Pure Maths Colloquium: Counting representations over finite fields.
24 April 2017 16:00 in CM221
Finite fields are, well, finite. This means that structures traditionally associated to fields, such as vector spaces, varieties, representations, etc., often have a finite number of points or elements when they are defined over a finite field. Determining this number is a problem that comes up quite often in geometry, combinatorics and number theory.
One particular instance of such counting problem over a finite field appears in algebra. Given a quiver (an oriented graph), one can define a representation as a certain generalization of the notion of vector space. Kac then proved that the number of indecomposable representations of a given dimension defined over a finite field is a polynomial in the cardinal of the field.
In this talk, we will present this result, which turns out to be rather surprising. We will define the objects we need (quiver, representation, isomorphism,...) along the way, using only basic linear algebra. Minimal knowledge of finite fields will help, but knowing how to count up to any prime number p should suffice.
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