nLab faithful representation

Idea

An action or representation of an algebraic object such as a group or an algebra is faithful when the action of two elements being equal implies that these two elements are already equal. When the representation of a group $G$ on some $V$ is thought of as a functor $\mathbf{B}G \to \mathbf{B}Aut(V)$, then it is faithful precisely if this functor is a faithful functor.

A representation of an algebraic object, such as a group or an algebra, is a way of studying it by making it act on some object. The hope being that the behaviour of the object makes it easier to see the structure of the acting object. Another way of thinking of such a representation is that it is a homomorphism of the acting object into some other object that is (presumably) better understood. Making a group or algebra act on a vector space is the same as giving a homomorphism into the corresponding general linear group or endomorphism algebra, making a group act on a set maps it into the corresponding permutation group.

When studying an object via its representation then we really only “see” that part of the object that the representation sees. Thus there is the potential for forgetting information when passing to a representation. This can be a good thing, but it might not be. It is important to classify the possible scenarios and the label faithful representation is used for when no information is lost. This is in line with other uses of the word faithful.

Thus when we have a faithful representation, we can distinguish two elements of the acting object by their actions: if they always do the same thing then they were the same element.

Definition

Traditional

Let $(\mathcal{C}, \otimes)$ be a closed monoidal category, $A$ a monoid object in $\mathcal{C}$ and

$$\rho \;\colon\; A \otimes V \longrightarrow V$$

an action/representation on some $V \in \mathcal{C}$. Equivalently this is given by its $((V \otimes -) \dashv [V,-] )$-adjunct $\tilde \rho$ (“currying”), which is a homomorphism

$$\tilde \rho \colon A \longrightarrow [V,V]$$

from $A$ to the object of endomorphism of $V$ (the internal hom).

Definition

The representation $\rho$ is called faithful if its adjunct $\tilde \rho$ is a monomorphism.

For $\infty$-Actions

Let $\mathbf{H}$ be an (∞,1)-topos, $G \in Grp(\mathbf{H})$ be an ∞-group object and $\rho$ an ∞-action on some $V \in \mathbf{H}$. By the discussion at ∞-action this is equivalently a homotopy fiber sequence of the form

$$\array{ V &\longrightarrow& V/G \\ && \downarrow \\ && \mathbf{B}G } \,.$$

Suppose that $V$ is $\kappa$-compact object for some cardinal $\kappa$. By the existence of the $\kappa$-small object classifier $Obj_\kappa \in \mathbf{H}$ the above homotopy fiber sequence is itself the homotopy pullback of the universal fibration $\widehat {Obj}_\kappa \to Obj_\kappa$ along some morphism $\mathbf{B}G \longrightarrow \mathbf{B}\mathbf{Aut}(V) \hookrightarrow Obj_\kappa$ to the delooping of the automorphism ∞-group of $V$

$$\array{ V &\longrightarrow& V/G &\longrightarrow& V/\mathbf{Aut}(V) &\longrightarrow& \widehat{Obj}_\kappa \\ && \downarrow && \downarrow && \downarrow \\ && \mathbf{B}G &\longrightarrow& \mathbf{B}\mathbf{Aut}(V) &\hookrightarrow& Obj_\kappa } \,.$$

There is the corresponding $\infty$-group homomorphism (see at looping and delooping)

$$\tilde \rho \colon G \longrightarrow \mathbf{Aut}(V) \,.$$

If this is an n-monomorphism for some $n$ one might call the action “$n$-faithful”. If $G$ and $V$ are 0-truncated then any ∞-action of $G$ on $V$ is an ordinary action in the underlying 1-topos and this is faithful in the traditional sense if it is $n$-faithful for $n = 1$ in this higher sense.

Properties

Existence

Proposition

If $G$ is a compact Lie group, then there exists a finite-dimensional faithful representation of $G$.

E.g. Kowalski 14, proof of theorem 6.1.2

Proposition

For $G$ any algebraic group, then the regular representation is faithful. Moreover, it has finite-dimensional faithful sub-representations.

(e.g. Milne 12, IX, theorem 9.1)

Subquotients

Proposition

If $V$ is a finite dimensional representation of an affine algebraic group $G$ over a field $k$, then every finite dimensional representation of $G$ is isomorphic to a subquotient of $\otimes^n(V \oplus V^\ast)$, where $V^\ast$ is the dual representation?.

(e.g. Milne 12, VIII, theorem 11.7)

Examples

Example

Let $G$ be a discrete group and $V$ a set (hence $\mathcal{C} =$ Set with its Cartesian product). The adjunct of an action map $\rho \colon G \times V \to V$ is a function

$$\tilde \rho \colon G \longrightarrow End(V)$$

from $G$ to the set of endofunctions of $V$.

The action is faithful if this function $\tilde \rho$ is injective.

Observe that $G$ being a group and $\rho$ being an action means that $\tilde \rho$ factors through the inclusion of the automorphism group $Aut(V) \hookrightarrow End(V)$. (If $V$ is a finite set then this is a symmetric group).

So, equivalently the action is faithful if

$$\tilde \rho \colon G \longrightarrow Aut(V)$$

is injective, hence is a monomorphism of groups.

More specifically, if $V$ is equipped with the structure of a vector space and the action is by linear functions, hence is a linear representation, then this means that $\tilde \rho$ factors through the general linear group $GL(V)$ of $V$ through a group homomorphism

$$\tilde \rho \colon G \longrightarrow GL(V) \,.$$

Again, $\tilde \rho$ is faithful if this is an injection, hence a group monomorphism.

The same applies when $G$ is equipped with extra geometric structure, such as being a topological group or Lie group.

Example

A Lie groupoid is called effective when the action of all its automorphism groups of objects on their germs is faithful. See at effective Lie groupoid for more on this.

References

• James Milne, Basic theory of affine group schemes, 2012 (pdf)

• Kowalski, An introduction to the representation theory of groups, 2014