#math.rt

arXiv:1004.5323 [math.RT]

arXiv:1009.1862 [math.RT]

arXiv:1103.4066 [math.RT]

arXiv:1012.0382 [math.NT]

"The orthosymplectic supergroup $OSp(m|2n)$ is introduced as the supergroup of isometries of flat Riemannian superspace $R^{m|2n}$ which stabilize the origin. It also corresponds to the supergroup of isometries of the supersphere $S^{m-1|2n}$. The Laplace operator and norm squared on $R^{m|2n}$, which generate $sl(2)$, are orthosymplectically invariant, therefore we obtain the Howe dual pair ($osp(m|2n), sl(2)$). This Howe dual pair solves the problems of the dual pair ($SO(m)\times Sp(2n), sl(2)$), considered in previous papers. In particular we characterize the invariant functions on flat Riemannian superspace and show that the integration over the supersphere is uniquely defined by its orthosymplectic invariance. The supersphere manifold is also introduced in a mathematically rigorous way. Finally we study the representations of $osp(m|2n)$ on spherical harmonics. This corresponds to the decomposition of the supersymmetric tensor space of the $m|2n$-dimensional super vectorspace under the action of $sl(2)\times osp(m|2n)$. As a side result we obtain information about the irreducible $osp(m|2n)$-representations $L_{(k,0,...,0)}^{m|2n}$. In particular we find branching rules with respect to $osp(m-1|2n)$."

"Recent studies in several interrelated areas -- from combinatorics and representation theory in mathematics to quantum field theory and topological string theory in physics -- have independently revealed that many classical objects in these fields admit a relatively novel one-parameter deformation. This deformation, known in different contexts under the names of Omega-background, refinement, or beta-deformation, has a number of interesting mathematical implications. In particular, in Chern-Simons theory beta-deformation transforms the classical HOMFLY invariants into Dunfield-Gukov-Rasmussen superpolynomials -- Poincare polynomials of a triply graded knot homology theory. As shown in arXiv:1106.4305, these superpolynomials are particular linear combinations of rational Macdonald dimensions, distinguished by the polynomiality, integrality and positivity properties. We show that these properties alone do not fix the superpolynomials uniquely, by giving an example of a combination of Macdonald dimensions, that is always a positive integer polynomial but generally is not a superpolynomial."