Implementing geometric complexity theory: On the separation of orbit closures via symmetries

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