Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 031, 27 pages      arXiv:1912.06488      https://doi.org/10.3842/SIGMA.2021.031

Representations of the Lie Superalgebra $\mathfrak{osp}(1|2n)$ with Polynomial Bases

Asmus K. Bisbo ^a, Hendrik De Bie ^b and Joris Van der Jeugt ^a
a) Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281-S9, B-9000 Gent, Belgium
^b) Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, Krijgslaan 281-S8, B-9000 Gent, Belgium

Received June 30, 2020, in final form March 10, 2021; Published online March 25, 2021

Abstract
We study a particular class of infinite-dimensional representations of $\mathfrak{osp}(1|2n)$. These representations $L_n(p)$ are characterized by a positive integer $p$, and are the lowest component in the $p$-fold tensor product of the metaplectic representation of $\mathfrak{osp}(1|2n)$. We construct a new polynomial basis for $L_n(p)$ arising from the embedding $\mathfrak{osp}(1|2np) \supset \mathfrak{osp}(1|2n)$. The basis vectors of $L_n(p)$ are labelled by semi-standard Young tableaux, and are expressed as Clifford algebra valued polynomials with integer coefficients in $np$ variables. Using combinatorial properties of these tableau vectors it is deduced that they form indeed a basis. The computation of matrix elements of a set of generators of $\mathfrak{osp}(1|2n)$ on these basis vectors requires further combinatorics, such as the action of a Young subgroup on the horizontal strips of the tableau.

Key words: representation theory; Lie superalgebras; Young tableaux; Clifford analysis; parabosons.

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