In this talk we make a (perhaps) unexpected link between the combinatorialnotion of partially ordered sets (posets) and certain families ofmatrices. Using this link, we give a simple, matrix-based proof of theMoebius Inversion Formula on a poset and a variant of it. It is thisvariant which is really at the heart of the talk, for it will allow us tosimultaneously analyze certain properties (determinant, inertia, etc.) ofentire families of matrices via their $LDL^T$-Factorizations.Refreshments will be provided

Let $G$ be a group and let $V$ be a finitedimensional $G$-module. Let $B$ denote the algebra of $G$-invariantpolynomial differential operators on $V$. It is natural to pose thefollowing questions:1) What is the representation theory of $B$? What are the primitiveideals of $B$?2) Does $B$ have finite dimensional representations? If so, are theycompletely reducible?medskip
oindentLittle is known about these questions when $G$ is noncommutative. Wegive answers for the adjoint representation of SL$_3(C)$, alreadyan interesting and difficult case.