---- Let $G$ be a finite [[group]], $V$ a finite-dimensional [[vector space]], and $(\rho, V)$ > [!definition] Definition. ([[irreducible group representation]]) > A [[group representation|representation]] $(\rho, V)$ of $G$ on $V$ is called **irreducible** (or an ** irrep**) if, with respect to $\rho$, $V$ has no proper, nontrivial [[group-invariant subspace|G-invariant]] [[linear subspace|subspace]]. > [!basicproperties] Properties. > - [[Maschke's Theorem]] > - [[Number of isomorphism classes of irreducible representations equals number of conjugacy classes]] > - [[order of group equals sum of squares of irrep dimensions]] > [!basicexample] > - Any 1-dimensional [[group representation|representation]] (e.g., the [[trivial group representation]] or the [[sign representation]] on $S_{n}$) is [[irreducible group representation|irreducible]]. > [!justification] Motivation. > If $V$ has proper nontrivial a $G$-[[group-invariant subspace|invariant]] [[linear subspace|subspace]] $W$, then [[group-invariant subspace admits group-invariant complement over C|W admits a]] proper nontrivial $G$-[[group-invariant subspace|invariant]] [[complement of a linear subspace|complement]] $W'$ such that $G = W \oplus W'$. Then $\rho=\rho |_{W} \oplus \rho |_{W'},$ and we have 'reduced' $\rho$ into a [[direct sum of representations|direct sum]] of simpler representations. An **irreducible representation** is one for which this 'reducing' cannot be performed. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```