Malkoun: the unrestricted subgroup permanent lower bound

Malkoun, Do these cousins of permanents satisfy the following inequality?, MathOverflow question 456248, revision 8, 382F6B28-9E4F-456B-9254-E0A041A674B5. The source note gives the identity, attribution, license and bounded literature comparison.

Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define $f_{G,K}(H)=\sum_{(\sigma,\tau)\in G\times K}\prod_{i=1}^n h_{\sigma(i),\tau(i)}$. Does there exist a positive constant $c(n,G,K)$ depending on $n$, $G$ and $K$ such that $\Re(f_{G,K}(H))\geq c(n,G,K)\prod_{i=1}^n h_{ii}$?

The complete quantifier order is

$$ \forall n\in\mathbb N\ \forall G,K\leq\operatorname{Sym}(n)\quad \exists c\in\mathbb R\quad c>0\ \land
\forall H\in M_n(\mathbb C),\quad (H=H^*\land H\succeq0)\Longrightarrow c\prod_i\Re(h_{ii})\leq\Re(f_{G,K}(H)). $$

claim has this proposition as its definition and result has type ¬ claim. Native index $i:\mathrm{Fin}(n)$ corresponds to source label $i+1$; every permutation acts on that same index set. crossSum sums over the actual subgroup types, without normalization. No Young-subgroup, rank, entry-sign or strictly positive diagonal condition is added to the universal assertion. diagonal_product_eq_real supplies the Hermitian real-diagonal bridge.

The source relates the expression to two subgroup symmetrizations of a pure tensor and to graph-associated functions. A lower bound would extend the positivity behind familiar permanent inequalities to a two-sided subgroup cross-sum. Positivity of a Gram matrix by itself does not imply positivity of a cross-sum over two different subgroup orbits.

The unrestricted question concerns actual permutation subgroups and matrices in the complex PSD cone. Negativity for arbitrary operators on a representation space leaves a realization gap. The counterexample below closes that gap with an explicit square Gram matrix in the original matrix domain.

Set $n=7$ and

$$ G=\langle(1\ 2\ 5\ 6\ 4\ 3\ 7)\rangle,\qquad K=\langle(1\ 2\ 7\ 5\ 6\ 4\ 3)\rangle. $$

In zero-based array form the generators are $(1,4,6,2,5,3,0)$ and $(1,6,0,2,5,3,4)$. Both have order seven. Their seven powers therefore enumerate the actual subgroups bijectively, using the pinned finEquivZPowers equivalence.

Let $V$ have columns $(1,2),(3,2),(1,-2),(3,2),(1,0),(1,-3),(3,-2)$ and embed its integer entries in $\mathbb C$. Then

$$ H=V^*V= \begin{pmatrix} 5&7&-3&7&1&-5&-1\ 7&13&-1&13&3&-3&5\ -3&-1&5&-1&1&7&7\ 7&13&-1&13&3&-3&5\ 1&3&1&3&1&1&3\ -5&-3&7&-3&1&10&9\ -1&5&7&5&3&9&13 \end{pmatrix}. $$

This is an actual $7$ by $7$ complex Hermitian positive semidefinite matrix, by Matrix.posSemidef_conjTranspose_mul_self. A rectangular Gram factor is sufficient: the source condition is on the resulting square matrix.

The diagonal is $(5,13,5,13,1,10,13)$ and has product $549250>0$. For powers $g^a$ and $k^b$, with $a,b=0,\ldots,6$, the seven row sums of the complete $49$-term sum are

$$ (568948,-90288,-206028,-182088,-90000,-100172,75684). $$

Their sum is $-23944$. Consequently every $c>0$ would require $549250c\leq-23944$, which is impossible. The same fixed matrix works for every positive $c$ for the fixed pair $G,K$.

D5/S3/Quantum/Matrix/MalkounSubgroupLowerBoundRefutation.result is the literal negation of the full claim. The native proof uses Subgroup.zpowers of actual Equiv.Perm (Fin 7) values, proves both generator orders, reindexes the complete subgroup sums by finEquivZPowers, and evaluates the resulting complex products exactly. The real diagonal equality is consumed in the diagonal-product calculation. The proof uses the pinned Gram-positivity supplier, rather than a positive-semidefiniteness assumption about the witness.

PROVED — Refutation of the complete main arbitrary-subgroup assertion. The native result proves ¬ claim using the displayed matrix and subgroups.

COMPUTED — Failure of the additional unrestricted strict-positivity assertion on the same family. The displayed matrix has positive diagonal and negative real cross-sum. The obstruction occurs already at the sign of the full cross-sum: positive semidefiniteness and positive diagonal do not ensure its nonnegativity for two arbitrary subgroups. A smaller positive constant cannot repair this sign failure. This consequence is the same counterexample, with no separate problem settlement claimed.

OPEN — The Young-subgroup question, including the author's rank-two restriction. The two cyclic subgroups here are not the specified products of symmetric groups on disjoint subsets.

OPEN — The separate trivial-subgroup question. Both subgroups in this counterexample are nontrivial.

The literature check has the finite scope stated in the source note. No worldwide-priority or exhaustive absence claim follows. The tensor formulation and arbitrary-operator follow-up are not used as substitutes for the original matrix domain. Independent review, repository integration and post-integration verification are distinct obligations from this mathematical statement.