The University of New South Wales
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:
The basic information can be arranged in two triangles, they are quite similar to the Pascal triangle of the binomial coefficients, 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.
Contact Maaike Wienk at firstname.lastname@example.org, with a cc to Tim Salmon at the University of New South Wales (email@example.com), one week in advance at the latest.
If you would like to attend this seminar in our access grid room then please check to see if the grid is already booked at this time and send an email to firstname.lastname@example.org to let the CSOs know that you would like to attend.