Representations of Semisimple Lie Algebras on Posets
Robert Donnelly
Murray State University
September 11,
4:15pm
refreshments at 3:45pm
2338
ABSTRACT

In this talk we'll describe one way to visualize the action of a
semisimple Lie algebra on a weight basis. The resulting picture will
actually be a poset with colored edges which we will call a
{\em supporting diagram}. (Often, many weight bases give rise to the
same supporting diagram.) Supporting diagrams have some notable
combinatorial structure: in particular, for irreducible
representations, they are always rank symmetric, rank unimodal, and
strongly Sperner posets.
Some wellknown distributive lattices, such as the ``boolean
lattice'' $B_{n}$ and the lattice $L(m,n)$ of all partitions that fit
inside an $m \times n$ box, arise naturally as supporting diagrams
for certain representations. In this talk we'll describe virtually
all of the posets that we know of that arise as supporting diagrams
for irreducible representations. However, this is a program that is
far from complete; the problem is that many irreducible
representations have not been explicitly constructed in such a way
that it is possible to easily write down the supporting diagrams.
We take as our model the explicit bases obtained by Gelfand and
Zetlin for the irreducible representations of $sl(n,\Bbb{C})$. We
will describe explicit bases (and the associated supporting diagrams)
for the fundamental representations of $sp(2n,\Bbb{C})$ and
$so(2n+1,\Bbb{C})$ that ``analogize'' the GZ bases in a certain
sense. We will also look at supporting diagrams for other special
families of representations.
This talk will be exampleoriented, and we will use these examples to
begin to explore some more general questions. Does every supporting
diagram contain the associated crystal graph? Which supporting
diagrams are the most ``efficient''? And when is a weight
basis``uniquely specified'' by its supporting diagram?

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