Tensor Products of Representations — Mathematics Knowledge System

Overview

The tensor product of two representations (V, rho) and (W, sigma) of a group G is the representation on V tensor W defined by (rho tensor sigma)(g) = rho(g) tensor sigma(g). Understanding how tensor products decompose into irreducibles is a central problem in representation theory. For finite groups, characters provide the key: the character of the tensor product is the pointwise product of the characters. Clebsch-Gordan theory describes the decomposition for SU(2) and other Lie groups; the Littlewood-Richardson rule handles symmetric groups via Young tableaux.

Intuition

If you have two quantum mechanical systems with symmetry group G, the combined system has both representations acting simultaneously on V tensor W. The decomposition into irreducibles tells you which 'multiplets' appear in the combined system. For SU(2) (spin), the Clebsch-Gordan series j1 tensor j2 = |j1-j2| + ... + j1+j2 is the mathematical statement of angular momentum addition: two spins j1 and j2 can combine to give total spin ranging from |j1-j2| to j1+j2.

Formal Definition

Definition

For representations rho: G -> GL(V) and sigma: G -> GL(W), the tensor product is (rho tensor sigma): G -> GL(V tensor W), (rho tensor sigma)(g)(v tensor w) = rho(g)v tensor sigma(g)w. If chi_rho and chi_sigma are the characters, the character of the tensor product is chi_{rho tensor sigma}(g) = chi_rho(g) * chi_sigma(g). The multiplicity of irreducible V_lambda in the tensor product is <chi_rho * chi_sigma, chi_lambda> = (1/|G|) sum_g chi_rho(g)*chi_sigma(g)*chi_lambda(g)^* .

(\rho \otimes \sigma)(g)(v \otimes w) = \rho(g)v \otimes \sigma(g)w

Tensor product action

\chi_{\rho \otimes \sigma}(g) = \chi_\rho(g)\cdot \chi_\sigma(g)

Character of tensor product

\rho \otimes \sigma \cong \bigoplus_\lambda m_\lambda V_\lambda,\quad m_\lambda = \langle \chi_\rho \chi_\sigma, \chi_\lambda \rangle

Decomposition into irreducibles

V_j \otimes V_{j'} = \bigoplus_{k=|j-j'|}^{j+j'} V_k

Clebsch-Gordan series for SU(2)

Notation

Notation	Meaning
$V \otimes W$	Tensor product of representations
$\chi_\rho$	Character of representation rho
$m_\lambda$	Multiplicity of irreducible V_lambda
$c^\lambda_{\mu\nu}$	Littlewood-Richardson coefficient

Theorems

Theorem 1: Character Product Formula

If rho and sigma are representations of a finite group G with characters chi_rho and chi_sigma, then chi_{rho tensor sigma} = chi_rho * chi_sigma (pointwise product). The multiplicity of each irreducible V_lambda in rho tensor sigma is the inner product <chi_rho * chi_sigma, chi_lambda>_G = (1/|G|) * sum_{g in G} chi_rho(g)*chi_sigma(g)*overline{chi_lambda(g)}.

Theorem 2: Clebsch-Gordan Theorem for SU(2)

The irreducible representations of SU(2) are V_j (dimension 2j+1) for j = 0, 1/2, 1, 3/2, ... The tensor product decomposes as V_j tensor V_{j'} = direct sum_{k=|j-j'|}^{j+j'} V_k. This is the mathematical form of angular momentum addition in quantum mechanics.

Theorem 3: Littlewood-Richardson Rule

For the symmetric group S_n with irreducibles V_lambda indexed by partitions lambda of n, the tensor product (induced product) V_mu tensor V_nu decomposes as direct sum_lambda c^lambda_{mu,nu} V_lambda where the Littlewood-Richardson coefficient c^lambda_{mu,nu} counts the number of skew tableaux of shape lambda/mu and content nu satisfying the LR condition (reading word is a ballot sequence).

Worked Examples

1
The standard rep has j = 1/2 (spin-1/2). V_{1/2} has dimension 2.
2
By the Clebsch-Gordan theorem: V_{1/2} tensor V_{1/2} = V_0 direct-sum V_1.
$V_{1/2} \otimes V_{1/2} = V_0 \oplus V_1$
3
V_0 is the trivial (singlet) representation, dimension 1. V_1 is the adjoint (triplet), dimension 3. Total dimension: 1+3 = 4 = 2*2, consistent.
4
Characters: chi_{1/2}(theta) = sin(theta)/sin(theta/2)... or more simply, chi_{1/2} = 2 at e. Check: chi(e)^2 = 4 = 1 + 3. Confirmed.

✓ Answer

V_{1/2} tensor V_{1/2} = V_0 + V_1 (singlet + triplet). This is the spin-singlet and spin-triplet decomposition of two spin-1/2 particles.

Practice Problems

Mediumproof writing

Prove that if chi_rho and chi_sigma are characters of representations rho, sigma of a finite group G, then chi_rho * chi_sigma is also a character (of the tensor product).

Common Mistakes

Common Mistake

Confusing direct sum and tensor product of representations.

The direct sum V + W has dimension dim V + dim W and acts block-diagonally: g sends (v,w) to (gv,gw). The tensor product V tensor W has dimension dim V * dim W and acts as g sends v tensor w to gv tensor gw. The character of the direct sum is chi_V + chi_W; the character of the tensor product is chi_V * chi_W.

Quiz

For SU(2), the tensor product V_1 tensor V_1 (two spin-1 particles) decomposes as:

V_0 + V_1V_0 + V_1 + V_2V_2 onlyV_0 + V_2

Historical Background

The decomposition of tensor products was studied by Clebsch and Gordan in the 19th century for SL(2,C). Weyl's 1925-26 work on representations of semisimple Lie groups systematized the tensor product theory. The Littlewood-Richardson rule (1934) gave a combinatorial algorithm for symmetric group tensor products. Kashiwara's crystal basis theory (1990s) provides a modern algebraic approach.

1872
Clebsch-Gordan coefficients introduced for angular momentum coupling
Alfred Clebsch, Paul Gordan
1925
Weyl develops the complete reducibility theorem for semisimple groups
Hermann Weyl
1934
Littlewood and Richardson give the rule for decomposing symmetric group tensor products
D.E. Littlewood, A.R. Richardson
1990
Kashiwara introduces crystal bases for quantum group representations
Masaki Kashiwara

Summary

The tensor product of reps (V,rho) and (W,sigma) is rho tensor sigma on V tensor W: (rho tensor sigma)(g) = rho(g) tensor sigma(g).
Character of tensor product = product of characters: chi_{rho tensor sigma} = chi_rho * chi_sigma.
Clebsch-Gordan for SU(2): V_j tensor V_{j'} = direct sum_{k=|j-j'|}^{j+j'} V_k.
Multiplicities computed by inner product: m_lambda = <chi_rho * chi_sigma, chi_lambda>.

References

BookFulton, W. and Harris, J. Representation Theory: A First Course. Springer, 1991.
BookSerre, J.P. Linear Representations of Finite Groups. Springer, 1977.
WebsiteWikipedia -- Tensor product of representations