16-dimensional compact projective planes with a large group by Hahl H., Salzmann H.

By Hahl H., Salzmann H.

Show description

Read Online or Download 16-dimensional compact projective planes with a large group fixing two points and two lines PDF

Similar symmetry and group books

Observation and control for operator semigroups

This ebook reviews remark and regulate operators for linear platforms the place the loose evolution of the kingdom could be defined by way of an operator semigroup on a Hilbert area. The emphasis is on well-posedness, observability and controllability homes. The summary effects are supported by way of a good number of examples coming in most cases from partial differential equations.

Representations of Real and P-Adic Group

The Institute for Mathematical Sciences on the nationwide college of Singapore hosted a learn application on "Representation conception of Lie teams" from July 2002 to January 2003. As a part of this system, tutorials for graduate scholars and junior researchers got by way of best specialists within the box.

Galois Theory for Beginners: A Historical Perspective

Galois concept is the end result of a centuries-long look for an answer to the classical challenge of fixing algebraic equations by way of radicals. during this ebook, Bewersdorff follows the historic improvement of the idea, emphasizing concrete examples alongside the best way. consequently, many mathematical abstractions at the moment are obvious because the common final result of specific investigations.

Additional resources for 16-dimensional compact projective planes with a large group fixing two points and two lines

Example text

Let w=Nlx1 +fl2x2+ ... + flaxa be another element of G. Then y = w if and only if a ; = /3 (i = 1, 2, ... 20) 1v=v. Furthermore, G c possesses a multiplicative structure. 21) where j (i, j) is a well-defined integer lying between 1 and g. i)• This multiplication is associative by virtue of the associative law for G. Thus, if u, y, w E G e , then (uv)w = u(vw). As we have already remarked (p. 27), a vector space which is endowed with an associative multiplication is called an algebra. Accordingly, we call G c the group algebra of G over C .

O+ U,, where UJ (i = 1, 2, ... , 1) are irreducible G-modules over K. Of course, the case 1= 1 refers to an irreducible representation A(x) (an irreducible G-module V). 3. Let G be a finite group of order g, and let K be a field whose characteristic is zero or else prime to g. Then (i) every matrix representation of G over K is completely reducible, or equivalently, (ii) every G-module over K is completely reducible. In subsequent chapters of this book we shall be exclusively concerned with finite groups and with ground fields of characteristic zero.

Prove that A (x) is irreducible. 5, dim ((A) = 1. 1. Orthogonality relations From now on we assume that G is a finite group of order g and that K is the field of complex numbers. Hence, by Maschke's Theorem, all representations of G are completely reducible. Furthermore, irreducibility henceforth means absolute irreducibility. Let A(x) and $(x) be inequivalent irreducible representations of degrees f and f' respectively, and write (i, j =1, 2, . 2) A (x ) = ( a (x)) ;; For brevity, we adopt the convention in this section that i and j always run from 1 to f, and that p and q run from 1 to f'.

Download PDF sample

Rated 4.82 of 5 – based on 5 votes