scholarly journals Projective representations of quivers

2002 ◽  
Vol 31 (2) ◽  
pp. 97-101 ◽  
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.

On the Smallest Degrees of Projective Representations of the Groups PSL(n, q)

1971 ◽  
Vol 23 (1) ◽  
pp. 90-102 ◽  
Author(s):  
Morton E. Harris ◽  
Christoph Hering
Keyword(s):  
Lower Bound ◽  
Higher Order ◽  
Projective Representation ◽  
Minimal Degree ◽  
Prime Integer ◽  
Order Of Magnitude ◽  
Projective Representations

In this paper, we obtain information about the minimal degree δ of any non-trivial projective representation of the group PSL(n, q) with n ≧ 2 over an arbitrary given field K. Our main results for the groups PSL(n, q) (Theorems 4.2, 4.3, and 4.4) state that, apart from certain exceptional cases with small n, we have the following rather surprising situation: if q = pf (where p is a prime integer) and char K = p, then δ = n, but if q = pf and char K ≠ p, then δ is of a considerably higher order of magnitude, namely, δ is at least qn–l – 1 or if n = 2 and q is odd. Note that for n = 2, this lower bound for δ is the best possible. However, for n ≧ 3, this lower bound can conceivably be improved.

Projective Representations of Minimum Degree of Group Extensions

1978 ◽  
Vol 30 (5) ◽  
pp. 1092-1102 ◽  
Author(s):  
Walter Feit ◽  
Jacques Tits
Keyword(s):  
Simple Group ◽  
Central Extension ◽  
Finite Simple Group ◽  
Minimum Degree ◽  
Projective Representation ◽  
Closed Field ◽  
Natural Question ◽  
Central Extensions ◽  
Ordinary Representation ◽  
Projective Representations

Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F-representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.

scholarly journals Groups whose Projective character degrees are powers of a prime

1988 ◽  
Vol 30 (2) ◽  
pp. 177-180 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Projective Representation ◽  
Character Degrees ◽  
Projective Character ◽  
Projective Representations ◽  
A Finite Group

Let Gbe a finite group, and P:G → GL(n, ) be such that for all x, y ∈ G(i) P(x)P(y) = α(x, y)P(xy), and(ii)P(l) = In,where α(x, y) ∈ *; then P is a projective representation of G with cocycle α and degree n. For other basic definitions concerning projective representations see [4].

scholarly journals Schur Products

2014 ◽  
Author(s):  
Corneliu Constantinescu
Keyword(s):  
Clifford Algebras ◽  
Projective Representation ◽  
Schur Function ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Third ◽  
Normal Representation ◽  
Projective Representations ◽  
Factor Set

The projective representation of groups was introduced in 1904 by Issai Schur. It differs from the normal representation of groups by a twisting factor, which we call Schur function in this book and which is called sometimes normalized factor set in the literature (other names are also used). It starts with a discret group T and a Schur function f for T. This is a scalar valued function on T^2 satisfying the conditions f(1,1)=1 and |f(s,t)|=1, f(r,s)f(rs,t)=f(r,st)f(s,t) for all r,s,t in T. The projective representation of T twisted by f is a unital C*-subalgebra of the C*-algebra L(l^2(T)) of operators on the Hilbert space l^2(T). This reprezentation can be used in order to construct many examples of C*-algebras. By replacing the scalars R or C with an arbitrary unital (real or complex) C*-algebra E the field of applications is enhanced in an essential way. In this case l^2(T) is replaced by the Hilbert right E-module E tensor l^2(T) and L(l^2(T)) is replaced by the C*-algebra of adjointable operators on E tensor l^2(T). We call Schur product of E and T the resulting C*-algebra (in analogy to cross products which inspired the present construction). It opens a way to creat new K-theories (see the draft "Axiomatic K-theory for C*-algebras"). In a first chapter we introduce some results which are needed for this construction, which is developed in a second chapter. In the third chapter we present examples of C*-algebras obtained by this method. The classical Clifford Algebras (including the infinite dimensional ones) are C*-algbras which can be obtained by projective representations of certain groups. The last chapter of this book is dedicated to the generalization of these Clifford Algebras as an example of Schur products.

Projective Representations of Quivers #

2005 ◽  
Vol 33 (10) ◽  
pp. 3467-3478 ◽  
Author(s):  
Edgar Enochs ◽  
Sergio Estrada
Keyword(s):  
Representations Of Quivers ◽  
Projective Representations

Separability criteria via wavelet transform on homogenous spaces and projective representations

Open Physics ◽  
2010 ◽  
Vol 8 (3) ◽  
Author(s):  
Mahdi Karamati ◽  
Mohammad Rezapour
Keyword(s):  
Group Theory ◽  
Quantum Entanglement ◽  
Projective Representation ◽  
Permutation Groups ◽  
Characteristic Functions ◽  
Compact Groups ◽  
Reconstruction Method ◽  
Separability Criteria ◽  
Homogenous Space ◽  
Projective Representations

AbstractThe intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and homogenous space, and the quantum entanglement description using group theory were both studied in our previous articles. Here, we present a universal description of quantum entanglement using group theory and non-commutative characteristic functions for homogenous space and projective representation of compact groups on Banach spaces for some well known examples, such as: Moyal representation for a spin; Dihedral and Permutation groups.

scholarly journals On tensor induction of group representations

1990 ◽  
Vol 49 (3) ◽  
pp. 486-501 ◽  
Author(s):  
L. G. Kovács
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Proper Subgroup ◽  
Projective Representation ◽  
Arbitrary Field ◽  
Group Representations ◽  
Tensor Factor ◽  
Irreducible Constituent ◽  
Finite Dimensional ◽  
Projective Representations

AbstractLet G be a (not necessarily finite) group and ρ a finite dimensional faithful irreducible representation of G over an arbitrary field; write ρ¯ for ρ viewed as a projective representation. Suppose that ρ is not induced (from any proper subgroup) and that ρ¯ is not a tensor product (of projective representations of dimension greater than 1). Let K be a noncentral subgroup which centralizes all its conjugates in G except perhaps itself, write H for the normalizer of K in G, and suppose that some irreducible constituent, σ say, of the restriction p↓K is absolutely irreducible. It is proved that then (ρ is absolutely irreducible and) ρ¯ is tensor induced from a projective representation of H, namely from a tensor factor π of ρ¯↓H such that π↓K = σ¯ and ker π is the centralizer of K in G.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

scholarly journals Dimension of the Exceptional Set in the Aronszajn–Donoghue Theorem for Finite Rank Perturbations

2020 ◽  
Author(s):  
Constanze Liaw ◽  
Sergei Treil ◽  
Alexander Volberg
Keyword(s):  
Finite Rank ◽  
Measure Zero ◽  
Adjoint Operator ◽  
Cyclic Vector ◽  
Spectral Measures ◽  
Hermitian Matrices ◽  
Exceptional Set ◽  
Direct Sum ◽  
Rank One Perturbation ◽  
Rank One

Abstract The classical Aronszajn–Donoghue theorem states that for a rank-one perturbation of a self-adjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)-1 = d^2-1$.

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