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We give an elementary proof of the first fundamental theorem of the invariant theory for the orthosymplectic supergroup by generalising the method of Atiyah, Bott and Patodi to the supergroup context. We use methods from super-algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space. In this way, a super Schur-Weyl-Brauer duality is established between the orthosymplectic supergroup of superdimension $(m|2n)$ and the Brauer algebra with parameter $m-2n$. The result may be interpreted in terms of the relevant Harish-Chandra super pair action (over the complex field), or equivalently, the orthosymplectic Lie supergroup over the infinite dimensional Grassmann algebra. We also state a corresponding theorem for the orthosymplectic Lie superalgebra, which involves an extra invariant generator, the super-Pfaffian.
We give a complete description of the finite-dimensional irreducible representations of the Yangian associated with the orthosymplectic Lie superalgebra $frak{osp}_{1|2}$. The representations are parameterized by monic polynomials in one variable, th
We give applications of Foliation Theory to the Classical Invariant Theory of real orthogonal representations, including: The solution of the Inverse Invariant Theory problem for finite groups. An if-and-only-if criterion for when a separating set is
We determine the Verma multiplicities of standard filtrations of projective modules for integral atypical blocks in the BGG category $mathcal{O}$ for the orthosymplectic Lie superalgebras $mathfrak{osp}(3|4)$ by way of translation functors. We then e
This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological ar
Suppose we have $n$ different types of self-replicating entity, with the population $P_i$ of the $i$th type changing at a rate equal to $P_i$ times the fitness $f_i$ of that type. Suppose the fitness $f_i$ is any continuous function of all the popula