# On vanishing of Kronecker coefficients

@article{Ikenmeyer2017OnVO, title={On vanishing of Kronecker coefficients}, author={Christian Ikenmeyer and Ketan Mulmuley and Michael Walter}, journal={computational complexity}, year={2017}, volume={26}, pages={949-992} }

We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood–Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood–Richardson coefficients, unless P = NP.We also show that there exists a #P-formula for a particular subclass of Kronecker coefficients whose positivity is NP-hard to decide. This is an evidence that…

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## References

SHOWING 1-10 OF 57 REFERENCES

Geometric complexity theory III: on deciding nonvanishing of a Littlewood–Richardson coefficient

- Mathematics
- 2012

We point out that the positivity of a Littlewood–Richardson coefficient $c^{\gamma}_{\alpha, \beta}$ for sln can be decided in strongly polynomial time. This means that the number of arithmetic…

Membership in Moment Polytopes is in NP and coNP

- Mathematics, Computer ScienceSIAM J. Comput.
- 2017

We show that the problem of deciding membership in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group is in NP and coNP. This is the…

Explicit Proofs and The Flip

- Computer Science, MathematicsArXiv
- 2010

Any proof of the arithmetic implication of the $P$ vs. $NP$ conjecture is close to an explicit proof in the sense that it can be transformed into an explicitProof by proving in addition that arithmetic circuit identity testing can be derandomized in a blackbox fashion.

Geometric Complexity Theory II: Towards Explicit Obstructions for Embeddings among Class Varieties

- Mathematics, Computer ScienceSIAM J. Comput.
- 2008

The results indicate that the fundamental lower bound problems in complexity theory are, in turn, intimately linked with explicit construction problems in algebraic geometry and representation theory.

Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

- Mathematics, Computer Science2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
- 2016

It is proved that the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial lower bounds on dc(perm), the determinantal complexity of the permanent polynomial.

Permanent versus determinant, obstructions, and Kronecker coefficients

- Computer Science, MathematicsArXiv
- 2015

The concept of a representation theoretic obstruction is introduced, which has close links to algebraic combinatorics, and some of the insights gained so far on the complexity of testing the positivity of Kronecker coefficients are addressed.

Semi-invariants of quivers and saturation for Littlewood-Richardson coefficients

- Mathematics
- 2000

Σ(Q,α) is defined in the space of all weights by one homogeneous linear equation and by a finite set of homogeneous linear inequalities. In particular the set Σ(Q,α) is saturated, i.e., if nσ ∈…

Kronecker coefficients for one hook shape

- Mathematics
- 2012

We give a positive combinatorial formula for the Kronecker coefficient g_{lambda mu(d) nu} for any partitions lambda, nu of n and hook shape mu(d) := (n-d,1^d). Our main tool is Haiman's \emph{mixed…

Computing Multiplicities of Lie Group Representations

- Mathematics, Computer Science2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
- 2012

It is shown that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures.