Subspace filters and polar bases for spaces of symmetric multilinear functions

摘要

Let V be a finite dimensional complex vector space, and for each positive integer n let Sn(V) denote the vector space whose elements are the complex valued symmetric n-linear functions on VxVx xV (ncopies). If f. S1(V), that is, f is a linear functional on V, thenwe define the symmetric n-linear function f n by f n(x1, x2,..., xn) = ni = 1 f (xi) for all x1, x2,..., xn. V. It is well known that the elements f n, known as polar elements, span Sn(V). A basis for Sn(V) consisting entirely of polar elements is a polar basis. We present a surprisingly simple recursive algorithm for the generation of polar bases. Explicit stand alone formulas for polar bases are also presented. One such explicit basis is so natural that it should probably be called the canonical polar basis for Sn(V). We introduce the notion of subspace filter and show that each subspace filter of order n provides polar bases for all of the spaces S1(V), S2(V),..., Sn(V) simultaneously, and we extend this result to the entire symmetric algebra S(V).