# Geometric complexity theory and tensor rank

@article{Brgisser2011GeometricCT, title={Geometric complexity theory and tensor rank}, author={Peter B{\"u}rgisser and Christian Ikenmeyer}, journal={ArXiv}, year={2011}, volume={abs/1011.1350} }

Mulmuley and Sohoni [GCT1, SICOMP 2001; GCT2, SICOMP 2008] proposed to view the permanent versus determinant problem as a specific orbit closure problem and to attack it by methods from geometric invariant and representation theory. We adopt these ideas towards the goal of showing lower bounds on the border rank of specific tensors, in particular for matrix multiplication. We thus study specific orbit closure problems for the group G =GL(W1) x GL(W2) x GL(W3) acting on the tensor product W=W1…

## 62 Citations

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It is a remarkable fact that two prominent problems of algebraic complexity theory, the permanent versus determinant problem and the tensor rank problem (matrix multiplication), can be restated as explicit orbit closure problems, and asymptotic versions of the the latter questions are of relevance in quantum information theory.

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