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A nonzero representation is reducible when it has a nonzero proper invariant subspace; otherwise it is an irreducible representation.

Tensor products $V_{j_L}\otimes V_{j_R}$ give the irreducible representations of the Spin(4) double cover. The simultaneous central sign acts by $(-1{)}^{2j_L+2j_R}$, so exactly the integer-sum pairs descend to SO(4). Their dimensions are $(2j_L+1)(2j_R+1)$. Under the diagonal spatial rotation subgroup, the Clebsch-Gordan decomposition for SU2 gives spins from $|j_L-j_R|$ through $j_L+j_R$ in steps of one.

For nonproportional roots $\gamma ,\delta$ of a reduced crystallographic root system, $S_{\gamma ,\delta }=\{\delta -p\gamma ,\dots ,\delta +q\gamma \}$ is consecutive, and $p-q=⟨\delta ,{\gamma }^{\vee }⟩$. Realize the crystallographic root system as the roots of a complex semisimple Lie algebra. The sum of the one-dimensional root spaces in this string is a module for the sl2 subalgebra associated with a root. Its weights are $⟨\delta ,{\gamma }^{\vee }⟩+2n$; they have one fixed parity. Complete reducibility gives a sum of irreducible representations of the sl2 Lie algebra. Two summands of this parity would overlap at weight zero when the parity is even, or at weights $\pm 1$ when it is odd, contradicting the one-dimensional root spaces. Thus there is one irreducible string, whose opposite endpoint weights sum to zero. This proves the relation. For distinct simple roots, $p=0$ and the cardinality is $1-⟨\delta ,{\gamma }^{\vee }⟩$. The nonproportional hypothesis excludes $\delta =\pm \gamma$, where zero is not a root and the consecutive-string statement fails.

A representation is semisimple when it is a direct sum of irreducible representations. Equivalently, every invariant subspace has an invariant complement.

If $\Vert f(x){\Vert }_{\mathrm{op}}\le 1$ and $\Vert f{\Vert }_{U^2}^4\ge cn>0$, select the singular values ${\lambda }_p\ge \sqrt{c/2}$ of the matrix Fourier blocks. Their weighted count $m={\sum }_p{d}_{{\rho }_p}$ lies between $cn/2$ and $2n/c$. Scaling unit right singular vectors and their corresponding unit left singular vectors by $\sqrt{d_{{\rho }_p}}$ gives matrices $U(p),V(p)$ satisfying ${\mathbb{E}}_{x}f(x)U(p){\rho }_p(x{)}^{\ast }={\lambda }_{p}V(p)$ and Hilbert-Schmidt inner product orthogonality within each chosen irreducible representation. The Schur averaging of rectangular matrices then gives ${\mathbb{E}}_x\Vert UP(x{)}^{\ast }b{\Vert }_2^2=\Vert b{\Vert }_2^2$ for the concatenated $U$ and the corresponding block diagonal matrix $P(x)$.

The regular group representation of a finite group over the complex numbers contains each irreducible representation with multiplicity equal to its dimension. Consequently the sum of the squared dimensions is the group order. Once a list of distinct irreducibles exhausts this sum, it is complete.

A unitary irreducible representation is both unitary and irreducible. Such representations are the building blocks in the harmonic analysis of groups and in quantum symmetry classifications.