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Macalester College

008 Kemeny Hall, 4 pm

Tea 3:30 pm, 300 Kemeny Hall

**Abstract: **
For a group $G$ and a vector space $V$, on which $G$ acts (a $G$-module),
we will produce a graph, called the representation graph, and we will look
at ways that the group gives information about the graph and ways that the
graph gives information about the group. The graph encodes the matrix
representations of $G$ which come from tensor products of $V$, and walks
on the graph enumerate multiplicities of irreducible modules. As an
important application, we will use the graph to understand the structure
of the algebra of operators which commute with the group on tensor space
(the tensor power centralizer algebra). \par We will illustrate these
ideas with many examples, including the symmetric, alternating, cyclic,
and dihedral groups. We will also examine $SU(2)$ and its finite subgroups
and see how the McKay correspondence shows up in this setting.

This talk will be accessible to graduate students.