Traces, symmetric functions, and a raising operator

Abstract

The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator extends the Heisenberg algebra so that the number operator becomes a Lie product. This study is motivated by natural appearance of these polynomials in the theory of invariants for Lax equations and in classical and topological field theories.

15 pages, 5 figures. The article is accepted for publication in Proc. of the Fifth International Workshop on Applied Category Theory, Graph-Operad-Logic, Mérida, May 2006, (Z. Oziewicz and V. Dvoeglazov, ed.)