Reducible and Irreducible Representations

Reducing a representation into irreducibles is the daily arithmetic of symmetry. In molecular spectroscopy the $3N$ Cartesian displacements of an $N$-atom molecule form a reducible representation of the point group; decomposing it by formula (7) classifies the vibrational normal modes, predicts infrared and Raman activity, and block-diagonalizes the dynamical matrix into small blocks. In coordination chemistry the crystal-field splittings of Worked Example 3 underlie the colours and magnetism of transition-metal compounds. In particle physics the same logic organizes the hadron spectrum: tensor products of $SU(3)$ representations are reducible, and their decomposition is the "Eightfold Way" that sorted mesons and baryons into multiplets and predicted the $\Omega^-$. Wherever a numerical eigenvalue problem carries a symmetry, the averaging projector $P_\mu=\tfrac{d_\mu}{|G|}\sum_g\chi^{(\mu)}(g)^\ast D(g)$ pre-block-diagonalizes it, turning symmetry into computational speed. The open frontier is modular representation theory — the characteristic-$p$ world where Maschke fails — now central to number theory and the classification of finite simple groups.

Watch the matrices $D(g)$ act on $V$. Suppose one subspace $W$ is a trap the group can never escape — every $D(g)$ sends $W$ back into $W$. Then the action restricted to $W$ is a smaller, self-contained representation, and in a basis whose first vectors span $W$ each matrix takes a definite shape: a top-left block that is the action on $W$, and a lower-left block forced to be zero, since a basis vector of $W$ can only map into $W$. That vanishing corner is block upper-triangular form — the whole visual content of "reducible", a representation partly but not fully come apart.

Full reducibility is the tidier world one block to the right: the leftover directions are themselves invariant, so the coupling block dies too and the matrix is block-diagonal — two independent representations with no cross-talk. The complement is invariant because for a finite or compact group you can smear any inner product over $G$ to make it $G$-invariant; with respect to it every $D(g)$ is unitary, and the orthogonal complement of an invariant subspace can never leak into it. That averaging is the engine behind Maschke's theorem and Weyl's unitary trick. With full reducibility in hand, irreducibles behave like prime numbers and the direct sum like multiplication: every representation factors uniquely into irreducible pieces, the multiplicities $m_\mu$ playing the role of prime exponents. The character $\chi(g)=\operatorname{tr}D(g)$ performs that factorization by inspection — the second figure shows it at work on $S_3$.

Every statement in this article follows from one definition — the invariant subspace. $W\subseteq V$ is invariant when the group action, applied to any vector of $W$, stays inside $W$; that is what lets $W$ carry a representation of its own.

where: $D(g)$ — the operator assigned to $g$; $W$ — a subspace of $V$; $G$ — the group. Reading: no group element maps a vector of $W$ outside $W$, so $D(g)|_W$ is a self-contained representation. Reducibility is nothing more than the existence of such a subspace, provided it is neither everything nor nothing. The trivial subspaces $\{0\}$ and $V$ always count as invariant and are excluded; a proper, nonzero invariant subspace is what makes a representation decomposable in principle.

where: $\{0\}\neq W \subsetneq V$ — $W$ is proper and nonzero; $\subsetneq$ — proper subset. Reading: if no such $W$ exists, $D$ is irreducible — an atom with no internal invariant structure. Choosing a basis whose first $k$ vectors span the invariant $W$ turns invariance into a matrix shape: the action of $W$ on itself fills the top-left block, and the lower-left block must vanish because a basis vector of $W$ cannot acquire components outside $W$.

where: $D_1(g)$ — the action on $W$, a $k$-dimensional representation; $D_2(g)$ — the induced action on the quotient $V/W$; $B(g)$ — the coupling block, in general nonzero. Reading: reducible means block upper-triangular, and the surviving $B(g)$ still blocks a clean split. Full reducibility kills that coupling block. It happens precisely when the complement $W'$ can also be chosen invariant, so the two blocks evolve independently and the representation is a direct sum.

where: $\oplus$ — direct sum on $V = W\oplus W'$; $W'$ — an invariant complement, $D(g)W'\subseteq W'$. Reading: block-diagonal is the honest statement that $D$ has come apart into two smaller representations with nothing left over. The mechanism that supplies an invariant complement, for a finite group, is averaging: take any Hermitian inner product on $V$ and average it over the whole group.

where: $|G|$ — order of the group; $\langle\cdot,\cdot\rangle$ — any Hermitian inner product; $\langle\cdot,\cdot\rangle_G$ — the $G$-invariant one, obeying $\langle D(h)u,D(h)v\rangle_G=\langle u,v\rangle_G$. Reading: with respect to $\langle\cdot,\cdot\rangle_G$ every $D(g)$ is unitary, so orthogonal complements of invariant subspaces stay invariant — the content of Maschke's theorem. With full reducibility guaranteed, iterating the split expresses any representation as a direct sum of irreducibles with integer multiplicities.

where: $D^{(\mu)}$ — the $\mu$-th irreducible representation; $m_\mu$ — its multiplicity; $\cong$ — equivalence of representations. Reading: the unique factorization of a representation into atoms. The multiplicities are computed, not guessed, by pairing the character of $D$ against each irreducible character — the projection formula owned by the companion article on character orthogonality, recalled here as the working tool.

where: $\chi(g)=\operatorname{tr}D(g)$ — character of $D$; $\chi^{(\mu)}(g)$ — character of $D^{(\mu)}$; $\ast$ — complex conjugation. Reading: characters are class functions, so the sum runs effectively over conjugacy classes and returns a non-negative integer. Pairing a character with itself gives a decisive irreducibility test: the norm-squared of the character equals the sum of squared multiplicities, which is $1$ exactly when a single irreducible is present.

where: the middle sum runs over the multiplicities in (6). Reading: unit character-norm certifies an atom; the integer $\sum_\mu m_\mu^2$ counts the pieces with weight, so a norm of $2$ signals two irreducible components.

Whether reducibility actually delivers a direct sum depends on the group and the field. The table collects the four regimes that matter.

1 · Decomposing the permutation representation of S₃ (fully worked)

Let $S_3$ act on $\mathbb{C}^3$ by permuting coordinates, so $D(g)$ is the $3\times3$ permutation matrix of $g$. Decompose it into irreducibles using characters alone. The irreducible characters of $S_3$ on the classes $\{e\}$, transpositions, $3$-cycles (sizes $1,3,2$) are $A_1=(1,1,1)$, $A_2=(1,-1,1)$, $E=(2,0,-1)$.

Step 1. Find the character of the permutation representation. The trace of a permutation matrix is the number of fixed points, so on the identity $\chi(e)=3$; a transposition such as $(12)$ fixes the single point $3$, giving $\chi=1$; a $3$-cycle fixes nothing, giving $\chi=0$. Hence $\chi_{\mathrm{perm}}=(3,1,0)$.

Step 2. Test whether it is already irreducible with the character-norm criterion (8):

The norm is $2$, not $1$, so the representation is reducible and made of pieces with $\sum_\mu m_\mu^2=2$ — either two distinct irreducibles each once, or one irreducible twice.

Step 3. Extract the multiplicities with the projection formula (7), remembering to weight each class by its size:

Step 4. Assemble and check. The decomposition is $D_{\mathrm{perm}}\cong A_1\oplus E$: dimensions add, $1+2=3$, and the multiplicity check matches Step 2, $1^2+0^2+1^2=2$. The reading is the familiar one — the permutation representation always contains the trivial direction $(1,1,1)$ where all coordinates are equal, and the leftover plane $x_1+x_2+x_3=0$ carries the irreducible standard representation. Result: $D_{\mathrm{perm}}\cong A_1\oplus E$, trivial $\oplus$ standard, $3=1+2$.

2 · Certifying an irreducible and reading off a projector

Confirm the standard representation $E=(2,0,-1)$ of $S_3$ is irreducible, and find the dimension of the subspace of $\mathbb{C}^3$ fixed by all permutations directly from a character.

Step 1. Apply the norm test to $E$:

Unit norm certifies $E$ is irreducible — over $\mathbb{C}$, no basis makes its $2\times2$ matrices block-triangular.

Step 2. Count the invariants of the permutation representation. The multiplicity of the trivial representation equals the dimension of the subspace fixed by every $D(g)$, and by (7) with $\chi^{(\mathrm{triv})}\equiv1$,

Step 3. Interpret. Exactly one dimension of $\mathbb{C}^3$ is left untouched by all of $S_3$ — the symmetric direction $\tfrac1{\sqrt3}(1,1,1)$ — and $P=\tfrac1{|G|}\sum_g D(g)$ is the projector onto it, with $\operatorname{tr}P=m_{A_1}=1$ confirming its rank. This is the averaging that powers Maschke's theorem, reused as a computational projector. Result: $E$ is irreducible (norm $1$); the symmetric subspace has dimension $1$.

3 · Crystal-field splitting of a d-level (symmetry-lowering)

A free atom's $d$-level transforms under the $5$-dimensional irreducible representation $D^{(2)}$ of $SO(3)$ ($\ell=2$). In an octahedral crystal field the symmetry drops to the point group $O$. What happens to the level?

Step 1. Restricted to the finite subgroup $O\subset SO(3)$, the once-irreducible $5$-dimensional representation becomes reducible — the coarser $O$ cannot keep the $5$-plane irreducible. Computing the character of $D^{(2)}$ on the classes of $O$ and applying (7) gives the ligand-field splitting

Step 2. Interpret. Lowering the symmetry group reduces the representation: one irreducible atom of $SO(3)$ resolves into a doubly degenerate $e_g$ pair ($d_{z^2},d_{x^2-y^2}$) and a triply degenerate $t_{2g}$ triplet ($d_{xy},d_{yz},d_{zx}$), split by the crystal-field gap $\Delta_{\mathrm{oct}}$ of transition-metal spectra. Result: the $5$-fold $d$-level splits as $e_g\,(2)\oplus t_{2g}\,(3)$.

The vocabulary of this article — representation, character, irreducible — was invented essentially from nothing in a five-year burst at the close of the nineteenth century, with one man at its centre.

Texts. Stone and Goldbart, Mathematics for Physics, group representation theory · Arfken and Weber, Mathematical Methods for Physicists, group theory · Hassani, Mathematical Physics, group representations · Tinkham, Group Theory and Quantum Mechanics · Hamermesh, Group Theory and Its Application to Physical Problems.

Provenance. The invariant-subspace and block-form arguments follow Stone and Goldbart, Arfken and Weber, and Hassani; Maschke's proof and the Burnside sum rule are cross-checked against Tinkham and Hamermesh. A verifier should confirm: (1) the regular-representation character $\chi^{\mathrm{reg}}(g)=|G|\,\delta_{g,e}$ and the Burnside sum $\sum_\mu d_\mu^2=|G|$ for $S_3$, where $1^2+1^2+2^2=6$; (2) the permutation-representation decomposition $\chi_{\mathrm{perm}}=(3,1,0)\to A_1\oplus E$, multiplicities $1,0,1$, character-norm $2$; (3) the irreducibility of $E$ (norm $1$). All numbers are standard textbook values.

Prerequisites. Group Representations · Groups, Subgroups, and Cosets · Homomorphisms and Quotient Groups. Leads to. Schur's Lemmas · Characters and Orthogonality Theorems · Character Tables and Decomposition of Representations · Tensor Products and Clebsch–Gordan Decomposition.

Ask the tutor: Ask me to decompose any representation you like — hand me a character or a set of matrices and I will walk you through the norm test, the projection formula, and the atoms it breaks into.