Permanent versus determinant: not via saturations

Peter Bürgisser; Christian Ikenmeyer; Jesko Hüttenhain

doi:10.1090/proc/13310

Home/Archive/arXiv

arXiv · 1501.05528

Permanent versus determinant: not via saturations

Peter Bürgisser·Christian Ikenmeyer·Jesko Hüttenhain

View original source ↗Read source document ↗Export metadata ↓

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).

Explore related subjects

cs.CC math.RT

Keep this discovery

BibTeX RIS

Peter Bürgisser, Christian Ikenmeyer, Jesko Hüttenhain. 2015-01-22. Permanent versus determinant: not via saturations. https://doi.org/10.1090/proc%2F13310

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Computational Complexity of Holant Problems on 4-regular Graphs from the Stable Subgroup Sequence of $SL(2,\mathbb{C})$

The Holant framework provides a general setting for studying counting problems and includes graph homomorphisms (\#GH) and counting constraint satisfaction problems (\#CSP) as special cases. Over the past twenty years, a series of computational complexity dichotomies have been established for Holant problems, but the classification for complex-valued signatures is still open. The main obstacle is the case in which all signatures have even arity. In this paper, we establish a dichotomy for Holant problems with a complex-valued 4-ary signature, which is a key base case for the full classification of Holant problems. We present a new strategy by introducing Schur's theorem, the classification of finite subgroups of $\mathrm{SL}(2,\mathbb{C})$ and stable subgroup sequences into the proof. These new techniques are of independent interest.

cs.CC↗

arXiv2026

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard--Milani\v{c}--\v{S}torgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

cs.CC↗

arXiv2026

Topology inside NC$^1$

We show that ACC$^0$ is precisely what can be computed with constant-width circuits of polynomial size and polylogarithmic genus. This extends a characterization given by Hansen, showing that planar constant-width circuits also characterize ACC$^0$. Thus polylogarithmic genus provides no additional computational power in this model. We consider other generalizations of planarity, including crossing number and thickness. We show that constant-width circuits of polynomial size and thickness two already suffice to capture all of NC$^1$.

cs.CC↗