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Domino tableaux, functions \(H\) and plethysms. (Tableaux de dominos, fonctions \(H\) et plethysmes.) (French) Zbl 0978.05539
Summary: We present a new family of symmetric functions, denoted by \(H_I(q)\), defined in terms of domino tableaux. These functions which depend on a parameter \(q\), specialize to Schur \(S\)-functions for \(q=0\), to a product of two \(S\)-functions for \(q=1\), and for \(q=-1\) (when \(I\) is a double partition) to a plethysm. This allows to split the square of a Schur function into its symmetric and antisymmetric parts, i.e., to give a combinatorial expression of the expansion of the basis of Schur functions of the phethysms \(S_2(S_I)\) and \(\Lambda_2 (S_I)\).
MSC:
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
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