Invariant (general notion)

This note is meant as a unified definition of “invariant” that covers the patterns used across the vault (cross-ratio, Minkowski interval, first integrals, tensor scalars, gauge invariance, moving frames/differential invariants, reduction by symmetries).

1. Invariants from a group action

Let $G$ be a group acting on a space $S$ (a set, topological space, manifold, vector space, space of fields, etc.) (see group action). Write the action as

The most basic meaning of “invariant” is simply fixed point. Let $G$ act on any space $X$. A point $x\in X$ is $G$-invariant if

Equivalently, its stabilizer subgroup is the whole group:

The main trick that unifies most meanings of “invariant” is:

many objects we care about (subsets, functions, tensors, equations, structures) can be viewed as points of another space $X$, and the original action on $S$ induces an action on $X$.
Then “invariant object” just means “invariant point of $X$”.

Start with an action $G\curvearrowright S$. Whenever you build a new space of objects out of $S$, there is usually an induced action.

1.1. Invariant subsets as fixed points in $\mathcal{P}(S)$

Let $\mathcal{P}(S)$ be the power set. Define the induced action

Then a subset $A\subseteq S$ is $G$-invariant (setwise invariant) iff it is a fixed point in this induced action:

It is often useful to distinguish:

• Setwise invariance: $g(A)=A$.
• Pointwise invariance: every point is fixed, $g\cdot a=a$ for all $a\in A$ and all $g$.

This is exactly the kind of “change of space of points” viewpoint we already use in Klein geometry (space of copies of a chosen feature under $G$).

1.2. Invariant functions as fixed points in a function space

Let $V$ be any set (often $\mathbb{R}$). Consider the function space $\mathrm{Map}(S,V)$. The induced action is the pullback/precomposition action

Then $f\in \mathrm{Map}(S,V)$ is a fixed point iff

which is the usual “numeric invariant” condition.

Equivalently, an invariant function is constant on $G$-orbits, hence it factors through the orbit map:

See also: quotient action (how actions descend to orbit spaces).

Invariant level sets. If $f$ is $G$-invariant, then each level set $f^{-1}(c)$ is an invariant subset of $S$ (a fixed point in $\mathcal{P}(S)$).

1.3. Invariants of a finite configuration

If you want an invariant of a finite set of points, you are really considering the induced action on configurations.

• Ordered $k$-tuples: $S^k$ with

  $$g\cdot (x_1,\dots ,x_k)=(g\cdot x_1,\dots ,g\cdot x_k).$$

  An invariant is a map $I:S^k\to V$ satisfying $I(g\cdot \mathbf{x})=I(\mathbf{x})$.

• Unordered $k$-element subsets: one can pass to the quotient by permutations, e.g.

  $${\mathrm{UConf}}_k(S):=S^k/{\mathfrak{S}}_k$$

  (often restricted to distinct points to avoid degeneracy).
  Then an invariant of an unordered configuration is a map

  $$I:{\mathrm{UConf}}_k(S)\to V$$

  constant on $G$-orbits.

Informal phrasing:

“Fix a set of elements of $S$; attach a number; apply a transformation; the number doesn’t change.”

1.4. Invariant tensors and (bi)linear forms

Many geometric “invariants” are not numbers attached to points of $S$, but structures on $S$ (metrics, symplectic forms, volume forms, connections, distributions, …). This is the same philosophy as above: tensors/forms are points in a tensor space, and invariance means fixed point for the induced action.

Suppose $S$ is a manifold and $G\subseteq \mathrm{Diff}(S)$ acts by diffeomorphisms. Let $\mathcal{T}$ be some space of tensor fields on $S$ (e.g. all $(0,2)$-tensor fields, or all differential 2-forms). Then $G$ acts on $\mathcal{T}$ by pullback:

So now $T\in \mathcal{T}$ is literally a “point”, and invariance is just fixed-point invariance in the sense of §1.3.

For example, a (0,2)-tensor field $B$ (a bilinear form at each tangent space) is $G$-invariant if it is a fixed point of this induced action:

Infinitesimally, if $X$ is an infinitesimal generator, this is equivalent to

See also: natural bundle (metrics/forms as points in natural bundles) and Killing vector field (infinitesimal invariance of a metric).

In the purely linear setting (a vector space $V$), one packages the same idea as an action of $\mathrm{GL}(V)$ on the set $\mathrm{Bil}(V)$ of bilinear forms by

so that “$B$ is invariant under a subgroup $G\le \mathrm{GL}(V)$” means exactly that $B$ is a fixed point for this action.
Equivalently, $G$ is contained in the stabilizer of $B$.

Euclidean, Minkowski, symplectic geometries as stabilizers

These three geometries can be summarized as: choose a preferred non-degenerate bilinear form (or 2-form), and look at the transformations that preserve it.

• Euclidean geometry. On $V={\mathbb{R}}^n$, the standard inner product $⟨\cdot ,\cdot ⟩$ is a symmetric positive definite bilinear form.
  Its stabilizer is the orthogonal group

  $$O(n)=\{A\in \mathrm{GL}(n):A^{T}A=I\}.$$

  Distances and angles arise as invariants built from $⟨\cdot ,\cdot ⟩$.

• Minkowski geometry / special relativity. On $V={\mathbb{R}}^{3,1}$, the Minkowski form $\eta$ is a symmetric non-degenerate bilinear form of signature $(3,1)$.
  Its stabilizer is the Lorentz group

  $$O(3,1)=\{A:A^T\eta A=\eta \}.$$

  The interval $s^2(p,q)=\eta (p-q,p-q)$ is an invariant of the induced action on 2-point configurations.

• Symplectic geometry. On $V={\mathbb{R}}^{2n}$, the structure is a non-degenerate skew-symmetric bilinear form (a symplectic form) $\omega$.
  Its stabilizer is the symplectic group

  $$Sp(2n,\mathbb{R})=\{A:A^T\omega A=\omega \}.$$

  In the manifold setting, one replaces $Sp(2n)$ by the group of symplectomorphisms $g$ with $g^{\ast }\omega =\omega$.

So “an invariant bilinear form” fits the note as an invariant point in the appropriate tensor space (fixed under pullback), and the corresponding geometry is organized by its stabilizer group.

2. Infinitesimal criterion (when $G$ is a Lie group)

If $G$ is a Lie group acting smoothly on a manifold $S$, each $\xi \in \mathfrak{g}$ yields a fundamental vector field ${\xi }_S$ on $S$.

Then $I:S\to \mathbb{R}$ is $G$-invariant iff

This is the bridge to the “PDE / first integral” viewpoint.

3. Flow/dynamics as a special case

If a dynamical system provides a flow ${\phi }_t:S\to S$, that is an action of $G=\mathbb{R}$ (or $\mathbb{Z}$) on $S$.

Then $F:S\to \mathbb{R}$ is an invariant iff

If the flow is generated by a vector field $X$, this becomes

i.e. $F$ is a first integral.

Remark: if one only has a semigroup action (e.g. time $t\ge 0$), one often uses forward invariance

since invertibility (and hence equality) may fail.

4. Variants you keep encountering

4.2. “Complete” invariants

A family $\{I_{\alpha }\}$ is complete (for a given class of objects) if it separates orbits:
two configurations lie in the same $G$-orbit iff all the invariants agree.
This is the classification goal behind moving frames and differential invariants.

See: moving frame, method of moving frames.

4.3. Gauge invariance and “symmetry of a theory”

In field theory, $S$ may be a space of fields and $G$ a group of gauge transformations.
Then “invariants” are functions constant on gauge orbits; a “symmetry of the theory” is typically a subgroup that preserves the action/criterion and hence preserves (or permutes) the set of solutions.

See: gauge theory, on theories, symmetries and gauge.

5. Canonical examples (for orientation)

• Projective geometry (cross-ratio on ${\mathbb{P}}^1$)

  Take $S={\mathbb{P}}^1(F)$ (projective line over a field $F$, see projective line) and

  $$G=\mathrm{PGL}(2,F)=\mathrm{GL}(2,F)/F^{\times }.$$

  (See also: projective linear group and projectivity.)
  The action is by fractional linear transformations: a class of matrices
  $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ acts on an affine coordinate $t\in F\cup \{\mathrm{\infty }\}$ by

  $$t⟼\frac{at+b}{ct+d}.$$

  The relevant configuration space is (an open subset of) 4-tuples of points,
  typically with the non-degeneracy condition “$A,B,C$ distinct” (often all four distinct).

  Definition (via normalization). Given distinct $A,B,C\in {\mathbb{P}}^1$, there exists a unique projectivity sending

  $$A\mapsto 0,B\mapsto 1,C\mapsto \mathrm{\infty }.$$

  The image of a fourth point $D$ under this unique projectivity is the cross-ratio $(A,B;C,D)$.

  Coordinate formula. In an affine chart (where subtraction/division makes sense),

  $$(A,B;C,D)=\frac{(C-A)(D-B)}{(C-B)(D-A)}.$$

  Invariance statement. For every $g\in \mathrm{PGL}(2,F)$,

  $$(A,B;C,D)=(gA,gB;gC,gD).$$

  In the language of §1.3, the cross-ratio is a $G$-invariant function on a suitable domain in $({\mathbb{P}}^1{)}^4$.
  Conceptually, it is a coordinate on (part of) the orbit space $({\mathbb{P}}^1{)}^4/\mathrm{PGL}(2)$.

• Special relativity (Minkowski interval)

  Take $S={\mathbb{R}}^{3,1}$ (see Minkowski space) with Minkowski bilinear form (signature convention may vary)

  $$\eta (x,x)=t^2-x^2-y^2-z^2.$$

  Let $G=O(3,1)$ (see Lorentz group) be the group of linear transformations preserving $\eta$:

  $$\eta (gx,gy)=\eta (x,y)\mathrm{\forall }g\in O(3,1).$$

  There are two very common “configuration” viewpoints:

  1. One-point invariant (vector norm). The map $I:S\to \mathbb{R}$ given by $I(x)=\eta (x,x)$ is $O(3,1)$-invariant.
     The sign of $\eta (x,x)$ (timelike/spacelike/lightlike) is also invariant.

  2. Two-point invariant (interval between events). For events $p,q\in S$, define the displacement $\mathrm{\Delta }=p-q$ and

     $$s^2(p,q):=\eta (\mathrm{\Delta },\mathrm{\Delta }).$$

     If one restricts to the linear Lorentz group acting on displacements, this is immediate; if one uses the full
     Poincaré group (Lorentz + translations) acting on events, the difference $p-q$ removes the translation.

  Physical interpretation.

  • If $s^2>0$ (timelike separation), the proper time between the events is $\tau =\sqrt{s^2}$ (in units where $c=1$).
  • If $s^2<0$ (spacelike separation), $\sqrt{-s^2}$ is the proper length measured in a frame where the events are simultaneous.
  • If $s^2=0$, the separation is lightlike and lies on the invariant light cone.

  So the “Minkowski interval” is exactly a $G$-invariant attached to a pair of points (a 2-configuration).

• Mechanics: $G=\mathbb{R}$ acting by a flow; invariants are first integrals.

• Linear algebra / tensors: $G$ acts by change of basis; invariants are basis-independent scalars (trace, determinant, eigenvalues, contractions).

Entry points

• Group actions & orbits: group action, Lie algebra action, quotient action, free action, transitive action, faithful action.
• Differential invariants / normalization: moving frame, method of moving frames.
• Projective invariants: projective line, cross-ratio, projective line, cross-ratio, and invariants, projectivity, projective linear group.
• Tensor / structure invariance: tensor#Invariants of a tensor, natural bundle, Killing vector field, Klein geometry.
• Dynamics / reduction by symmetries: first integral, symmetry group of a DE system, prolongation of a diffeomorphism.
• Physics (intervals, gauge, conservation): Minkowski space, Lorentz group, gauge theory, Noether's theorem, on theories, symmetries and gauge, Hermitian form, inner product and symplectic form relationship.