Linear transformations on symmetric spaces

M. H. Lim

Vol. 55 (1974), No. 2, 499–505

DOI: 10.2140/pjm.1974.55.499

Abstract

Let U be an n-dimensional vector space over an algebraically closed field F of characteristic zero, and let ∨^rU denote the r-th symmetric product space of U. Let T be a linear transformation on ∨^rU which sends nonzero decomposable elements to nonzero decomposable elements. We prove the following:

1. If n = r + 1 then T is induced by a nonsingular transformation on T.
2. If 2 < n < r + 1 then either T is induced by a nonsingular transformation on U or T(∨^rU) = ∨^rW for some two dimensional subspace W of U.

The result for n > r+1 was recently obtained by L. J. Cummings.

Mathematical Subject Classification 2000

Primary: 15A69

Milestones

Received: 3 October 1973

Revised: 18 September 1974

Published: 1 December 1974