Tensor product of group representations

For representations $V$ and $W$ of a group $G$, the tensor product representation is the tensor product $V\otimes W$ with diagonal action

For finite-dimensional group representations, the character of their tensor product is the pointwise product of their characters. In a tensor-product basis, the diagonal entries of $A\otimes B$ are $A_{ii}{B}_{jj}$, so their sum is $(\mathrm{tr}A)(\mathrm{tr}B)$. This gives a direct route from a character table to tensor-product decompositions.

The symmetric cube of a group representation $V$ is the symmetric part of $V^{\otimes 3}$. Over the complex numbers, its character isThe projection onto symmetric tensors averages the six permutations of the three factors. The identity permutation contributes $\chi (g{)}^3$, the three transpositions each contribute $\chi (g)\chi (g^2)$, and the two three-cycles each contribute $\chi (g^3)$. Cubic polynomial invariants are fixed vectors in ${\mathrm{Sym}}^3(V^{\ast })$, whose dimension follows by averaging this character for a finite group.

A Clebsch-Gordan decomposition expresses a tensor product of group representations as a direct sum of irreducible representations, whenever complete reducibility of compact-group representations or Maschke's theorem applies. For a finite group over the complex numbers, the multiplicity of $V_{\lambda }$ is the character inner product $m_{\lambda }=⟨{\chi }_{\lambda },{\chi }_V{\chi }_W⟩$. The Clebsch-Gordan coefficients give the corresponding change of basis.

The symmetric square of a group representation on $V$ is the quotient vector spacewith the induced tensor product representation. If the characteristic of a field is different from two, $vw\mapsto (v\otimes w+w\otimes v)/2$ identifies it with the invariant subspace fixed by the factor-exchange map $\tau$. Over the complex numbers, the projection operator $(I+\tau )/2$ givesIn characteristic two, the quotient definition still works, but this averaging projection operator does not exist. Real coordinate-quadratic perturbations of a nonmagnetic electronic multiplet transform in this symmetric square.

For a finite-dimensional irreducible complex representation $V$, invariant tensors in $V\otimes V$ correspond to equivariant maps $V^{\ast }\to V$. Schur's lemma makes their dimension one if $V$ is self-dual and zero otherwise. For a finite group, in particular, a real-character irreducible representation has precisely one trivial copy in its tensor square. For arbitrary complex irreducibles, the unconditional statement is that $V\otimes V^{\ast }$ contains the trivial representation once.

For an arbitrary group, a real-valued character need not imply self-duality: the one-dimensional group representation $n\mapsto 2^n$ of $\mathbb{Z}$ has real character but its tensor square has action $n\mapsto 4^n$, so contains no trivial representation.

For a $G$-representation $V$ and an $H$-representation $W$ over the same field, $V⊠W$ is the representation of $G\times H$ on $V\otimes W$ given by $(g,h)(v\otimes w)=gv\otimes hw$. Its character is $(g,h)\mapsto {\chi }_V(g){\chi }_W(h)$. For finite groups $G,H$, over a splitting field for finite group representations in characteristic zero the external products of irreducible representations form all irreducible representations of $G\times H$: character orthogonality gives irreducibility, and their squared degrees sum to $|G||H|$, proving exhaustion. Unlike the tensor product of two representations of one group, the two factors here act independently.

Identify the product with endomorphisms of the defining space. Special unitary group conjugation fixes scalar matrices and preserves the complementary traceless subspace. For $N\ge 2$, the latter is the complex Adjoint representation of a Lie algebra, which is irreducible because ${\mathfrak{sl}}_N$ is simple. With fundamental Dynkin index $1/2$, the trace-Casimir identity gives $C_2(\mathbf{N})=(N^2-1)/(2N)$; the tensor-product Casimir trace identity then gives $C_2(\mathbf{Adj})=N$.

A $(j,k)$ tensor has one fundamental representation factor for every upper index and a dual factor for every lower index. Under $A\in SU(n)$, upper factors transform by $A$ and lower factors by $A^{-1}$; unitarity identifies the latter with conjugate fundamental matrices. Thus complex conjugation exchanges $(j,k)$ and $(k,j)$. Index permutations and contraction with an invariant tensor commute with the action, producing invariant subspaces and decomposition maps.

An invariant tensor is fixed by every element of the represented group, so it spans or belongs to a trivial group representation. The special unitary group preserves its identity tensor and the upper and lower volume tensors. Such tensors make tensor contraction equivariant and permit invariant inner products and real structures.

The tensor-product weight diagram has six distinct nonzero differences of the three defining weights and a zero weight of multiplicity three. Identify the tensor product with endomorphisms of ${\mathbb{C}}^3$ under conjugation. Its scalar line is a singlet, while the traceless matrices give the irreducible adjoint octet, with two zero weights. The root commutators connect all of the traceless matrix units, proving irreducibility rather than inferring it solely from a drawing.