Complexity and Completeness of Immanants
<|reference_start|>Complexity and Completeness of Immanants: Immanants are polynomial functions of n by n matrices attached to irreducible characters of the symmetric group S_n, or equivalently to Young diagrams of size n. Immanants include determinants and permanents as extreme cases. Valiant proved that computation of permanents is a complete problem in his algebraic model of NP theory, i.e., it is VNP-complete. We prove that computation of immanants is VNP-complete if the immanants are attached to a family of diagrams whose separation is $\Omega(n^\delta)$ for some $\delta>0$. We define the separation of a diagram to be the largest number of overhanging boxes contained in a single row. Our theorem proves a conjecture of Buergisser for a large variety of families, and in particular we recover with new proofs his VNP-completeness results for hooks and rectangles.<|reference_end|>
@article{brylinski2003complexity,
    title={Complexity and Completeness of Immanants},
    author={Jean-Luc Brylinski and Ranee Brylinski},
    journal={arXiv preprint arXiv:cs/0301024},
    year={2003},
    archivePrefix={arXiv},
    eprint={cs/0301024},
    primaryClass={cs.CC math.CO}
}