  Understanding the difference between group orbits and their closures is a key
difficulty in geometric complexity theory (GCT): While the GCT program is set
up to separate certain orbit closures, many beautiful mathematical properties
are only known for the group orbits, in particular close relations with
symmetry groups and invariant spaces, while the orbit closures seem much more
difficult to understand. However, in order to prove lower bounds in algebraic
complexity theory, considering group orbits is not enough.
  In this paper we tighten the relationship between the orbit of the power sum
polynomial and its closure, so that we can separate this orbit closure from the
orbit closure of the product of variables by just considering the symmetry
groups of both polynomials and their representation theoretic decomposition
coefficients. In a natural way our construction yields a multiplicity
obstruction that is neither an occurrence obstruction, nor a so-called
vanishing ideal occurrence obstruction. All multiplicity obstructions so far
have been of one of these two types.
  Our paper is the first implementation of the ambitious approach that was
originally suggested in the first papers on geometric complexity theory by
Mulmuley and Sohoni (SIAM J Comput 2001, 2008): Before our paper, all existence
proofs of obstructions only took into account the symmetry group of one of the
two polynomials (or tensors) that were to be separated. In our paper the
multiplicity obstruction is obtained by comparing the representation theoretic
decomposition coefficients of both symmetry groups.
  Our proof uses a semi-explicit description of the coordinate ring of the
orbit closure of the power sum polynomial in terms of Young tableaux, which
enables its comparison to the coordinate ring of the orbit.

Computational Complexity;Algebraic Geometry;Representation Theory