Tensors have numerous applications in areas such as complexity theory and data analysis, where it is often necessary to understand ‘decompositions’ and/or ‘canonical forms’ of tensors in certain tensor product spaces. Such problems are often studied over the complex numbers, but there are also reasons to to study them over finite fields, including connections with classifications of semifields. In this talk, I will discuss the following problem. Consider the vector space V of 3x3 matrices over a finite field F, i.e. the tensor product of F^3 with itself. The 1-dimensional subspaces spanned by the fundamental (or rank-1) tensors in V form the so-called Segre variety in the projective space PG(V), and the setwise stabiliser G in PGL(V) of this variety may be identified with PGL(3,F) acting via g in G taking a matrix representative A to g^TAg. The G-orbits of points and lines in the ambient projective space PG(V) were determined by Michel Lavrauw and John Sheekey (Linear Algebra Appl. 2015). I will discuss joint work with Michel Lavrauw in which we determine which of the G-line orbits can be represented by symmetric 3x3 matrices, i.e. we classify the orbits of lines in PG(V) under the setwise stabiliser K of the so-called Veronese variety. Interestingly, several of the G-line orbits that have such ‘symmetric representatives’ split under the action of K, and in many cases this splitting depends on the characteristic of F. Connections are also drawn with old work of Jordan, Dickson and Campbell on the classification of ternary quadratic forms.
The symmetric representation of lines in PG(F^3 ⊗ F^3)
Tomasz Popiel (QMUL)
Mon, 04/12/2017 - 16:30