Noncommutative symmetric functions

TL;DR

This paper develops a noncommutative theory of symmetric functions using quasi-determinants, establishing a Hopf algebra structure and exploring various applications including noncommutative polynomials and matrices.

Contribution

It introduces a formal noncommutative symmetric functions framework with a Hopf algebra structure and diverse applications, extending classical symmetric function theory.

Findings

Established a Hopf algebra structure for noncommutative symmetric functions

Developed new methods for computing in descent algebras

Explored applications to noncommutative polynomials and matrices

Abstract

This paper presents a noncommutative theory of symmetric functions, based on the notion of quasi-determinant. We begin with a formal theory, corresponding to the case of symmetric functions in an infinite number of independent variables. This allows us to endow the resulting algebra with a Hopf structure, which leads to a new method for computing in descent algebras. It also gives unified reinterpretation of a number of classical constructions. Next, we study the noncommutative analogs of symmetric polynomials. One arrives at different constructions, according to the particular kind of application under consideration. For example, when a polynomial with noncommutative coefficients in one central variable is decomposed as a product of linear factors, the roots of these factors differ from those of the expanded polynomial. Thus, according to whether one is interested in the construction of a polynomial with given roots or in the expansion of a product of linear factors, one has to consider two distinct specializations of the formal symmetric functions. A third type appears when one looks for a noncommutative generalization of applications related to the notion of characteristic polynomial of a matrix. This construction can be applied, for instance, to the noncommutative matrices formed by the generators of the universal enveloping algebra $U(gl_n)$ or of