MA2001 prosjekt nr 7 - våren 2006

Young diagrams and the symmetry types of cubic matrices

For square matrices we have two symmetry types: symmetric and anti-symmetric, that corresponds to symmetric and anti-symmetric bilinear forms/functions.

What it would be for cubic matrices? Or, better to say, what if we start to consider trilinear forms? What it would mean anti-symmetric or symmetric then? Could it be more types than just properly degeneralized "symmetric" and "anti-symmetric"?

The answer is in fact known and it is "yes", there are more types! For n-forms (or for n-valent tensors) the description of types is closely related to famous Young diagrams, Young tables and the representations of the permutation groups.

To learn some of this means to step into a fascinating domain of mathematics, that lays on the crossroad of Group theory, Linear Algebra and Combinatorics. One of the more specific goals of the project could be to understand and describe the symmetry types for 3-linear and 4-linear forms.

Veileder: Alexei Rudakov
Rom:  650 SBII
Telefon:  (735) 91703
E-post:  rudakov (at) math.ntnu.no