Meditatio algebraica noncommutativa : Newton-Girard and Waring-Lagrange theorems for two non-commuting variables

Nicholas Young
Vendredi, 8 Mars, 2019 - 14:00 - 15:00
In 1629 Albert Girard, a French Huguenot exiled to the Netherlands, 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 Waring proved in his {\em Meditationes algebraicae} 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 studied the analogous questions for non-commuting variables. She showed that there is no finite algebraic basis for the algebra of symmetric functions in $d > 1$ 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.