bibkey: malkoun2023cousins
authors: Malkoun
year: 2023
title: Do these cousins of permanents satisfy the following inequality?
doi: null
url: https://mathoverflow.net/questions/456248/do-these-cousins-of-permanents-satisfy-the-following-inequality
claim: The main question asks for a positive constant for every pair of permutation subgroups, uniformly over all complex Hermitian positive semidefinite matrices of the fixed size.
strata_touched:
  - D5/S3/Quantum/Matrix/MalkounSubgroupLowerBoundRefutation
license: CC-BY-SA-4.0
triage: anchor

Malkoun’s two-sided cousins of permanents

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• Source: https://mathoverflow.net/questions/456248/do-these-cousins-of-permanents-satisfy-the-following-inequality.
• MathOverflow question 456248, revision 8, revision GUID 382F6B28-9E4F-456B-9254-E0A041A674B5.
• Author: Malkoun.
• The source was published in 2023. Its statement and the attributed excerpt below are used under CC BY-SA 4.0. The mathematical counterexample and Lean proof are repository work; no external source software is incorporated.

Main assertion

For an $n$ by $n$ complex Hermitian positive semidefinite matrix $H=(h_{ij})$ and subgroups $G,K\le {\Sigma }_n$, the source defines

$$f_{G,K}(H)=\sum _{(\sigma ,\tau )\in G\times K}\prod _{i=1}^n{h}_{\sigma (i),\tau (i)}.$$

It asks:

Does there exist a positive constant $c(n,G,K)$ depending on $n$, $G$ and $K$ such that the following holds? $\Re (f_{G,K}(H))\ge c(n,G,K){\prod }_{i=1}^n{h}_{ii}$.

The constant is chosen before the universally quantified matrix. The diagonal entries of a Hermitian matrix are real. Accordingly, the native real inequality uses ${\prod }_i\Re (h_{ii})$; diagonal_product_eq_real proves that its real embedding equals the source’s complex diagonal product.

The main assertion allows arbitrary subgroups. The author’s separate interest in products of symmetric groups on disjoint subsets, together with rank-two matrices, is a restriction and is not the assertion refuted here. The separate question with a trivial subgroup also remains outside this settlement.

Counterexample in the source domain

Use the source’s labels $1,\dots ,7$, the cyclic subgroups generated by

$$g=(1256437),k=(1275643),$$

and the complex Gram matrix $H=V^{\ast }V$, where

$$V=\left(\begin{array}{ccccccc}1& 3& 1& 3& 1& 1& 3\\ 2& 2& -2& 2& 0& -3& -2\end{array}\right).$$

Both generators have order seven. The complete double sum over their subgroups is $-23944$, whereas the diagonal product is $549250$. Thus the proposed lower bound fails for every positive real constant for this fixed subgroup pair. The additional strict-positivity question fails on this same matrix; this is one counterexample family and one main-assertion settlement.

Literature scope and unresolved boundaries

The source qualification inspected the revision-8 body, the supplied complete revision history, the primary comments, the linked question MO456435, and David Speyer’s answer 456542. The latter refutes a further arbitrary-operator generalization and explicitly leaves the original pure-tensor question uncertain. An arbitrary-operator counterexample alone supplies no Gram-matrix counterexample to the present statement.

The checked literature also includes arXiv:2508.13472v3, §4, Conjecture D and Proposition 4.2; Zhang, arXiv:1608.02844v1; Wanless, arXiv:2202.01867v1; and Tran, arXiv:2101.03428v5. Conjecture D restricts to Young subgroups and the proposition treats nested partitions. The Schur-power spectral and related permanent inequalities in the other references do not state a settlement of the arbitrary-subgroup rectangular cross-sum identified above.

The bounded primary and literature checks found no earlier equivalent complete settlement in that corpus. Zero answers on the inspected main page is only a page observation. Exhaustive citation coverage, worldwide priority and absence of a settlement outside the checked corpus are not established. The Young subgroup restriction and the trivial-subgroup question are not decided here.