Symmetry and Groups

Equivariance states that a transformation of a physical object can be carried through a map without changing the result.

This chapter develops the general language. Rotations and their irreducible representations are treated in Representations of SO(3) and O(3).

Transformations and Group Actions

Definition (group). A group \(G\) is a set with an associative composition, an identity \(e\), and an inverse \(g^{-1}\) for each \(g\in G\).

Definition (group action). An action of \(G\) on a set \(\mathcal X\) assigns an object \(g\mathbin{\cdot}x\) to every \(g\in G\) and \(x\in\mathcal X\), such that

\[ e\mathbin{\cdot}x=x, \qquad g_1\mathbin{\cdot}(g_2\mathbin{\cdot}x) =(g_1g_2)\mathbin{\cdot}x. \]

We use active transformations: \(g\mathbin{\cdot}x\) is the transformed physical object, described in the same coordinate frame as \(x\). For example, a proper rotation sends a position to

\[ \boldsymbol r' = R\boldsymbol r. \]

Euclidean transformations

An element of the three-dimensional Euclidean group is a pair

\[ g=(Q,\boldsymbol t), \qquad Q\in \mathrm{O}(3), \quad \boldsymbol t\in\mathbb R^3, \]

acting on a point as

\[ g\mathbin{\cdot}\boldsymbol r =Q\boldsymbol r+\boldsymbol t. \]

The group law is

\[ (Q_1,\boldsymbol t_1)(Q_2,\boldsymbol t_2) = \left( Q_1Q_2,\, \boldsymbol t_1+Q_1\boldsymbol t_2 \right). \]

Thus \(\mathrm{E}(3)=\mathbb R^3\rtimes\mathrm{O}(3)\): translations and orthogonal transformations do not form an ordinary direct product.

For a scalar field \(f:\mathbb R^3\to\mathbb F\), where \(\mathbb F=\mathbb R\) or \(\mathbb C\), the corresponding active field action is

\[ \left[g\mathbin{\cdot}f\right](\boldsymbol r) = f\left(Q^{-1}(\boldsymbol r-\boldsymbol t)\right). \]

More generally, if the value of a field belongs to a space \(V\) carrying a spatial representation \(\rho_V(Q)\), then

\[ \left[g\mathbin{\cdot}f\right](\boldsymbol r) = \rho_V(Q) f\left(Q^{-1}(\boldsymbol r-\boldsymbol t)\right). \]

The transformation of the argument and the transformation of the field value are distinct. A scalar field has only the former; a vector or tensor field has both.

An internal value such as electronic spin may instead carry an action of a separate group. Spatial and spin actions, and the conditions under which they become joint, are developed in Spatial and Spin Symmetries.

Atomic centers and relabeling

For atom-centered data, a Euclidean transformation moves every center,

\[ \boldsymbol R_i' =Q\boldsymbol R_i+\boldsymbol t, \]

while an attached angular object transforms under \(Q\). Once all spatial arguments are written relative to their centers, a global translation leaves the relative coordinates unchanged. This is why the nontrivial local transformation type is usually an \(\mathrm{O}(3)\) representation rather than a finite-dimensional representation of the full \(\mathrm{E}(3)\).

A relabeling of atoms acts on the carrier index \(i\); permutation symmetry additionally requires the corresponding atomic attributes to be permuted. This does not replace the \(\mathrm{O}(3)\) action on the angular components. Spatial transformations and relabeling may be considered together, but they are different actions and should remain distinguishable in notation.

Representations

When the transformed objects form a vector space, a linear group action is a representation.

Definition (representation). A representation of \(G\) on a vector space \(V\) is a map

\[ \rho_V:G\longrightarrow \operatorname{GL}(V) \]

such that

\[ \rho_V(e)=I_V, \qquad \rho_V(g_1g_2)=\rho_V(g_1)\rho_V(g_2). \]

The pair \((V,\rho_V)\), not the dimension of \(V\) alone, specifies the transformation type. Two three-component objects need not transform in the same way: a polar vector and an axial vector agree under \(\mathrm{SO}(3)\) but differ under inversion.

The representation is the basis-independent linear action. After choosing a basis \(B\) of \(V\), it is described by matrices

\[ D_B(g)=[\rho_V(g)]_B. \]

If another basis \(B'\) uses component vectors related by

\[ [v]_{B'}=T[v]_B, \]

then

\[ D_{B'}(g)=T D_B(g)T^{-1} . \]

The matrices have changed, but the representation has not. This distinction is central to Wigner matrices, spherical harmonics, and Clebsch–Gordan coefficients.

Equivariant Maps

Let \(G\) act on both \(\mathcal X\) and \(\mathcal Y\).

Definition (equivariance). A map \(F:\mathcal X\to\mathcal Y\) is \(G\)-equivariant if

\[ F(g\mathbin{\cdot}x) = g\mathbin{\cdot}F(x) \qquad \text{for every }g\in G. \]

For representation spaces this becomes

\[ \boxed{ F\left(\rho_X(g)x\right) = \rho_Y(g)F(x) }. \]

An invariant map is the special case in which the output carries the trivial action:

\[ F\left(\rho_X(g)x\right)=F(x). \]

Proposition (composition). If \(F:\mathcal X\to\mathcal Y\) and \(H:\mathcal Y\to\mathcal Z\) are \(G\)-equivariant, then \(H\circ F\) is \(G\)-equivariant.

Indeed,

\[ (H\circ F)(g\mathbin{\cdot}x) =H(g\mathbin{\cdot}F(x)) =g\mathbin{\cdot}(H\circ F)(x). \]

This closure under composition is a structural fact about equivariant maps; it does not depend on how a map is implemented.

Intertwiners

Definition (intertwiner). A linear map \(A:V\to W\) between two representations is an intertwiner if

\[ A\rho_V(g)=\rho_W(g)A \qquad \text{for every }g\in G. \]

Thus an intertwiner is precisely a linear equivariant map. A change of basis between two coordinate realizations of the same representation satisfies

\[ T D_B(g)=D_{B'}(g)T. \]

The same equation is later used to characterize angular couplings. It is also the correct rule for converting components between software conventions: the conversion must commute with the physical group action.