Schur Superpolynomials: Combinatorial Definition and Pieri Rule
Abstract
Schur superpolynomials have been introduced recently as limiting cases of the Macdonald superpolynomials. It turns out that there are two natural superextensions of the Schur polynomials: in the limit q=t=0 and q=t→∞, 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 semistandard 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.
 Publication:

SIGMA
 Pub Date:
 March 2015
 DOI:
 10.3842/SIGMA.2015.021
 arXiv:
 arXiv:1408.2807
 Bibcode:
 2015SIGMA..11..021B
 Keywords:

 symmetric superpolynomials;
 Schur functions;
 super tableaux;
 Pieri rule;
 Mathematical Physics;
 High Energy Physics  Theory;
 Mathematics  Combinatorics
 EPrint:
 SIGMA 11 (2015), 021, 23 pages