On Schur multiplier and projective representations of Heisenberg groups

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.

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Real projective representations of finite groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100068341 ◽

1990 ◽

Vol 107 (1) ◽

pp. 27-32 ◽

Cited By ~ 3

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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On the degrees of projective representations

Glasgow Mathematical Journal ◽

10.1017/s001708950000714x ◽

1988 ◽

Vol 30 (2) ◽

pp. 133-135 ◽

Cited By ~ 13

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.

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Some Inequalities for the Order of the Schur Multiplier of a Pair of Groups

Communications in Algebra ◽

10.1080/00927870802070033 ◽

2008 ◽

Vol 36 (7) ◽

pp. 2481-2486 ◽

Cited By ~ 3

Author(s):

Mohammad Reza R. Moghaddam ◽

Ali Reza Salemkar ◽

Taghi Karimi

Keyword(s):

Schur Multiplier

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Counting and equidistribution in quaternionic Heisenberg groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000426 ◽

2021 ◽

pp. 1-38

Author(s):

JOUNI PARKKONEN ◽

FRÉDÉRIC PAULIN

Keyword(s):

Hyperbolic Geometry ◽

Quaternion Algebra ◽

Rational Points ◽

Hyperbolic Spaces ◽

Heisenberg Groups ◽

Arithmetic Groups ◽

Hermitian Forms ◽

Dimension 2 ◽

Counting Formula ◽

The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

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Projective representations of quivers

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202108192 ◽

2002 ◽

Vol 31 (2) ◽

pp. 97-101 ◽

Cited By ~ 1

Author(s):

Sangwon Park

Keyword(s):

Projective Representation ◽

Representations Of Quivers ◽

Direct Sum ◽

Projective Representations

We prove thatP1 →f P2is a projective representation of a quiverQ=•→•if and only ifP1andP2are projective leftR-modules,fis an injection, andf (P 1)⊂P 2is a summand. Then, we generalize the result so that a representationM1 →f1 M2 →f2⋯→fn−2 Mn−1→fn−1 Mnof a quiverQ=•→•→•⋯•→•→•is projective representation if and only if eachMiis a projective leftR-module and the representation is a direct sum of projective representations.

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