Classical invariant theory has essentially addressed the action of the general linear group on homogeneous polynomials. Yet the orthogonal group arises in applications as the relevant group of transformations, especially in 3 dimensional space. Having a complete set of invariants for its action on ternary quartics, i.e. degree 4 homogeneous polynomials in 3 variables, is, for instance, relevant in determining biomarkers for white matter from diffusion MRI.
We characterize a generating set of rational invariants of the orthogonal group acting on even degree forms by their restriction on a slice. These restrictions are invariant under the octahedral group and their explicit formulae are given compactly in terms of equivariant maps. The invariants of the orthogonal group can then be obtained in an explicit way, but their numerical evaluation can be achieved more robustly using their restrictions. The exhibited set of generators futhermore allows us to solve the inverse problem and the rewriting.
Central in obtaining the invariants for higher degree forms is the preliminary construction, with explicit formulae, for a basis of harmonic polynomials with octahedral symmetry, dif- ferent, though related, to cubic harmonics.
This is joint work with Paul Görlach (now at MPI Leipzig), in a joint project with Téo Papadopoulo (Inria Méditerranée).[-]

Superpolynomials are formed with $N$ commuting and anti-commuting (skew) variables. By considering the space of skew variables of fixed degree as a module of the symmetric group $\mathcal{S}_{N}$ the theory of generalized Jack polynomials constructed by S Griffeth can be used to define nonsymmetric Jack superpolynomials. We present the theory, give details about the structure and derive norm formulas. Denote the parameter by $\kappa$ then the norm is positive-definite for $-\frac{1}{N}<\kappa<\frac{1}{N}$. Analogously there is a structure as Hecke algebra $\mathcal{H}_{N}(t)$-module on the skew polynomials and this allows the use of the theory of vectorvalued $(q, t)$-Macdonald polynomials studied by J-G Luque and the author. We outline the theory and present norm formulas and evaluations at special points. The norm is positive-definite for $q>0$ and min $(q^{1 / N}, q^{-1 / N}) < t < max (q^{1 / N}, q^{-1 / N} )$. As in the scalar case the evaluations use $(q, t)$-hook products.[-]

We establish a new connection between moments of n×n random matrices $X_{n}$ and hypergeometric orthogonal polynomials. Specifically, we consider moments $\mathbb{E}\mathrm{Tr} X_n^{-s}$ as a function of the complex variable $s\in\mathbb{C}$, whose analytic structure we describe completely. We discover several remarkable features, including a reflection symmetry (or functional equation), zeros on a critical line in the complex plane, and orthogonality relations. In each of the classical ensembles of random matrix theory (Gaussian, Laguerre, Jacobi) we characterise the moments in terms of the Askey scheme of hypergeometric orthogonal polynomials. We also calculate the leading order n→∞ asymptotics of the moments and discuss their symmetries and zeroes. We discuss aspects of these phenomena beyond the random matrix setting, including the Mellin transform of products and Wronskians of pairs of classical orthogonal polynomials. When the random matrix model has orthogonal or symplectic symmetry, we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials. This is work in collaboration with Fabio Cunden, Neil O' Connell and Nick Simm.[-]