Symmetric Functions and the Theorem of the Arithmetic and Geometric Means

Vesselin Gasharov 1
1 Department of Mathematics University of Michigan Ann Arbor, MI 48109-1003, USA

Abstract

We show that results analogous to the theorem of the arithmetic and geometric means hold for the three multiplicative fundamental bases of the vector space of symmetric functions – the elementary symmetric functions, the homogeneous symmetric functions, and the power sum symmetric functions. We give examples to demonstrate that no such results hold for the two non-multiplicative fundamental bases – the Schur functions and the monomial symmetric functions.