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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:
 500000 Symmetric functions and generalizations 5e+10 Combinatorial aspects of representation theory
SYMMETRICA
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