= Representation theory
{wiki}
Representation theory describes abstract symmetries by <linear maps> on <vector spaces>. A <group representation> assigns an invertible <linear map> to each <group> element, respecting multiplication. The payoff is that a symmetry problem becomes a problem about <matrices>, <invariant subspaces>, and <eigenvalues>. Decomposing the action can reveal combinations of variables that a symmetric system cannot mix.
Take the <cyclic group> $C_3=\{e,g,g^2\}$, with $g^3=e$. On three real coordinates, let $g$ send $(a,b,c)$ to $(c,a,b)$:
$$
\rho(g)=P=\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix},\qquad P^3=I.
$$
This <permutation representation> does more than relabel a list: it acts linearly on every possible assignment of values to the three positions. The <vector> $(1,1,1)$ stays fixed. The <vector subspace> $W=\{(a,b,c):a+b+c=0\}$ is also preserved, since a permutation does not change the sum.
Every <vector> separates uniquely into these two types:
$$
(a,b,c)=\bar a(1,1,1)+(a-\bar a,b-\bar a,c-\bar a),\qquad
\bar a=\frac{a+b+c}{3}.
$$
For example, $(4,1,1)=2(1,1,1)+(2,-1,-1)$. The first term belongs to a <trivial representation>; the second is a contrast with zero coordinate sum. This is a <direct sum> decomposition $\mathbb R^3=\mathbb R(1,1,1)\oplus W$. The symmetry preserves the mean and rotates the contrasts.
\Image[/cyclic-representation-decomposition.png]
{title=Three-coordinate cyclic symmetry preserves the mean and rotates the zero-sum contrasts by 120 degrees}
The rotation is literal after choosing the <orthonormal basis> $u=(2,-1,-1)/\sqrt6$, $v=(0,1,-1)/\sqrt2$ of $W$. In this basis,
$$
P|_W=\begin{pmatrix}-1/2&-\sqrt3/2\\\sqrt3/2&-1/2\end{pmatrix}.
$$
It is a rotation through $120$ degrees, and three applications return to the starting point. There is no nonzero invariant real line, because this rotation has no real <eigenvector>. Thus the two-dimensional action is an <irreducible representation> over the <real numbers>. Over the <complex numbers> it instead splits into two <eigenlines> with <eigenvalues> $e^{2\pi i/3}$ and $e^{-2\pi i/3}$. The coefficient <field> is part of an irreducibility claim.
Why would this decomposition matter for a physical calculation? Consider three equal masses with scalar displacements $x_i$ and the symmetric quadratic coupling energy
$$
V(x)=\frac{k}{2}\left[(x_1-x_2)^2+(x_2-x_3)^2+(x_3-x_1)^2\right],\qquad k>0.
$$
The <equations of motion> are $m\ddot x=-kLx$, where
$$
L=\begin{pmatrix}2&-1&-1\\-1&2&-1\\-1&-1&2\end{pmatrix}=3I-J
$$
is a <graph Laplacian> and $J$ is the all-ones <matrix>. The constant <vector> has <eigenvalue> zero: translating all coordinates equally costs no coupling energy. On $W$, $J$ vanishes and $L=3I$, so both independent contrast directions have <angular frequency> $\sqrt{3k/m}$. A three-coordinate coupled problem reduces to a free mean and two identical <harmonic oscillators>.
\Image[/symmetry-separated-oscillations.png]
{title=Equal pair couplings leave the mean fixed and make two independent zero-sum modes oscillate at the same frequency}
For initial displacement $x(0)=(4,1,1)$ and initial <velocity> zero, the exact solution is $x(t)=2(1,1,1)+\cos(\sqrt{3k/m}\,t)(2,-1,-1)$. The fixed mean and oscillating contrast can be read directly from the decomposition. More generally, a <linear operator> commuting with the group action carries its fixed-vector subspace into itself; with an invariant <inner product>, it preserves the corresponding orthogonal complement too.
For a <finite group>, averaging over the action constructs an invariant <inner product> over the <real numbers> or <complex numbers>. The resulting invariant complements explain <Maschke's theorem>: a finite-dimensional <group representation> decomposes into <irreducible representations>. <Characters> compress the action into traces and help identify those pieces. The assumptions matter; in <modular representation theory> the coefficient <field> can have <characteristic> dividing the group order, and such decompositions can fail. Chapter 4 of Pavel Etingof and coauthors' notes develops these mechanisms. https://math.mit.edu/~etingof/reprbook.pdf[Etingof and coauthors: Introduction to representation theory]{ref}.
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