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A Littlewood-Richardson miscellany. (English) Zbl 0796.05091
The main subject of this paper are pictures, which are certain bijective maps between skew Ferrer’s diagrams. As is well known, pictures generalize Littlewood-Richardson fillings which combinatorially describe e.g. the decomposition of the tensor product of two irreducible representations of the symmetric group. The main result is a beautiful bijection between pictures and pairs of pictures which best is illustrated by a diagram (which appears in the head of the paper), and hence has to be omitted here. This diagram exhibits important features of the Robinson-Schensted correspondence, the jeu de taquin, and the Littlewood-Richardson rule, in a single picture. Among the corollaries are bijective proofs of identities featuring the Littlewood-Richardson coefficients \(\langle \theta, \sigma \rangle\) (where \(\theta, \sigma\) are skew Ferrer’s diagrams), such as \(f_ \sigma = \sum_ \lambda \langle \lambda, \sigma \rangle f_ \lambda\), and \(\langle \theta, \sigma \rangle = \sum_ \lambda \langle \theta, \lambda \rangle \langle \sigma,\lambda \rangle\), where \(\lambda\) runs through Ferrer’s diagrams, and where \(f_ \lambda\) is the number of standard Young tableaux of shape \(\lambda\). The main result is proved in two different ways, first by Robinson-Schensted techniques, and second by poset-theoretic arguments. The second approach has the advantage that a certain independence is inherent in the proof while it has to be established separately in the first approach. Furthermore, the authors unify and bring into relation previous work around the Littlewood-Richardson rule.

MSC:
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
20C30 Representations of finite symmetric groups
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