Charles W. Curtis

Representation theory of finite groups and associative algebras

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A norm map for endomorphism algebras of Gelfand-Graev representations

Let G be a connected, reductive algebraic group defined over a finite field, with Frobenius endomorphism F: G → G. In a previous article by the first author [3], the irreducible representations, in an algebraically closed field K of characteristic zero, of the Hecke algebra Ή of a Gelfand-Graev representation γ of the finite reductive group G F were determined. They are parametrized by pairs (T, θ), with T an F-stable maximal torus of G, and θ an irreducible character of the finite torus T F . Each irreducible representation f T, θ can be factored, \({{f}_{{T,{\kern 1pt} \theta }}}{\mkern 1mu} = {\mkern 1mu} \tilde{\theta } {\mkern 1mu} ^\circ {{f}_{T}} \), with f T a homomorphism of algebras, independent of θ, from H to the group algebra KTF, and \({\tilde{\theta }} \) the extension of θ to an irreducible representation of the group algebra of the torus T F .

Representation Theory of Finite Groups: from Frobenius to Brauer

The representation theory of finite groups began with the pioneering research of Frobenius, Burnside, and Schur at the turn of the century. Their work was inspired in part by two largely unrelated developments which occurred earlier in the nineteenth century. The first was the awareness of characters of finite abelian groups and their application by some of the great nineteenth-century number theorists. The second was the emergence of the structure theory of finite groups, beginning with Galois’ brief outline of the main ideas in the famous letter written on the eve of his death and continuing with the work of Sylow and others, including Frobenius himself.

On the Endomorphism Algebras of Gelfand-Graev Representations

The Gelfand-Graev representations of a finite reductive group G F are multiplicity free representations, each of which contains a large share of the irreducible representations of G F .The G F -endomorphism algebra of a Gelfand-Graev representation γ is a commutative algebra, whose irreducible representations correspond to the irreducible components of γ. This note contains announcements of some results on the connection between the irreducible representations of these endomorphism algebras and the virtual representations R T,θ of Deligne and Lusztig.

Vector Spaces and Systems of Linear Equations

This chapter contains the basic definitions and facts about vector spaces, together with a thorough discussion of the application of the general results on vector spaces to the determination of the solutions of systems of linear equations. The chapter concludes with an optional section on the geometrical interpretation of the theory of systems of linear equations. Some motivation for the definition of a vector space and the theorems to be proved in this chapter was given in Section 1.

Dual Vector Spaces and Multilinear Algebra

This chapter begins with two sections on some important constructions on vector spaces, leading to quotient spaces and dual spaces. The section on dual spaces is based on the concept of a bilinear form defined on a pair of vector spaces. The next section contains the construction of the tensor product of two vector spaces and provides an introduction to the subject of what is called multilinear algebra. The last section contains an application of the theory of dual vector spaces to the proof of the elementary divisor theorem, which was stated in Section 25.

Vector Spaces with an Inner Product

This chapter begins with an optional section on symmetry of plane figures, which shows how some natural geometrical questions lead to the problem of studying linear transformations that preserve length. The concept of length in a general vector space over the real numbers is introduced in the next section, where it is shown how length is related to an inner product. The language of orthonormal bases and orthogonal transformations is developed with some examples from geometry and analysis. Beside the fact that the real numbers form a field, we shall use heavily in this chapter the theory of inequalities and absolute value, and the fact that every real number a ≥ 0 has a unique nonnegative square root \(\sqrt a\).

Some Applications of Linear Algebra

This chapter begins with a section on the classification of symmetry groups in three dimensions, continuing the discussion of symmetry groups begun in Section 14. The next two sections are an introduction to analytic methods in matrix theory. In the first, a system of first order linear differential equations with constant coefficients is translated into a single vector differential equation, and solved using the exponential function of a matrix. The second contains a theorem on the eigenvalues and eigen-vectors of a matrix whose entries are positive real numbers. An application to stochastic matrices and Markov chains is outlined. Finally, we take up an approach, using linear algebra, to a problem in classical algebra, on sums of squares.

The Theory of a Single Linear Transformation

The main topic of this chapter is an introduction to the theory of a single linear transformation T ∈L (V,V). The goal is to find a basis of the vector space V such that the matrix of the linear transformation T with respect to this basis is as simple as possible. Since the end result, often called a normal form, is used in different ways, there are several versions of what the normal form of the matrix should be. Some are given in this chapter, and others, for orthogonal and unitary transformations, etc., in Chapter 9.

Introduction to Linear Algebra

Mathematical theories are not invented spontaneously. The theories that have proved to be useful have their beginnings, in most cases, in special problems, which are difficult to understand or to solve without a grasp of the underlying principles. This chapter begins with two such problems, which have a common set of underlying principles. Linear algebra, which is the study of vector spaces, linear transformations, and matrices, is the result of trying to understand the common features of these and other similar problems.