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Splitting the square of a Schur function into its symmetric and antisymmetric parts. (English) Zbl 0853.05081
A new combinatorial description of the product of two Schur functions is considered. The key idea is to use domino tableaux (i.e. made up of \(1 \times 2\) rectangular boxes filled with integers increasing along the rows and strictly increasing along the columns) and to express the Littlewood-Richardson coefficients in terms of so-called Yamanouchi domino tableaux. In particular this approach gives a simple method of discriminating the tableaux in \(S_I S_I\) coming from symmetric parts \(S^2 (S_I)\). The \(q\)-analogues of the coefficients have also interesting combinatorial interpretations. It is shown that domino tableaux may be seen as the elements of a monoid, called the superplastic monoid. A new family of symmetric functions \((H\)-functions) are considered.
The article is self-contained, has many examples and algorithms and can be used as a good introduction into the domino tableaux method.

MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
Software:
SYMMETRICA
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