# When is this cyclic representation irreducible?

Let $G$ be a finite group, and let $(\rho, W)$ be a representation of $G$ on $W$. We assume that $W = \bigoplus_i W_i$ is a direct sum of equivalent irreducible representations $W_i$. There are many ways to decompose $W$ into irreducibles, as is explained here, so this decomposition is far from unique.

If we take a vector $w \in W$, the subspace $$S = \operatorname{span}\left\{\rho_g w \, | \, g \in G\right\}$$ is an invariant subspace of $W$, and hence defines a subrepresentation (right?). My question is: when is this representation irreducible?

If $S$ is not always irreducible, how can I explicitly construct a representation containing $w$ that is irreducible?

Any answer is appreciated, but I would prefer an elementary explanation.

• $S$ is by definition a cyclic representation (or cyclic module), so your question actually is whether a cyclic module can have non-trivial submodules. Maybe this way of formulating the problem can give you some inspiration. Commented Apr 29, 2016 at 9:01
• Thanks. Do you have any recommended sources to read more about cyclic modules? I only have a vague idea of what a module is. Commented Apr 29, 2016 at 9:18
• Here are some great notes on representations of finite groups: maths.gla.ac.uk/~ajb/dvi-ps/groupreps.pdf. Chapters 4 to 10 of these notes: web.maths.unsw.edu.au/~danielch/modules12/beeren_notes.pdf cover the basics of modules, which is very easy if you think of a module as a sort of vector space over a ring (so not a field as you normally would). Commented Apr 29, 2016 at 9:28

## 1 Answer

The nonzero vector $(x_1,x_2,\ldots,x_n)$ of $X$ generates an irreducible submodule isomorphic to $W_1$ if and only if there are $FG$-isomorphisms $\phi_i:W_1 \to W_i$ with $\phi_i(x_1) = x_i$ for all $i$. Otherwise it generates a reducible submodule isomorphic to a number of copies of $W_1$.

The number will be equal to the maximum number of the $x_i$ that are pairwise distinct under $FG$-isomorphisms of the $W_i$.