Fourier transform on lattice strip

Question

I am working with a triangular lattice $L=\{n_1 a_2 + n_2 a_2 : n\in\mathbb{Z}^2 \}$ and $a_1 = \pmatrix{1 \\ 0}$ and $a_2 = \frac{1}{2} \pmatrix{-1 \\ \sqrt{3}}$, and I want to compute the Pontryagin dual of a strip of this lattice, i.e.

$$L_N := \{n_1 a_2 + n_2 a_2 : n_1 \in \mathbb{Z}, n_2 = 0,...,N \} \subset L.$$

It is known that the Pontryagin dual of the triangular lattice is the set of all characters $\chi_k : L \to S^1$ such that $$ \chi_k(x) = e^{ik\cdot x}, \quad x\in L $$ with $k\in B$, where $B$ is the first Brillouin zone of the triangular lattice (or any retiling of it).

It is also know that for a LCA group $G$ and a closed subgroup $H$ of $G$. The Pontryagin dual of $H$ is the quotient $\hat{H} = \hat{G}/H^{\perp}$, where $H^{\perp} = \{\chi \in \hat{G} : \chi(x) = 1\ \forall x\in G\}$.

I was hoping that I could use this train of thought to determine the Pontryagin dual of $L_N$.

It is true that $L_N \subset L$ but $(L_N,+)$ is not a closed subgroup of $(L,+)$ (as thankfully pointed out by @MateuszKwaśnicki) so one cannot apply the above statement to determining the Pontryagin dual of $L_N$.

Question: If $H$ is a quotient group of $G$, i.e. $H = G/Q$ for some $Q$, is there a clear relationship between $\hat{G}$ and $\hat{H}$, as is the case when $H$ is a closed subgroup of $G$?

Any hints on computing the dual of the lattice strip would be much appreciated, as would any references that deal with this sort of theory.

Edit notes: $L_N$ is not a closed subgroup of $L$ but rather a quotient.

Answer 1

There's a perfect duality here:

• If $H$ is a closed subgroup of $G$, then $\hat H$ is canonically isomorphic with $\hat G / H^\perp$, where $H^\perp$ is the set of all characters on $G$ which are equal to one on $H$.

• If $H = G / Q$ with $Q$ a closed subgroup of $G$, then $\hat H$ is canonically isomorphic with $Q^\perp$, the set of all characters on $G$ which are equal to one on $Q$ (and this set is a closed subgroup of $\hat G$).

See, for example, 1 and the references therein.

---

Regarding $L_N$, if we identify $(N+1)a_2$ with zero (that is, we consider $L_N$ as the quotient group $L / \{k (N+1) a_2 : k \in \mathbb Z\}$), then $L_N$ is isomorphic to the product group $\mathbb Z \times \mathbb Z_{N+1}$, and so the dual is isomorphic to $\mathbb T \times \mathbb Z_{N+1}$, where $\mathbb T$ is the circle group. The characters on $L_N$ are precisely those characters on $L$ which are equal to one at $(N+1)a_2$. The dual of $L_N$ is graphically represented as follows:

In the above image, $N = 2$, vector $a_1$ is shown in purple, $a_2$ in blue, gray area is the Brillouin zone (left) or the fundamental region (right), and three black lines (solid, dashed and dotted) correspond to characters on $\hat L_N$. (Auxiliary red lines demarkate four tiles that can be used to make both pictures.)

Note, however, that we can equally well identify $(N+1)a_2$ with, say, $a_1$ (so that $L_N$ is the quotient group $L / \{k (N+1) a_2 - k a_1 : k \in \mathbb Z\}$). Then $L_N$ is simply isomorphic to $\mathbb Z$, and $\hat L_N$ is isomorphic to $\mathbb T$. The picture now is:

In the above image again $N = 1$, and there is only one black line.

---

Reference:

1 Pontryagin duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pontryagin_duality&oldid=49720