Fibonacci numbers and linear algebra

Abstract: 

The famous Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, ... have attracted a lot of interest in and outside of mathematics. They play an important role in applications in biology, but also in computer science and other areas. The lecture will draw the attention to questions in linear algebra which lead to Fibonacci numbers: we will give a categorical interpretation of some well-known combinatorial identities, but also exhibit new partition formulas.

We will consider triples of matrices (with entries in some field k) of the same shape, or, equivalently, triples of linear transformations say from V to W, where V, W are fixed vector spaces over k. Such triples are called 3-Kronecker modules. Trying to classify them, it turns out that certain 3-Kronecker modules play an exceptional role: we call these modules Fibonacci modules, since the dimension of the vector spaces involved are Fibonacci numbers. Given suitable 3-Kronecker modules, there is a corresponding linear representation of the 3-regular tree. In particular, this happens for the Fibonacci modules and it displays the Fibonacci numbers by integral functions on the tree. In this way we obtain the new partition formulas. Here is the display for the Fibonacci numbers 8 and 21:

http://www.mathematik.uni-bielefeld.de/~ringel/abs/fibonacci.gif

The basic information can be arranged in two triangles, they are quite similar to the Pascal triangle of the binomialcoefficients, but in contrast to the additivity rule for  the Pascal triangle, we now deal with additivity along hooks. There are intriguing relations between the two triangles. These relations correspond to certain exact sequences involving Fibonacci modules, but they can be verified also recursively.

The lecture is based on joint investigations with Philipp Fahr.

This seminar is part of the AMSI AGR National Seminar Series.