On Characters and Superdimensions of Some Infinite-Dimensional Irreducible Representations of osp ( m n )
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ELEMENTARY PARTICLES AND FIELDS Theory
On Characters and Superdimensions (m|n)∗ of Some Infinite-Dimensional Irreducible Representations of osp osp(m|n) N. I. Stoilova1), J. Thierry-Mieg2), and J. Van der Jeugt3)** Received April 17, 2018
Abstract—Chiral spinors and self dual tensors of the Lie superalgebra osp(m|n) are infinite-dimensional representations belonging to the class of representations with Dynkin labels [0, . . . , 0, p]. We show that the superdimension of [0, . . . , 0, p] coincides with the dimension of a so(m − n) representation. When the superdimension is finite, these representations could play a role in supergravity models. Our technique is based on expansions of characters in terms of supersymmetric Schur functions. In the process of studying these representations, we obtain new character expansions. DOI: 10.1134/S1063778818060285
1. INTRODUCTION Models of supergravity theory [1, 2] are often implicitly or explicitly based upon tensor representations of the orthosymplectic Lie superalgebra osp(m|n) [3, 4]. Chiral spinors and self-dual tensors of osp(m|n) play an important role in such models. These tensors are, however, infinite-dimensional. Nonetheless, the so-called superdimension of these tensors corresponds to the dimension of a finitedimensional tensor of so(m − n) [5] (to be interpreted appropriately when m − n is negative [6]), thus paving the way for new covariant quantization schemes. In [5] we initiated the study of this correspondence between certain infinite-dimensional representations of osp(m|n) and finite-dimensional representations of so(m − n). Let us be more precise. In terms of (the distinguished) Dynkin diagrams of osp(m|n), the spinor representation has Dynkin labels [0, 0, . . . , 0, 1] and the self-dual tensor [0, 0, . . . , 0, 2]. In [5], we treated the irreducible representations (irreps) with Dynkin labels [0, 0, . . . , 0, p], where p is a positive integer (a convention followed throughout this paper). In the present paper, we shall first review some of the results of [5], and for this we need to recall some ∗
The text was submitted by the authors in English. Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, 1784 Bulgaria. 2) NCBI, National Library of Medicine, National Institute of Health, Bethesda, MD20894 USA. 3) Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Belgium. ** E-mail: [email protected] 1)
definitions and notation. For all these developments, characters of a class of representations of osp(m|n) play a prominent role. Since the Lie superalgebras osp(2m + 1|2n) and osp(2m|2n) both contain the general linear Lie superalgebra gl(m|n) as a subalgebra, it is convenient to express the characters of the infinite-dimensional osp-irreps as an infinite sum of gl(m|n) characters (given by supersymmetric Schur functions). In [5] this was done for the irreps [0, 0, . . . , 0, p] of osp(2m(+1)|2n) (leading to a new character formula for the case of osp(2m|2n)). In the current paper, we can exte
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