It is an action of the group on a vector space by means of linear maps. It can be seen like a functor if we think of a group as a category.
Given a group $G$, a representation is then a homomorphism $\rho: G \to GL(V)$ where $V$ is a vector space. If we fix a basis we can say it is a matrix representation.
A vector subspace $W\subset V$ is $G$-invariant if $\rho(g)v\in W$ for every $g\in G$ and $v\in W$. It can lead to a subrepresentation or a quotient representation.
A representation $\rho:G\to GL(V)$ is irreducible if do not have non-trivial subrepresentations. Otherwise it is called reducible.
If the action of $G$ on $V-\{0\}$ is transitive then the representation is irreducible.
Since $\rho(G)$ is a subring of $End(V)$ then $V$ is a $\rho(G)$-module.
Related: spin representation.
Related: projective representation.
There is exactly 1 irreducible representation of $SU(2)$ in $GL(n)$ for every $n\in \mathbb N$. See this video. SU(2)
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Author of the notes: Antonio J. Pan-Collantes
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