Permanent versus determinant: not via saturations
Computational Complexity
2017-03-01 v2 Representation Theory
Abstract
Let Det_n denote the closure of the GL_{n^2}(C)-orbit of the determinant polynomial det_n with respect to linear substitution. The highest weights (partitions) of irreducible GL_{n^2}(C)-representations occurring in the coordinate ring of Det_n form a finitely generated monoid S(Det_n). We prove that the saturation of S(Det_n) contains all partitions lambda with length at most n and size divisible by n. This implies that representation theoretic obstructions for the permanent versus determinant problem must be holes of the monoid S(Det_n).
Cite
@article{arxiv.1501.05528,
title = {Permanent versus determinant: not via saturations},
author = {Peter Bürgisser and Christian Ikenmeyer and Jesko Hüttenhain},
journal= {arXiv preprint arXiv:1501.05528},
year = {2017}
}
Comments
12 pages; shortened title, corrected error in proof, added bound on stretching factor, provided explicit examples