# The Zak transform and representations induced from characters of an abelian subgroup

## TL;DR

This paper introduces a variant of the Zak transform for finite groups relative to an abelian subgroup, linking it to induced representations and applications in frame theory.

## Contribution

It develops a new Zak transform variant for finite groups and explores its connection to induced representations from abelian subgroups.

## Key findings

- Establishes a relationship between the Zak transform and induced representations.
- Demonstrates how the Zak transform analyzes right translations in $L^2(G)$.
- Provides applications to equiangular tight frames.

## Abstract

We consider a variant of the Zak transform for a finite group $G$ with respect to a fixed abelian subgroup $H$, and demonstrate a relationship with representations of $G$ induced from characters of $H$. We also show how the Zak transform can be used to study right translations by $H$ in $L^2(G)$, and give some examples of applications for equiangular tight frames.