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Schur Superpolynomials: Combinatorial Definition and Pieri Rule
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural super-extensions of the Schur polynomials: in the limit $q=t=0$ and $q=t\rightarrow\infty$, corresponding respectively to the Schur superpolynomials and their dual. However, a direct definition is missing. Here, we present a conjectural combinatorial definition for both of them, each being formulated in terms of a distinct extension of semi-standard tableaux. These two formulations are linked by another conjectural result, the Pieri rule for the Schur superpolynomials. Indeed, and this is an interesting novelty of the super case, the successive insertions of rows governed by this Pieri rule do not generate the tableaux underlying the Schur superpolynomials combinatorial construction, but rather those pertaining to their dual versions. As an aside, we present various extensions of the Schur bilinear identity.
Keywords: symmetric superpolynomials; Schur functions; super tableaux; Pieri rule. 
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     author = {Olivier Blondeau-Fournier and Pierre Mathieu},
     title = {Schur {Superpolynomials:} {Combinatorial} {Definition} and {Pieri} {Rule}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a20/}
}
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Olivier Blondeau-Fournier; Pierre Mathieu. Schur Superpolynomials: Combinatorial Definition and Pieri Rule. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a20/

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