Los Encuentros en Teoría de Grupos son organizados bianualmente por la Red Ibérica en Teoría de Grupos. Esta décima edición tendrá lugar en el Instituto Universitario de Investigación de Matemáticas de la Universidad de Sevilla (IMUS), entidad que colabora con la organización.


On the linear characters of finite algebra groups
Carlos André
(Universidade de Lisboa)

If \(A\) is a finite-dimensional nilpotent associative algebra over a field \(F\), then \(G = 1+A\) is called an algebra group over \(F\). Describing the irreducible (complex) characters of the algebra groups over finite fields is known to be a wild problem. In the case where the nilpotency degree of \(A\) is smaller than the characteristic of \(F\), then the method of Kirillov applies and all irreducible representations are parametrised by coadjoint orbits. In the general case, this seems not to be true, mainly because it depends on the definition of the exponential map. In this talk, we realise every finite algebra group as a quotient of an algebra group defined over the Witt ring of the finite field \(F\), and explain how the Artin-Hasse exponential may be used to apply Kirillov's method in the general situation. To illustrate we describe the linear characters of every finite algebra group and answer a question of I.M. Isaacs expressing the number of linear characters as a power of the number of elements of \(F\).

(Joint work with A.L. Branco-Correia and A.P. Nicolás.)

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