Description
The Weyr
canonical form of a matrix is a little known cousin of
the Jordan canonical form. The Weyr form is dual to the
Jordan normal form in the sense that while the Jordan form
of an $n\times n$ matrix (over $\mathbb{C}$, say)
corresponds to a partition of $n$, the Weyr form corresponds
to the dual partition. While the Jordan form is closer to
being diagonal, the Weyr form has several other advantages
which make it superior in various problems in algebra. For
example, there are recent results in various areas of
algebra (including representation theory) where solutions to
problems were obtained using the Weyr form, but where it is
not known whether the Jordan form can be used.
After quickly introducing the Weyr form and its basic
properties, we will look at all or some of the following
applications, each of which will require the study of new
theory:
- Centralisers of
matrices (i.e., the ring of all matrices
commuting with a given matrix). Centralisers are
fundamental for many things in group and representation
theory and if the given matrix is in Weyr form, its
centraliser takes a much simpler form than if the matrix
were in Jordan form.
- The Gerstenhaber
theorem, namely that if $A$ and $B$ are two
commuting $n\times n$ matrices over a field $F$, then
the ring $F[A,B]$ generated by the two matrices over $F$
has dimension at most $n$ (as a vector space over $F$).
A similar statement (much easier to prove; try it!) is
that $F[A]$ has dimension at most $n$. On the other
hand, if we use four matrices, then there are examples
where the dimension is more than $n$. For three
commuting matrices, it is an intriguing open problem
whether the ring has dimension at most $n$. So far, no
proof or counter-example has ever been found. The Weyr
form gives an approach an approach to this problem.
- Basic notions of
algebraic geometry. We will look at various
proofs, which require the notions of dimension of an
algebraic variety, the Zariski topology and
corresponding irreducibility, density etc. We will study
these notions with the aim of understanding the
arguments in the applications to Gerstenhaber's theorem
and the following:
- Guralnick's theorem
for $\mathcal{C}(3,n)$. Let $\mathcal{C}(k,n)$
denote the set of $k$-tuples of commuting $n\times n$
matrices over an algebraically closed field $F$. This is
an algebraic variety. It has been known for some time
that it is irreducible for $k=1,2$ and for any $k$ when
$n<4$, but that it is reducible for all $k\geq 4$
when $n\geq 4$. The only open case is therefore $k=3$,
that is, the irreducibility of the variety of commuting
triples of matrices. For $n\geq 8$ it has been checked
(with increasingly complicated arguments) that
$\mathcal{C}(k,n)$ is irreducible. In 1992, Guralnick
proved the striking result that $\mathcal{C}(k,n)$ is
reducible for $n\geq 32$! (that's 32 with an
exclamation, not 32 factorial). Later refinements have
narrowed this down to $n\geq 30$, but we still don't
know for which $n$ $\mathcal{C}(k,n)$ goes from being
irreducible to reducible. We will study a version of
Guralnick's elegant argument, but simplified using Weyr
matrices.
Prerequisites
Algebra II is necessary and you must know Linear Algebra I
like the back of your hand.
Resources
- K. C. O'Meara, J. Clark, C. I. Vinsonhaler, Advanced Topics in Linear
Algebra.
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