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Abstract: We will consider the case where the complex orthogonal group, O(n,C) acts on the
n by n matrices by restricting the adjoint representation of GL(n,C). This provides an action on
the ring of complex valued polynomial functions on the matrices. A combinatorial
description of an initial segment of the Hilbert series for the invariant
polynomials under this action will be described. The combinatorics
relates certain sums over Littlewood-Richardson coefficients to the
enumeration of cyclic graphs with directed edges. This relationship is
established via representation theory of the symmetric group.
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