`machenry@mathstat.yorku.ca`

Isobaric polynomials are symmetric polynomials written on the elementary symmetric polynomial basis. This basis represents symmetric functions in terms of partitions of integers, that is, in terms of Young diagrams. This way of representing symmetric polynomials was discovered by Redfield in the 1920's, and first used by him to prove a 1924 conjecture of MacMahon, and later used by Polya in his Counting Theorem. The ring of isobaric polynomials also comes equipped with a convolution operator which gives rise to a group structure on this ring, which in turn yields group theoretical representations of the multiplicative and additive groups of the ring (UFD) of arithmetic functions. In addition we have defined "Logarithm" and "Exponential " operators in terms of the convolution product. We use these operators to define "trigonometric" functions on the isobaric ring, which gives rise to a hyperbolic geometry, providing a host of new identities for symmetric polynomials. There are applications to arithmetic and to algebraic number theory.