On the Modular Representation Theory of the Two-Dimensional Special Linear Group Over an Algebraically Closed Field

1977 ◽  
Vol s2-16 (2) ◽  
pp. 237-252 ◽  
Author(s):  
P. W. Winter
Keyword(s):  
Representation Theory ◽  
Linear Group ◽  
Special Linear Group ◽  
Modular Representation ◽  
Closed Field ◽  
Two Dimensional ◽  
Modular Representation Theory ◽  
Special Linear ◽  
Algebraically Closed Field

NEW ICA ALGORITHMS BASED ON SPECIAL LINEAR GROUP

2012 ◽  
Vol 04 (01) ◽  
pp. 1250028
Author(s):  
HANG TAN ◽  
XIANHE HUANG ◽  
YING TANG ◽  
HUACHUN TAN
Keyword(s):  
Representation Theory ◽  
Linear Group ◽  
Equilibrium Points ◽  
Classical Problem ◽  
Special Linear Group ◽  
Orthogonal Matrices ◽  
Special Linear ◽  
New Methods ◽  
The Matrix ◽  
New Algorithms

Unlike the traditional independent component analysis (ICA) algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices, in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening. The new algorithms are based on the special linear group SL(n). In order to achieve our target, we first provide a representation theory for any matrix in SL(n), which only simply uses the product of multiple exponentials of traceless matrices. Based on the matrix representation theory, two novel ICA algorithms are developed along with simple analysis on their equilibrium points. Moreover, we apply our methods to the classical problem of signal separation. The experimental results indicate that the superior convergence of our proposed algorithms, which can be expected as two viable alternatives to the ICA algorithms available in publications.

Trace polynomials of words in special linear groups

1979 ◽  
Vol 28 (4) ◽  
pp. 401-412 ◽  
Author(s):  
J. B. Southcott
Keyword(s):  
Linear Group ◽  
Characteristic Zero ◽  
Special Linear Group ◽  
Linear Groups ◽  
Two Dimensional ◽  
Arbitrary Mapping ◽  
Special Linear ◽  
Special Linear Groups ◽  
Integer Coefficients

AbstractIf w is a group word in n variables, x1,…,xn, then R. Horowitz has proved that under an arbitrary mapping of these variables into a two-dimensional special linear group, the trace of the image of w can be expressed as a polynomial with integer coefficients in traces of the images of 2n−1 products of the form xσ1xσ2…xσm 1 ≤ σ1 < σ2 <… <σm ≤ n. A refinement of this result is proved which shows that such trace polynomials fall into 2n classes corresponding to a division of n-variable words into 2n classes. There is also a discussion of conditions which two words must satisfy if their images have the same trace for any mapping of their variables into a two-dimensional special linear group over a ring of characteristic zero.

Relative characters for H-projective RG-lattices

1988 ◽  
Vol 104 (2) ◽  
pp. 207-213 ◽  
Author(s):  
Peter Symonds
Keyword(s):  
Representation Theory ◽  
Direct Summand ◽  
Modular Representation ◽  
Dedekind Domain ◽  
Modular Representation Theory ◽  
Trivial Group ◽  
Ordinary Character ◽  
Projective Lattices

If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

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