Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols

In the settings of The Real Numbers: Standing Notation and Background and The Flat Torus: Standing Notation, with the dimension $n$, the initial segment $[n]$ and the integer lattice $\mathbb{Z}^{n}$, let $M\in\mathbb{N}$, and let the side $L=2M+1$, the cube $\Gamma_{M}$, the set $W_{L}$ and the lattice torus $\mathbb{L}_{M}$ be those of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field. For $m\in\mathbb{Z}$, $C_{m}:\mathbb{R}\to\mathbb{R}$ is the trigonometric monomial $C_{m}(t)=\cos(2\pi mt)$ of Cell Integrals of the Trigonometric Monomials. Finite products of real numbers indexed by $[n]$ are those of Finite Product Notation in a Field, and sums over finite index sets those of Sum over a Finite Index Set. For $i\in[n]$, $R_{i}:\mathbb{Z}^{n}\to\mathbb{Z}^{n}$ is the map changing the sign of the $i$-th component and keeping the others.

1. (Interaction)¶ A reflection-symmetric periodic pair interaction of side $L$ is a map $\mathfrak{j}:\mathbb{Z}^{n}\to\mathbb{R}$ such that

$$\mathfrak{j}(w+Ll)=\mathfrak{j}(w)\qquad\text{and}\qquad\mathfrak{j}(R_{i}w)=\mathfrak{j}(w)\qquad\text{for all }w,l\in\mathbb{Z}^{n}\text{ and }i\in[n].$$

2. (Interaction matrix)¶ The interaction matrix of a reflection-symmetric periodic pair interaction $\mathfrak{j}$ of side $L$ is the map $J:\mathbb{L}_{M}\times\mathbb{L}_{M}\to\mathbb{R}$, $J(y,z)=\mathfrak{j}\bigl(L(y-z)\bigr)$. Here $L(y-z)=w-w'\in\mathbb{Z}^{n}$ for $y=L^{-1}w$ and $z=L^{-1}w'$ with $w,w'\in W_{L}$, and every site is of this form by The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §lattice.

3. (Symbol)¶ The symbol of $\mathfrak{j}$ is the map $\hat{J}:\Gamma_{M}\to\mathbb{R}$,

$$\hat{J}(k)=\sum_{w\in W_{L}}\mathfrak{j}(w)\prod_{i=1}^{n}C_{k_{i}}\bigl(L^{-1}w_{i}\bigr)\qquad(k\in\Gamma_{M}),$$

a sum of real numbers over the nonempty finite set $W_{L}$ of The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field §lattice, the product being over $[n]$ with $k_{i}\in\mathbb{Z}$.