Models for Some Irreducible Representations of \(\mathfrak{so}(m, \mathbb{C})\) in Discrete Clifford Analysis
In this paper we work in the ‘split’ discrete Clifford analysis setting, i.e., the m-dimensional function theory concerning null-functions of the discrete Dirac operator ∂, defined on the grid \( \mathbb{Z} \) m , involving both forward and backward diffe
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Models for Some Irreducible Representations of so(m, C) in Discrete Clifford Analysis Hilde De Ridder and Tim Raeymaekers Abstract. In this paper we work in the ‘split’ discrete Clifford analysis setting, i.e., the m-dimensional function theory concerning null-functions of the discrete Dirac operator ∂, defined on the grid Zm , involving both forward and backward differences. This Dirac operator factorizes the (discrete) StarLaplacian (Δ∗ = ∂ 2 ). We show how the space Hk of discrete k-homogeneous spherical harmonics, which is a reducible so(m, C)-representation, may explicitly be decomposed into 22m isomorphic copies of irreducible so(m, C)representations with highest weight (k, 0, . . . , 0). The key element is the inthediscrete Clifford algebra in 22m troduction of 22m idempotents, k+m−1dividing k+m−3 subalgebras of dimension − . k k Mathematics Subject Classification (2010). 17B15, 47A67, 20G05, 15A66, 39A12. Keywords. Discrete Clifford analysis, irreducible representation, orthogonal Lie algebra, harmonic functions.
In classical Clifford analysis, the infinitesimal ‘rotations’ are given by the angular momentum operators, in our function theoretical setting denoted by the differential operators La,b = xa ∂xb − xb ∂xa . These operators satisfy the commutation relations [La,b , Lc,d] = δb,c La,d − δb,d La,c − δa,c Lb,d + δa,d Lb,c , which are the defining relations of the orthogonal Lie algebra so(m). Since these are endomorphisms of the space Hk (m, C) of scalar-valued harmonic polynomials homogeneous of degree k, this polynomial space is a model for an (irreducible) so(m, C)-representation [see, e.g., [9, 1]]. Classically, to establish Mk , the space of spinor-valued monogenics, homogeneous of degree k, as so(m, C)-representation, the following operators are considered 1 Mk → La,b + ea eb Mk . dR(ea,b ) : Mk → Mk , 2 The first author acknowledges the support of the Research Foundation – Flanders (FWO), grant no. FWO13\PDO\039.
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H. De Ridder and T. Raeymaekers
These operators are endomorphisms of Mk which also satisfy the defining relations of so(m, C): [dR(ea,b ), dR(ec,d )] = δb,c dR(ea,d ) − δb,d dR(ea,c ) − δa,c dR(eb,d ) + δa,d dR(eb,c ). In [5], we developed discrete counterparts of the operators La,b and dR(ea,b ) in the discrete Clifford analysis setting. Definition 1. The (discrete) angular momentum operators are discrete operators La,b = ξa ∂b + ξb ∂a , 1 a = b m, acting on the discrete functions. Here ξa and ∂a , a = 1, . . . , m are the discrete vector variables and co-ordinate differences in the discrete Clifford analysis setting (see Section 1). For a = b, we define La,a = 0. Furthermore, let the operator Ωa,b act on discrete functions f as Ωa,b f = La,b f eb ea . The discrete angular momentum operators also satisfy the defining relations of the orthogonal Lie algebra so(m) (see, e.g., [11]): [Ωa,b , Ωc,d ] = δb,c Ωa,d − δb,d Ωa,c − δa,c Ωb,d + δa,d Ωb,c . Furthermore, they are endomorphisms of the space Hk of Clifford algebra-valued , k-homogeneous harmonics since Ωa,b commutes with sl2 = Δ, ξ 2
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