scholarly journals On the degrees of projective representations

1988 ◽  
Vol 30 (2) ◽  
pp. 133-135 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Cohomology Class ◽  
Conjugacy Classes ◽  
Complex Numbers ◽  
Schur Multiplier ◽  
Projective Representations

All representations and characters studied in this paper are taken over the complex numbers, and all groups considered are finite. For basic definitions concerning projective representations see [1].If G is a group and or is a cocycle of G we denote by Proj(G, α) = {ξ1, …, ξt} the set of irreducible projective characters of G with cocycle α, where of course t is the number of α-regular conjugacy classes of G; ξ1, (1) is called the degree of ξ1. Also as normal, M(G) will denote the Schur multiplier of G, [α] the cohomology classof α, and [1] the cohomology class of the trivial cocycle of G.

The projective characters of the Mathieu group M12 and of its automorphism group

1980 ◽  
Vol 87 (3) ◽  
pp. 401-412 ◽  
Author(s):  
J. F. Humphreys
Keyword(s):  
Automorphism Group ◽  
Cohomology Class ◽  
Schur Multiplier ◽  
Mathieu Group ◽  
Projective Representations

In (1), Burgoyne and Fong have shown that the Schur multiplier of the Mathieu group M12 is of order 2. It is shown in Theorem 2·4 that the Schur multiplier of Aut M12, the automorphism group of M12, is also of order 2. It is therefore possible to choose a complex 2-cocycle α of Aut M12, taking only the values 1 and − 1, such that the cohomology class of α is of order 2 and the cohomology class of the restriction of α to M12 is of order 2. The characters of the irreducible α-projective representations of Aut M12 are calculated in § 2.

scholarly journals On projective characters of the same degree

1998 ◽  
Vol 40 (3) ◽  
pp. 431-434 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Complex Numbers ◽  
Central Type ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results. Let Proj (G, α) denote the set of irreducible projective characters of a group G with cocyle α. In previous papers (for exampe [2], [4], and [6]) numerous authors have considered the situation when Proj(G, α) = 1 or 2; such groups are said to be of α-central type or of 2α-central type, respectively. In particular in [4, Theorem A] the author showed that if Proj(G, α) = {ξ1, ξ2}, then ξ1(1)=ξ2(1). This result has recently been independently confirmed in [8, Corollary C].

scholarly journals Computing the Nonabelian Tensor Square of Metacyclic p–Groups of Nilpotency Class Two

2013 ◽  
Vol 61 (1) ◽  
Author(s):  
A. M. Basri ◽  
N. H. Sarmin ◽  
N. M. Mohd Ali ◽  
J. R. Beuerle
Keyword(s):  
Commutator Subgroup ◽  
Group Structure ◽  
Nilpotency Class ◽  
Conjugacy Classes ◽  
Schur Multiplier ◽  
Group Order ◽  
Nonabelian Tensor ◽  
Current Article ◽  
Tensor Square ◽  
Compute Structure

In this paper, we develop appropriate programme using Groups, Algorithms and Programming (GAP) software enables performing different computations on various characteristics of all finite nonabelian metacyclic p–groups, p is prime, of nilpotency class 2. Such programme enables to compute structure of the group, order of the group, structure of the center, the number of conjugacy classes, structure of commutator subgroup, abelianization, Whitehead’s universal quadratic functor and other characteristics. In addition, structures of some other groups such as the nonabelian tensor square and various homological functors including Schur multiplier and exterior square can be computed using this programme. Furthermore, by computing the epicenter order or the exterior center order the capability can be determined. In our current article, we only compute the nonabelian tensor square of certain order groups, as an example, and give GAP codes for computing other characteristics and some subgroups.

scholarly journals On projective characters of prime degree

1991 ◽  
Vol 33 (3) ◽  
pp. 311-321 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Prime Number ◽  
Structural Information ◽  
Alternating Group ◽  
Complex Numbers ◽  
Prime Degree ◽  
Counter Example ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

Groups which act transitively on the projective characters of a normal subgroup

1988 ◽  
Vol 104 (3) ◽  
pp. 429-434 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Normal Subgroup ◽  
Complex Numbers ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [4] for basic definitions and elementary results.

Groups whose projective characters are Galois conjugate

1989 ◽  
Vol 106 (2) ◽  
pp. 193-197 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Complex Numbers ◽  
Projective Representations

All groups G considered in this paper are finite and all representations of G are defined over the complex numbers. The reader unfamiliar with projective representations is referred to [6] for basic definitions and elementary results.

Groups with two projective characters

1988 ◽  
Vol 103 (1) ◽  
pp. 5-14 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Complex Numbers ◽  
Projective Representations

All representations and characters studied in this paper are taken over the field of complex numbers, and all groups considered are finite. The reader unfamiliar with projective representations is referred to [8] for basic definitions and terminology.

scholarly journals A pair of homotopy-theoretic version of TQFT’s induced by a Brown functor

2021 ◽  
pp. 2150053
Author(s):  
Minkyu Kim
Keyword(s):  
Dimension Reduction ◽  
Path Integral ◽  
Cohomology Class ◽  
Hopf Algebras ◽  
Path Integrals ◽  
Homology Theory ◽  
Projective Representations ◽  
Theoretic Version

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT’s starting from a [Formula: see text]-valued Brown functor where [Formula: see text] is the category of bicommutative Hopf algebras over a field [Formula: see text] : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT’s. In another application, we reproduce the Dijkgraaf–Witten TQFT and the Turaev–Viro TQFT from an ordinary [Formula: see text]-valued homology theory.

Real projective representations of finite groups

1990 ◽  
Vol 107 (1) ◽  
pp. 27-32 ◽  
Author(s):  
P. N. Hoffmann ◽  
J. F. Humphreys
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Schur Multiplier ◽  
Representations Of Finite Groups ◽  
Projective Representations ◽  
A Finite Group

The projective representations of a finite group G over a field K are divided into sets, each parametrized by an element of the group H2(G, Kx). The latter is the Schur multiplier M(G) when K = ℂ.

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