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Skew quasisymmetric Schur functions and noncommutative Schur functions. (English) Zbl 1214.05170

This paper is about generalisations of symmetric functions; the aim is to introduce a new family – the skew quasisymmetric Schur functions – and to prove a noncommutative version of the Littlewood–Richardson rule, though in fact the authors do a good job of keeping the paper self-contained, so that it serves as an excellent introduction to the area, and in particular to the combinatorics of compositions and partitions underlying symmetric functions.
The paper begins with an introduction to Young diagrams and tableaux, generalising several well-known definitions to (skew) compositions. Symmetric functions are introduced in Section 2.3, with the key definition being Definition 2.19, in which the skew quasisymmetric Schur functions are introduced.
In Section 3 the main result – the noncommutative Littlewood–Richardson rule – is stated, and the largest section of the paper is devoted to its proof. Sections 5 and 6 present some applications and further avenues, though these are sensibly kept reasonably short.
Overall, the paper is superbly structured and very well written, with helpful examples. The English is a little shaky in places, and the typesetting is odd (the journal’s house style seems to be to italicise \(\Omega\) but not \(\mho\), which looks bizarre). The only serious criticism I have of this paper is that the authors have chosen far too many MSC classifications, and given far too many keywords, which compromises the usefulness of these tools in searching the literature.

MSC:

05E05 Symmetric functions and generalizations
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