The article should be considered as one of the basic work in a new area– -theory of noncommutative symmetric functions. It contains a lot of deep ideas, illustrated by examples, proofs and references. One of the main instruments here is the notion of quasi-determinant. The reader can find both elementary properties, noncommutative versions of commutative notions, theorems (e.g. quasi-minors, Sylvester’s and Bazin’s theorem) and quite new points of view even for the commutative case. In any case he should enjoy the beauty of several formulas, expressed in terms of quasi-determinants (e.g. transition formulas or Giambelli’s formula for quasi-Schur functions).
The approach to the noncommutative invariant functions is based on generating power series. Let \(\{\Lambda_k\mid k = 1,2, \ldots \}\) be an infinite set of noncommuting indeterminates. Then the generating series for elementary symmetric functions is \(\lambda (t) = 1 + \sum_{k \geq 1} t^k \Lambda_k\) (so the \(\Lambda_k\) themselves are considered as elementary symmetric (!) functions). The generating series for complete homogeneous symmetric functions is defined as in the commutative case: \(\sigma (t) = (\lambda - t)^{-1}\). There are two different analogous for the power sums series. The first of them, \(\psi (t) = \sum_{k \geq 1} t^{k - 1} \Psi_k\), can be obtained from the equation \({d \over dt} \sigma (t) = \sigma (t) \psi (t)\), the second one from the equation \(\phi (t) = \sum_{k \geq 1} t^{k - 1} \Phi_k = {d \over dt} \log \sigma (t)\). The difference between them, arising in the noncommutative case, leads to some interesting Lie relations between \(\Phi_k\) and \(\Psi_k\). Of course, the expected Hopf algebra structure appears immediately too.
Wonderful connections with Solomon’s descent algebra establish the following nontrivial idea. After that noncommutative Eulerian polynomials and trigonometric functions are introduced. Duality and specializations are the next important topics. One more nice idea here is the appearance of automata theory. The end of the article is devoted to the noncommutative rational functions: noncommutative continued \(J\)- fractions, Padé approximation—even this is sufficient to attract attention to this article. But in reality it contains much more and deserves to be familiar with.

Reviewer: V.A.Ufnarovski (Lund)