Pieri rules for Schur functions in superspace

Abstract

Abstract. The Schur functions in superspace s Λ ands Λ are the limits q = t = 0 and q = t = ∞ respectively of the Macdonald polynomials in superspace. We prove Pieri rules for the bases s Λ ands Λ (which happen to be essentially dual). As a consequence, we derive the basic properties of these bases such as dualities, monomial expansions, and tableaux generating functions.

Bernstein operators and super-Schur functions: combinatorial aspects

“…Table 2, note that the first four Pieri rules were first conjectured in [5]. These together with the following two were then proved in [10]. The remaining two Pieri rules have not been considered before.…”

“…Hence, ρ is an involution (ρ 2 = 1). From [10], when acting on a super-Schur function of Type I, it gives…”

“…For an element f ∈ A Q , the derivative w.r.t. x n under the homomorphism ρ is such that ρ(∂ x n f ) (−1) n−1 ∂ x n ρ( f ) since otherwise this would be equivalent to considering an action of the form ρ ⊥ • x n • ρ ⊥ which cannot be the case because it is known [10] that ρ ⊥ is not a homomorphism. Another way of seing this is from the map ρ : θ n →ẽ n−1 , which now has a non-trivial dependence on the x n 's.…”

Hopf algebra structure of symmetric and quasisymmetric functions in superspace

“…, θ N (such that θ i θ j = −θ j θ i , and θ 2 i = 0). Natural generalizations of the monomial, power-sum, elementary and homogeneous symmetric functions, as well as of the Schur [2,13], Jack and Macdonald polynomials have been studied. To illustrate the surprising richness of the theory of symmetric functions in superspace, we could mention that there is even an extension to superspace of the original Macdonald positivity conjecture.…”

The norm and the Evaluation of the Macdonald polynomials in superspace

“…It was established in the proof of Lemma 4 in[10] thate r O sym | θ1···θm+1 = (−1) m A m+1 e (m+1) r U + m c , 27…”