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Two questions.

  1. How do I find all finite-dimensional linear representations of $\mathbb{Z}$, the additive group of the integers? I know that all finite-dimensional differentiable representations of the reals under addition are given by sending $t$ to $\exp(tA)$ where $A$ is a linear operator on the vector space of given dimension. So I can restrict those representations to the subgroup $\mathbb{Z}$, but are there more?

  2. What about the integers $\mod m$ under addition? How do I find all their finite-dimensional linear representations?

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Figuring this out for $\Bbb Z$ is easy because the group $\Bbb Z$ is generated by the single element $1$. In particular, for any representation $\rho:\Bbb Z \to \Bbb C^{n \times n}$, the following holds:

  • $\rho(1)$ is an invertible matrix
  • for any $k \in \Bbb Z$, we have $$ \rho(k) = \rho(k \cdot 1) = \rho(1)^k $$

Thus, every representation of $\Bbb Z$ can be written in the form $$ \rho(k) = A^k $$ where $A$ is an invertible matrix. Moreover, if we can find a matrix $B$ for which $\exp(B) = A$ (which we can, as long as $A$ is invertible), then this becomes $$ \rho(k) = \exp(kB) $$ so indeed every representation has the form you mentioned.

$\Bbb Z_m$ also is generated by $1$, but it has the additional property that $m \cdot 1 = 0$. Thus, the representations of $\Bbb Z_m$ are precisely those of the form $$ \rho(k) = A^k $$ for which $A^m = I$ ($I$ is the identity matrix). If $C$ is one of the "logarithms of the identity" (i.e. $\exp(C) = I$), then we can get the representation $$ \rho(k) = [\exp(C/m)]^k = \exp[(k/m)C] $$