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An overview of \(\Lambda\)-type operations on quasi-symmetric functions. (English) Zbl 0990.05131

Let \(X=\{x_1<x_2<\ldots < x_n <\ldots\}\) be an infinite totally ordered set of commutative variables. A function \(f\in\mathbb{C}[X]\) is quasi-symmetric, if for every fixed \((k_1,k_2,\ldots,k_m)\) all monomials \(x_{i_1}^{k_1}x_{i_2}^{k_2}\ldots x_{i_m}^{k_m}\) with \(i_1<i_2<\ldots<i_m\) have the same coefficient in \(f\). The algebra of quasi-symmetric functions is related with other algebraic and combinatorial objects. In particular, it is the graded dual of the algebra of noncommutative symmetric functions. In this direction, an important question is to know whether there exists a good generalization of the notion of \(\Lambda\)-ring for which the algebra of quasi-symmetric functions is an universal object, exactly as in the case of the usual algebra of commutative symmetric functions with respect to the usual structure of \(\Lambda\)-ring. This question is still open and the main difficulty seems to be that one does not know a representation theoretic interpretation of the natural notion of plethysm of quasi-symmetric functions introduced by C. Malvenuto and C. Reutenauer [Discrete Math. 193, No. 1-3, 225-233 (1998; Zbl 1061.05506)]. The paper under review explores the \(\Lambda\)-type operation that can be defined on quasi-symmetric functions, as it was already done in the context of noncommutative symmetric functions; see D. Krob, B. Leclerc and Y.-Y. Thibon [Int. J. Algebra Comput. 7, No. 2, 181-264 (1997; Zbl 0907.05055)]. The authors survey all known definitions of \(\Lambda\)-ring type operations on the quasi-symmetric functions and focus on the plethysm which is the most complicated operation of this kind. They present also several new algorithms based on lattice theory for efficiently computing of plethysms.

MSC:

05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
06A07 Combinatorics of partially ordered sets
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