The Grassmann Manifold

Abstract

Let

$$ {C_{N + 1}} = Cx \ldots xC,\;N + 1\,factors $$

((8.1))

be the complex number space of N + 1 dimensions. Let GL(N+1, C) be the general linear group in N + 1 complex variables, which we identify with the group of all (N+1) × (N+1) non-singular matrices with complex elements. Suppose GL(n+l,C) acts on C_N+1 to the right, as described by

$$ \left( {{z^0}, \ldots ,{z^N}} \right) \to \left( {{z^0}, \ldots ,{z^N}} \right)g,\;gGL\left( {n + l,C} \right) $$

((8.2))

Among the subgroups of GL(N+1, C) are: (1) the unitary group U(N+1), which consists of all matrices g satisfying

$$ {t_{g\bar g}} = I $$

((8.3))

where I is the identity matrix; (2) the group GL(k+1, N-k, C), consisting of all non-singular matrices of the form

$$ \left( {\underbrace {\begin{array}{*{20}{c}} x \\ x \\\end{array}}_{k + 1}\,\underbrace {\begin{array}{*{20}{c}} 0 \\ x \\\end{array}}_{N - k}} \right)\,\begin{array}{*{20}{c}} {\} \,k + 1} \\ {\} \,N - k} \\\end{array} $$

((8.4))

where the elements at the upper-right corner are zero. The group GL(k+l, N-k, C) is the subgroup of all elements of GL(N+1, C) leaving fixed the (k+1)-dimensional subspace of C_N+1 spanned by the first k+1 coordinate vectors.