The classification of subspaces in Hermitean vector spaces (Q1087965)

This is another contribution to a field of bilinear algebra which flourished especially in the 1970's. The object of study here are pairs (E,F), where E is a finite-dimensional, nondegenerate \(\epsilon\)- Hermitian space over a division ring k with an antiautomorphism \(a\mapsto \bar a\), and F a k-linear subspace of E. As is well-known, Witt's theorem solves the classification problem in the case when all \(\epsilon\)- symmetric elements of k are traces \(a+\epsilon \bar a\). To settle also the exceptional case, the authors stress the role of the subspace \(E^*\) of trace-valued vectors in E. They introduce notions of orthogonal and isotypic decomposition which refer to E together with its two subspaces \(E^*\) and F, and to the lattice generated by all three. The main result then is a set of fourteen (almost independent, more or less complicated) invariants which determine the pair (E,F) up to isometry. To demonstrate its usefulness, known results on extension of isometries and congruence of subspaces in Hermitian spaces are derived.