Special functions in Clifford analysis and axial symmetry. (English) Zbl 0634.30042

Sommen, F.

J. Math. Anal. Appl. 130, No. 1, 110-133 (1988).

In our previous paper [Rend. Circ. Mat. Palermo, II Ser. Suppl. 6, 259- 269 (1984; Zbl 0564.30036)], we proved a general Laurent expansion for monogenic functions in symmetric domains of \({\mathbb{R}}^{m+1}\), depending on the kind of symmetry involved. In this paper we consider axial symmetric domains and we apply our results in order to introduce new speciat \({\mathcal H}\) be a principal series of G. The Gaschütz-module A of G is defined as the direct product of all splitting p-chief factors of \({\mathcal H}\). A is in a natural manner a G-(right)-module.
There exists a “canonical” crossed homomorphism \(\psi\) of G onto A; ker \(\psi\) \(=D\) is a p-prefrattini-group of G (i.e. the intersection of a set of complements, one for each complemented p-chief-factor of \({\mathcal H})\). The group ring \({\mathbb{F}}_ p[A]\) is made a G-module by \(a\circ g:=a^ g\psi (g)\), \(a\in A\), \(g\in G\) where \(a^ g\) indicates the natural operation of G on A. This module is called \({\mathbb{F}}_ p[A]_{\psi}.\)
Then the following holds: a) There is a G epimorphism \({\hat \psi}\): \(P_ 1\to {\mathbb{F}}_ p[A]_{\psi}\), ker \({\hat \psi}\cong P_ 1(D)J_ D\otimes_ D{\mathbb{F}}_ p[G]\), where \(J_ D\) is the radical of \({\mathbb{F}}_ p[D]\). b) \({\hat \psi}\) maps \(J_ G\) onto \(J_{A,\psi}\) the augmentation ideal of \({\mathbb{F}}_ p[A]_{\psi}\). c) The mapping \(\pi\) : \(J_{a,\psi}\to A\), given by a-1\(\to^{\pi}a\) is a G-epimorphism. d) \(\tau =\pi\). \({\hat \psi}\) maps \(P_ 1J_ G\) onto A and ker \(\tau\) \(=P_ 1J^ 2_ G.\)
Gaschütz’s theorem: \(P_ 1J_ G/P_ 1J^ 2_ G\cong A\) is a consequence. Moreover, because of the fact that the composition factors of \({\mathbb{F}}_ p[A]_{\psi}\) coincide with the composition factors of the natural G-module \({\mathbb{F}}_ p[A]\) all these are composition factors of \(P_ 1\). As a further application one gets a simple proof of the theorem of Green and Hill. More information on \(P_ 1(G)\) is given.

Reviewer: A.Brandis

MSC:

30G35	Functions of hypercomplex variables and generalized variables
32A30	Other generalizations of function theory of one complex variable
30A05	Monogenic and polygenic functions of one complex variable
33C45	Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

Clifford analysis; Laurent expansion; monogenic functions; principal series; Gaschütz-module; splitting p-chief factors; crossed homomorphism; p-prefrattini-group; augmentation ideal; Gaschütz’s theorem; composition factors

Citations:

Zbl 0564.30036

References:

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