AN INEQUALITY FOR NON-DECOMPOSABLE ELEMENTS IN THE GRASSMANN ALGEBRA

ABSTRACT. Let V be a unitary space of dimension n and let z and x be non-zero homogeneous elements in AV of degrees r and n-r respectively. The main results concern the inequality IIz,x xll •< Ilzll Ilxll and an analysis of the conditions for equality, particularly when z and x are not necessarily decomposable. The specialization of z and x to decomposable lements reduces to the classical Fischer inequality for positive semi-definite hermitian matrices. 1. Statements. Let V be a unitary space, dim V = n, with an orthonormal (o.n.) basis el,...,e n and let fl,...,fn be the dual basis in V*. The Hodge operator [2; page 130] Hf: ^V •/XV * is defined by (1) Hf(x) = x _1 f" where x __1 f^denotes the right interior product and f " = fl,x--- ^ fn ' A change in the o.n. basis of V simply replaces Hf by a scalar multiple of Hf of modulus 1; thus we identify all such Hf by the single notation H. It is well known that H is a bijection that maps the set of decomposable homogeneous elements in/NV of degree r onto the set of decomposable homogeneous elements of degree n-r in/XV * [4; page 21]. Moreover r if z is decomposable in /XV (i.e., z is an exterior product of r vectors in V) then the (n-D-dimensional space determined by H(z), [H(z)], is the annihilator of [z], i.e., [H(z)l = [z] o It is simple to see that if (2) z = 2;o•GQr,nCo•eo• the