last one generalizing to $G _ {m} ( V)$ with $V$ a vector space.
In the setting of algebraic geometry one defines the projective scheme $G _ {n,m}$ defined over $\mathbf Z$ whose $k$- points form $G _ {n,m} ( k)$.