Representations of Semisimple Lie Algebras on Posets

Robert Donnelly

Murray State University

September 11,
refreshments at 3:45pm


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 well-known 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 example-oriented, 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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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)

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