For a field F of characteristic zero, let operatornameImm_(2^m)(X)=∑_sigma∈ S_2mchi_(2^m)(sigma)∏_i=1^2mx_i,sigma(i), where chi_(2^m) is the irreducible character of S_2m indexed by the rectangular Young diagram (2^m). Is the family operatornameImm_(2^m):m≥ 1 VNP-complete under polynomial-size projections?

Bürgisser’s width-two rectangular immanant problem

Bürgisser’s problem asks whether width-two rectangular immanant families are $\mathrm{VNP}$ -complete in the arithmetic model. Earlier work established hardness results but left the stronger classification unresolved.

Known results

Brylinski and Brylinski (2003): partition families with separation $\Omega(n^\delta)$ , for some $\delta>0$ , are $\mathrm{VNP}$ -complete under c-reductions.
Mertens and Moore (2011): variable-width- $2$ immanants are $\#\mathrm{P}$ -hard; diagrams containing only the entry $2$ give $\oplus\mathrm{P}$ -hardness.
Mertens and Moore (2013): the family of all partitions of width $2$ is $\#\mathrm{P}$ -hard.

August 25, 2026 claimed classification

A manuscript dated in the latest report claims $\mathrm{VNP}$ -completeness under p-projections for both width-two and length-two rectangular Young-diagram immanants, directly addressing Bürgisser’s question. This is a complete-resolution claim, but the preprint is unrefereed and no independent verification or reported proof-gap analysis was found.

Current status (as of August 2026): The manuscript claims the width-two rectangular case is settled with $\mathrm{VNP}$ -completeness under p-projections, but that claim remains unverified.

Bürgisser’s width-two rectangular immanant problem

Primary source

Additional references

Progress summary

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