Newton-Girard and Waring-Lagrange theorems for two non-commuting variables

Professor Nicholas Young (University of Leeds, Newcastle University) will present his research in Analysis.

In 1629 Albert Girard gave formulae for the power sums of several commuting variables in terms of the elementary symmetric functions; his result was subsequently often attributed to Newton.

Over a century later Edward Waring proved that an arbitrary symmetric polynomial in finitely many commuting variables could be expressed as a polynomial in the elementary symmetric functions of those variables.

In 1939 Margarete Wolf showed that there is no finite algebraic basis for the algebra of symmetric functions in $d > 1$ {\em non-commuting} variables, so there is no finite set of `elementary symmetric functions' in the non-commutative case.

Nevertheless, Jim Agler, John McCarthy and I have proved analogues of Girard's and Waring's theorems for symmetric functions in {\em two} non-commuting variables. We find three free polynomials $f, g, h$ in two non-commuting indeterminates $x, y$ such that every symmetric polynomial in $x$ and $y$ can be written as a polynomial in $f, g, h$ and $1/g$. In particular, power sums can be written explicitly in terms of $f,g$ and $h$. To do this we developed the notion of a non-commutative manifold.