  We show that most arithmetic circuit lower bounds and relations between lower
bounds naturally fit into the representation-theoretic framework suggested by
geometric complexity theory (GCT), including: the partial derivatives technique
(Nisan-Wigderson), the results of Razborov and Smolensky on $AC^0[p]$,
multilinear formula and circuit size lower bounds (Raz et al.), the degree
bound (Strassen, Baur-Strassen), the connected components technique (Ben-Or),
depth 3 arithmetic circuit lower bounds over finite fields
(Grigoriev-Karpinski), lower bounds on permanent versus determinant
(Mignon-Ressayre, Landsberg-Manivel-Ressayre), lower bounds on matrix
multiplication (B\"{u}rgisser-Ikenmeyer) (these last two were already known to
fit into GCT), the chasms at depth 3 and 4 (Gupta-Kayal-Kamath-Saptharishi;
Agrawal-Vinay; Koiran), matrix rigidity (Valiant) and others. That is, the
original proofs, with what is often just a little extra work, already provide
representation-theoretic obstructions in the sense of GCT for their respective
lower bounds. This enables us to expose a new viewpoint on GCT, whereby it is a
natural unification and broad generalization of known results. It also shows
that the framework of GCT is at least as powerful as known methods, and gives
many new proofs-of-concept that GCT can indeed provide significant asymptotic
lower bounds. This new viewpoint also opens up the possibility of fruitful
two-way interactions between previous results and the new methods of GCT; we
provide several concrete suggestions of such interactions. For example, the
representation-theoretic viewpoint of GCT naturally provides new properties to
consider in the search for new lower bounds.
