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

Comments

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) $.

References

[a1]	P.A. Griffiths, J.E. Harris, "Principles of algebraic geometry" , 1–2 , Wiley (Interscience) (1978) MR0507725 Zbl 0408.14001
[a2]	R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980) MR0608414 Zbl 0435.32004