Semisymmetric polynomials and the invariant theory of matrix vector pairs

In this paper we investigate a new family of multivariable polynomials. These polynomials, denoted Rλ(z1, . . . , zn; r), depend on a parameter r and are indexed by a partition λ of length n. Up to a scalar, Rλ is characterized by the following elementary properties: • Rλ is symmetric in the odd variables z1, z3, z5, . . . as well as in the even variables z2, z4, z6, . . .. Polynomials having this kind of symmetry are called semisymmetric. • For the partition λ = (λ1, . . . , λn) define the odd degree as |λ|odd := ∑ i odd λi. Then the degree of Rλ(z) is |λ|odd. • Consider the vector % := ((n−1)r, (n−2)r, . . . , r, 0). Then Rλ(z) vanishes at all points of the form z = %+ μ where μ is any partition with μ 6= λ and |μ|odd ≤ |λ|odd. The simplest nontrivial example comes from the partition (1) = (1, 0, . . . , 0) in which case Rλ(z; r) = ∑n i=1(−1)zi − bn/4c. It is clearly semisymmetric, has degree |λ|odd = 1 and vanishes at z = %+ μ where μ = (0) or μ = (1). The polynomials Rλ(z) are analogous to the polynomials Pλ(z; r) which were previously introduced in[KS1]. In fact, the definition of Pλ is the same except that Pλ is symmetric in all variables z1, . . . , zn and the odd degree |λ|odd is replaced by the (full) degree |λ| = ∑ i λi. The Pλ are called shifted Jack polynomials since their highest degree components are the Jack polynomials. This is in contrast to their semisymmetric counterparts: even their highest degree components form a genuinely new class of multivariable homogeneous polynomials. All the polynomials mentioned above have a representation theoretic origin. LetG be a connected reductive group acting on a finite dimensional vector space V . We are interested in the case when this action is multiplicity free, i.e., every simple G-module occurs at most