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

Analyse Fonctionnelle

Salle Kampé de Fériet M2
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 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 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.