Representations of the Lie Superalgebra (1|2n) with Polynomial Bases

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2021
• Main Authors: Bisbo, Asmus K., De Bie, Hendrik, Van der Jeugt, Joris
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2021
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211318
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Representations of the Lie Superalgebra (1|2) with Polynomial Bases. Asmus K. Bisbo, Hendrik De Bie and Joris Van der Jeugt. SIGMA 17 (2021), 031, 27 pages

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