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A Lie algebra of Grassmannian Dirac operators and vector variables. (English) Zbl 1507.17011

Summary: The Lie algebra generated by \(m \; p\)-dimensional Grassmannian Dirac operators and \(m\; p\)-dimensional vector variables is identified as the orthogonal Lie algebra \(\mathfrak{so}(2m+1)\). In this paper, we study the space \(\mathcal{P}\) of polynomials in these vector variables, corresponding to an irreducible \(\mathfrak{so}(2m+1)\) representation. In particular, a basis of \(\mathcal{P}\) is constructed, using various Young tableaux techniques. Throughout the paper, we also indicate the relation to the theory of parafermions.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
05E10 Combinatorial aspects of representation theory
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
15A66 Clifford algebras, spinors
15A75 Exterior algebra, Grassmann algebras
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