Introduction to Geometric Complexity Theory I
summary
Compares determinant and permanent by giving explicit constructions and bounds: a graph-theoretic adjacency-weighted graph with 2^n−1 vertices yields a formula expressing perm_n as a determinant of size ≤ 2^n−1, while a geometric Taylor-expansion argument studies the tangent and degree‑2 part (a quadratic form of rank ≤ 2n) to prove the Mignon–Ressayre lower bound determinantal complexity ≥ n^2/2. Uses cyclic‑covering combinatorics, tangent/dual variety geometry, and explicit matrix examples.
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