Independent multistructures, flag structures, and sets of linearly independent polar elements

Abstract

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 V×V×⋯×V (n-copies). If f∈S1(V), that is, f is a linear functional on V, then we define the symmetric n-linear function fn by fn(x1,x2,…,xn)=∏i=1nf(xi) for all x1,x2,…,xn∈V. It is well known that the elements fn, known as polar elements, span Sn(V). Our focus is the simple generation of sets of linearly independent polar elements of given order n. A basis for Sn(V) consisting entirely of polar elements is a polar basis. We introduce several new structures that are closely connected to the problem of generating independent sets of polar elements. In particular, we introduce the notion of independent multistructure of order n, n≥0, and depth p, p≥1, and show how such structures generate sets of linearly independent polar elements of cardinality (n+pp). When p=dim⁡(V) we obtain a basis for Sn(V) consisting entirely of polar elements. We also introduce the dual concept of flag structure of order n and depth p. Both of these concepts evolved from a consideration of subspace filters which were defined previously in [6].