Applications to Representation Theory

Tensor products play a central role in representation theory, allowing construction of new representations from old. This connects abstract algebra to geometry and physics.

TheoremTensor Product of Representations

If $V$ and $W$ are representations of group $G$ (i.e., homomorphisms $\rho_V: G \to GL(V)$ and $\rho_W: G \to GL(W)$ ), then $V \otimes W$ becomes a representation via: $\rho_{V \otimes W}(g)(\mathbf{v} \otimes \mathbf{w}) = \rho_V(g)(\mathbf{v}) \otimes \rho_W(g)(\mathbf{w})$

This construction is fundamental: it allows building complex representations from simple ones.

ExampleAngular Momentum in Quantum Mechanics

In quantum mechanics, spin states form representations of $SU(2)$ . Two spin- $\frac{1}{2}$ particles (each with 2-dimensional state space) combine via tensor product: $\mathbb{C}^2 \otimes \mathbb{C}^2 = \mathbb{C}^4$

This decomposes into spin-0 (singlet) and spin-1 (triplet): $\mathbb{C}^2 \otimes \mathbb{C}^2 \cong \mathbb{C}_0 \oplus \mathbb{C}_1^3$

The Clebsch-Gordan coefficients describe this decomposition.

TheoremSchur's Lemma and Tensor Products

For irreducible representations $V$ and $W$ of finite group $G$ :

If $V \not\cong W$ : $\text{Hom}_G(V, W) = 0$ If $V \cong W$ : $\text{Hom}_G(V, W) = \mathbb{C} \cdot \text{id}$

The tensor product $V \otimes W$ generally decomposes into irreducibles: $V \otimes W \cong \bigoplus_i m_i U_i$

where $U_i$ are irreducible representations and $m_i$ are multiplicities (Clebsch-Gordan series).

ExamplePolynomial Representations

For $GL_n(\mathbb{C})$ , the standard representation is $\mathbb{C}^n$ . Tensor powers decompose:

$\text{Sym}^k(\mathbb{C}^n)$ : homogeneous polynomials of degree $k$ (irreducible)
$\Lambda^k(\mathbb{C}^n)$ : alternating forms (irreducible for $k \leq n$ )

These generate all polynomial representations via tensor products and decompositions.

TheoremTensor Categories

The category of representations of $G$ with tensor product forms a tensor category (monoidal category):

Associativity: $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$
Unit object: trivial representation $\mathbb{C}$
Braiding: commutativity constraint $V \otimes W \cong W \otimes V$

This structure governs how representations combine and interact.

Remark

Tensor products of representations are ubiquitous in physics: combining particles (quantum mechanics), adding angular momenta, and constructing field theories. In mathematics, they enable induction/restriction functors, Frobenius reciprocity, and character theory. The decomposition of tensor products into irreducibles (Clebsch-Gordan theory) is computationally crucial in quantum chemistry and particle physics.