Schur Superpolynomials: Combinatorial Definition and Pieri Rule

Symmetry Integrability and Geometry-Methods and Applications

article

Olivier Blondeau-Fournier¹  Pierre Mathieu¹ 

[1] Département de physique, Université Laval

关键词: symmetric superpolynomials;    Schur functions;    super tableaux;    Pieri rule;   

DOI  :  10.3842/SIGMA.2015.021

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown