Hyperpfaffians and Geometric Complexity Theory

The hyperpfaffian polynomial was introduced by Barvinok in 1995 as a natural generalization of the well-known Pfaffian polynomial to higher order tensors. We prove that the hyperpfaffian is the unique smallest degree SL-invariant on the space of higher order tensors. We then study the hyperpfaffian's computational complexity and prove that it is VNP-complete. This disproves a conjecture of Mulmuley in geometric complexity theory about the computational complexity of invariant rings.

[1]  Volker Strassen,et al.  Algebraic Complexity Theory , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[2]  Markus Bläser,et al.  Generalized matrix completion and algebraic natural proofs , 2018, Electron. Colloquium Comput. Complex..

[3]  Leslie G. Valiant,et al.  Completeness classes in algebra , 1979, STOC.

[4]  Avi Wigderson,et al.  Search problems in algebraic complexity, GCT, and hardness of generators for invariant rings , 2020, CCC.

[5]  Ketan Mulmuley,et al.  Geometric Complexity Theory V: Efficient algorithms for Noether Normalization , 2012 .