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Induced projective representations

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Abstract

The left‐ (right‐) regular projective representation of a finite group G and the corresponding ’’projective’’ representation of the left‐group algebra are defined for a given standard factor system, and special features of these constructions are discussed. Starting from a given projective unitary irreducible representation of a normal (but not necessarily Abelian) subgroup N of G, we obtain by induction the matrix elements of the projective unitary irreducible representations of G, where the corresponding group algebra is used as aid. These considerations are of interest for the construction of projective unitary irreducible representations of little cogroups of nonsymmorphic space groups. For the present method allows us to construct, for a q lying on the ’’surface’’ of the Brillouin zone, these projective representations out from unitary irreducible representations belonging to q’s of ’’lower’’ symmetry. This method is used to determine for all little cogroups of the nonsymmorphic space group Pn3n complete sets of projective unitary irreducible representations.

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... However, the value of any physical observable, which can be expressed in terms of the CG coefficients, is invariant under a change of basis (Birman et al 1976). In the past few years much attention has been devoted to the theoretical problem of calculating CG coefficients (Litvin and Zak 1968, Card 1973c, Birman 1974a, Sakata 1974, Berenson and Birman 1975, van den Broek and Cornwell 1978, Dirl 1979a, c, 1981, 1982, Chen et a1 1983 and some hand calculations have been done on a few space groups (Berenson et a1 1975, Suffczynski and Kunert 1978, Dirl 1979d, Kunert and Suffczynski 1980, Kunert 1983. Dirl (1979aDirl ( , c, 1981Dirl ( , 1982 exploits the fact that the columns of the CG matrix may be seen as symmetry adapted vectors which may then be constructed by projection operator techniques. ...
... However, the value of any physical observable, which can be expressed in terms of the CG coefficients, is invariant under a change of basis (Birman et al 1976). In the past few years much attention has been devoted to the theoretical problem of calculating CG coefficients (Litvin and Zak 1968, Card 1973c, Birman 1974a, Sakata 1974, Berenson and Birman 1975, van den Broek and Cornwell 1978, Dirl 1979a, c, 1981, 1982, Chen et a1 1983 and some hand calculations have been done on a few space groups (Berenson et a1 1975, Suffczynski and Kunert 1978, Dirl 1979d, Kunert and Suffczynski 1980, Kunert 1983. Dirl (1979aDirl ( , c, 1981Dirl ( , 1982 exploits the fact that the columns of the CG matrix may be seen as symmetry adapted vectors which may then be constructed by projection operator techniques. ...
... In the past few years much attention has been devoted to the theoretical problem of calculating CG coefficients (Litvin and Zak 1968, Card 1973c, Birman 1974a, Sakata 1974, Berenson and Birman 1975, van den Broek and Cornwell 1978, Dirl 1979a, c, 1981, 1982, Chen et a1 1983 and some hand calculations have been done on a few space groups (Berenson et a1 1975, Suffczynski and Kunert 1978, Dirl 1979d, Kunert and Suffczynski 1980, Kunert 1983. Dirl (1979aDirl ( , c, 1981Dirl ( , 1982 exploits the fact that the columns of the CG matrix may be seen as symmetry adapted vectors which may then be constructed by projection operator techniques. This elegant method correctly deals with the 'multiplicity problem' which arises from the fact that space groups are not simply reducible. ...
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our papers [11] and [12] we shall abbreviate the term "continuous unitary representation of (~" to "representation of (~". If X is a proper closed subgroup of (~ whose representations are in a suitable sense "all known" one may pose the following two questions. (a) Which representations of 3( are the restrictions to it of irreducible representations of ~? (b) Given such a representation of 3( how can one construct all irreducible representations of (~ of which it is the restriction? When 3( is the identity subgroup question (a) has a trivial answer (apart from questions of dimension) and question (b) is essentially the same as that of determining all irreducible representations of (~? However, for other choices of 3(, questions (a) and (b) can furnish a useful breakdown of the problem of determining all irreducible representations of (~ into two more accessible components. It is the primary purpose of this paper to discuss questions (a) and (b) and their application to the determination of the representations of (~ in the special case in which 3( is normal. Actually we shall find it more convenient to deal with the slight variation in which we identify representations of 3( which are quasi equivalent in the sense defined on page 195 of [12]. For the special case in which 3( is not only normal but commutative and in which ~ is a semi direct product of 3( and (~/3( this program has been carried out in outline in our paper
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