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 $$ f_{G, K}(H) = \sum_{(\sigma, \tau) \in G \times K} \prod_{i = 1}^n h_{\sigma(i), \tau(i)}, $$ where $H = (h_{ij})$. Does there exist a positive constant $c(n, G, K)$ depending on $n$, $G$ and $K$ such that the following holds? $$ \Re(f_{G, K}(H)) \geq c(n, G, K) \prod_{i=1}^n h_{ii}$$ In the previous formula, $\Re(.)$ denotes the real part.
I am interested in a more special case of the above, where $G$ and $K$ are each products of symmetric groups on disjoint subsets of $\{1, \ldots, n\}$ (like for example the symmetric group on $\{1, 3 \}$ times the symmetric group on $\{2, 4\}$). Moreover, in the special case I am interested in, $H$ has rank $2$. That being said, the question does make sense in general. I did not yet run numerical tests. I wonder if anyone has seen such inequalities, whether proved or conjectured. If $G$ is trivial, then these have definitely been studied. They include the known Marcus's inequality for the permanent if $K$ is the full symmetric group $\Sigma_n$. I will have to dig in the literature to see if they have been proved in general or not if $K$ is an arbitrary subgroup of $\Sigma_n$. But what about these two-sided cousins? Have they been studied too, please?
Motivation: they do occur when studying some special functions on the configuration space of $n$ distinct points that are associated to graphs, in the special case where the graph is a tree, though the groups $G$ and $K$ are each products of symmetric groups (on disjoint sets of indices between $1$ and $n$, inclusively).
Additional questions:
- Are the inequalities true, known to be true/false or conjectured to be true if one of the two groups is trivial?
- Is it true that the real part of $f_{G, K}$ is positive on the cone of all positive semidefinite matrices with positive entries on their diagonal?
Note that on the previous cone, the ratio of the real part of $f_{G, K}$ divided by the product of the diagonal elements of $H$ is bounded, since it descends to a continuous (actually smooth) function on the product of $n$ copies of $\mathbb{C}P^{n-1}$. Indeed, if we write $H = V^*V$, for some complex $n$ by $n$ matrix $V$, then both $f_{G, K}$ and the product of the diagonal elements of $H$ have the same homogeneity as functions of the columns of $V$. So by a continuity/compactness argument, we get that the ratio is bounded (so the real part of the ratio is also bounded).
So the main inequality would follow for a given $(G, K)$ provided the real part of $f_{G, K}$ does not vanish anywhere on the cone of $n$ by $n$ hermitian positive semidefinite matrices with positive diagonal elements.
Edit: the problem can be reformulated using the same ideas that show that the permanent of an hermitian positive semidefinite matrix is the norm squared of a certain tensor. I hope that this remark will excuse me from writing up all the details. If someone wants me to write more, please let me know.
Here is the equivalent problem. Let $v_i \in \mathbb{C}^n$ be nonzero vectors, for $i = 1, \ldots, n$ and let $G$ and $K$ be two subgroups of the symmetric group $\Sigma_n$. Define $$\psi_G = \sum_{\sigma \in G} \bigotimes_{i=1}^n v_{\sigma(i)}$$ and similarly $$\psi_K = \sum_{\sigma \in K} \bigotimes_{i=1}^n v_{\sigma(i)}.$$ If $\langle ., . \rangle$ denotes the standard hermitian inner product on $\mathbb{C}^n$, my question is whether or not $$ \Re \langle \psi_G, \psi_K \rangle > 0, $$ no matter what non-zero vectors $(v_i)$ in $\mathbb{C}^n$ we start with.
This is a more coordinate-free and geometric (sort of) formulation of my question.
Note that the case where, say, one of $G$ and $K$ is trivial and the other is a subgroup, say $H$, of $\Sigma_n$ are equivalent, and are also equivalent to the case of $G$ and $K$ both equal to $H$. This can be easily proved by "transferring" one index to the other side and using that the left translate of $H$ using an element of $H$ is nothing but a permutation of $H$.
As a remark, the questions asked in this post led to a kind of interesting other post (in my humble opinion): A question regarding symmetrizing the tensor product of vectors in two different ways.
