Let $f\in \mathbb{R}[x\_1,x\_2,x\_3,x\_4]$ defined by
$$f\_a(x\_1,x\_2,x\_3,x\_4)=\prod\_{1\leqslant i<j\leqslant4}(x\_i-x\_j)^{2a\_{ij}}$$
where $a=(a\_{12},a\_{13},a\_{14},a\_{23},a\_{24},a\_{34})\in \mathbb{N}^6$.
Define a $4\times4$ matrix $A\_f$ as follow:
$$A\_{f\_a}=\begin{bmatrix}
 L(1) & L(\frac{x\_2}{x\_1}) & L(\frac{x\_4}{x\_3}) & L(\frac{x\_2x\_4}{x\_1x\_3})\\
 L(\frac{x\_1}{x\_2}) & L(1) & L(\frac{x\_1x\_4}{x\_2x\_3}) & L(\frac{x\_4}{x\_3})\\
 L(\frac{x\_3}{x\_4}) & L(\frac{x\_2x\_3}{x\_1x\_4}) & L(1) & L(\frac{x\_2}{x\_1})\\
 L(\frac{x\_1x\_3}{x\_2x\_4}) & L(\frac{x\_3}{x\_4}) & L(\frac{x\_1}{x\_2}) & L(1)
\end{bmatrix}$$
where $L(\frac{x\_{i\_1}\cdots x\_{i\_k}}{x\_{j\_1}\cdots x\_{j\_k}})$ denotes the coefficient of 
$$\left(\prod\_{1\leqslant i<j\leqslant4}(x\_ix\_j)^{a\_{ij}}\right)\frac{x\_{i\_1}\cdots x\_{i\_k}}{x\_{j\_1}\cdots x\_{j\_k}}$$
in the expansion of $f$.


For example, when $a\_1=(a,0,0,0,0,b)$,
\begin{align}
f\_{a\_1}(x\_1,x\_2,x\_3,x\_4)&=(x\_1-x\_2)^{2a}(x\_3-x\_4)^{2b},\\
\\
(-1)^{a+b}A\_{f\_{a\_1}}&=
\begin{bmatrix}
 \binom{2a}{a}\binom{2b}{b} & -\binom{2a}{a-1}\binom{2b}{b} & -\binom{2a}{a}\binom{2b}{b-1} & \binom{2a}{a-1}\binom{2b}{b-1}\\
 -\binom{2a}{a-1}\binom{2b}{b} & \binom{2a}{a}\binom{2b}{b} & \binom{2a}{a-1}\binom{2b}{b-1} & -\binom{2a}{a}\binom{2b}{b-1}\\
 -\binom{2a}{a}\binom{2b}{b-1} & \binom{2a}{a-1}\binom{2b}{b-1} & \binom{2a}{a}\binom{2b}{b} & -\binom{2a}{a-1}\binom{2b}{b}\\
 \binom{2a}{a-1}\binom{2b}{b-1} & -\binom{2a}{a}\binom{2b}{b-1} & -\binom{2a}{a-1}\binom{2b}{b} & \binom{2a}{a}\binom{2b}{b}
\end{bmatrix}.
\end{align}
It is not difficult to verify that the determinant of $A\_{f\_{a\_1}}$ is nonzero. In fact,$$\det(A\_{f\_{a\_1}})=\binom{2a}{a}^4\binom{2b}{b}^4\frac{(2a+1)^2(2b+1)^2}{(a+1)^4(b+1)^4}.$$
Moreover, I verified that $\det(A\_{f\_a})\neq0$ for many simple $a\in \mathbb{N}^6$. 


**My question**


Does $\det(A\_{f\_a})\neq0$ hold for all $a\in \mathbb{N}^6$? Is there any significance for the matrix $A\_{f\_a}$? Any idea is welcome!



The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
For a given monomial $Y=\frac{x\_{i\_1}\cdots x\_{i\_k}}{x\_{j\_1}\cdots x\_{j\_k}}$ the coefficient $L(Y)$ multiplied by the constant $(-1)^{\sum\_{i<j} a\_{ij}}$ equals $$[Y]\prod\_{i,j}(1-x\_i/x\_j)^{a\_{ij}}=\int Y^{-1}d\mu,$$
where $d\mu$ is the measure on the $4$-dimensional torus $\mathbb{T}^4=\{(x\_1,x\_2,x\_3,x\_4)\in \mathbb{C}^4:|x\_1|=|x\_2|=|x\_3|=|x\_4|=1\}$ which density w.r.t. normalized Lebesgue measure equals to $\prod\_{i,j}(1-x\_i/x\_j)^{a\_{ij}}$ (the key observation is that this is always real and almost always positive). Your matrix is then (up to aforementioned sign) Gram matrix of the functions $1,x\_2/x\_1,x\_4/x\_3,x\_2x\_4/x\_1x\_3$ in $L^2(\mu)$. They are linearly independent, hence the determinant is strictly positive.


