[Paper Review] Noncommutative symmetric functions

Israel M. Gelfand, Daniel Krob+4 more|ArXiv.org|Jul 20, 1994

Algebraic structures and combinatorial modelsMathematics24 references160 citations

TL;DR

This paper introduces a comprehensive theory of noncommutative symmetric functions using noncommutative analogs of elementary, complete, and power-sum symmetric functions, defined over a free associative algebra. It establishes connections with quasi-determinants, descent algebras, and rational power series, demonstrating that ribbon Schur functions form a polynomial basis and that noncommutative analogs of classical identities—such as the Cayley-Hamilton theorem and Padé approximants—hold via automata-theoretic and Hopf algebraic methods.

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

Motivation & Objective

• To develop a noncommutative analog of the classical symmetric function theory by reinterpreting symmetric functions in a free associative algebra over noncommuting generators.
• To establish a noncommutative version of the fundamental relationship between symmetric functions and representation theory, using the descent algebra as the noncommutative analog of the character ring.
• To extend classical identities—such as determinantal relations, Eulerian polynomials, and the Cayley-Hamilton theorem—into the noncommutative setting using quasi-determinants and rational power series.
• To connect noncommutative symmetric functions with automata theory and rational series via the Schützenberger theorem, showing that recognizable and rational series coincide in this context.
• To provide a unified framework for Lie idempotents, Eulerian idempotents, and continuous Baker-Campbell-Hausdorff formulas through the Hopf algebra structure of noncommutative symmetric functions.

Proposed method

• Define noncommutative elementary, complete, and power-sum symmetric functions as elements in the free associative algebra $\mathbf{Sym} = K\langle \Lambda_1, \Lambda_2, \ldots \rangle$, graded by weight instead of degree.
• Introduce ribbon Schur functions as noncommutative analogs of Schur functions, constructed via quasi-determinants, and prove they form a polynomial basis of $\mathbf{Sym}$.
• Establish transition matrices between different bases (e.g., $S$, $\Lambda$, $\Psi$, $\Phi$, $R$) using noncommutative analogs of classical symmetric function identities.
• Construct a Hopf algebra structure on $\mathbf{Sym}$, enabling the definition of an internal product isomorphic to the product in Solomon's descent algebra $\Sigma_n$.
• Use automata-theoretic models to represent rational noncommutative power series, with the behavior of a $K$-automaton over an alphabet $A$ yielding recognizable series.
• Apply the Schützenberger theorem to equate rational and recognizable series, and realize the star of a matrix via path enumeration in a labeled directed graph, leading to a noncommutative Cayley-Hamilton theorem.

Experimental results

Research questions

• RQ1 How can the classical theory of symmetric functions be generalized to a noncommutative setting where the generators do not commute?
• RQ2 What is the noncommutative analog of the Schur function basis, and which of these functions remain polynomial in the generators?
• RQ3 How do transition matrices between noncommutative bases (e.g., $S$ and $R$) generalize classical identities like those in the Jacobi-Trudi formula?
• RQ4 In what way do noncommutative symmetric functions relate to the descent algebra and Lie idempotents in the group algebra of the symmetric group?
• RQ5 Can rational noncommutative power series be characterized via automata, and how does this lead to a noncommutative version of the Cayley-Hamilton theorem?

Key findings

• The ribbon Schur functions, indexed by ribbon shapes, form a linear basis of the noncommutative symmetric functions algebra $\mathbf{Sym}$, and are the only quasi-Schur functions that are polynomials in the $\Lambda_k$ generators.
• The internal product on $\mathbf{Sym}$, induced by the Hopf algebra structure, corresponds to the product in Solomon's descent algebra $\Sigma_n$, establishing a noncommutative analog of the Kronecker product of representations.
• Noncommutative analogs of classical identities—such as the Jacobi-Trudi and dual Jacobi-Trudi formulas—hold when determinants are replaced by quasi-determinants.
• The noncommutative Eulerian polynomials and trigonometric functions arise naturally from the theory, with the Eulerian idempotents in $\Sigma_n$ admitting a simple interpretation in terms of noncommutative symmetric functions.
• The star of a generic $n \times n$ matrix $A = (a_{ij})$ is given by a path-sum formula over a labeled automaton, yielding a noncommutative version of the Cayley-Hamilton theorem via $A^* = I + A A^*$.
• Rational noncommutative power series in $K\langle\langle A\rangle\rangle$ are equivalent to recognizable series via the Schützenberger theorem, and the behavior of a $K$-automaton computes entries of $A^*$ as formal sums over paths in the graph.