nLab
faithful representation

Faithful representations

Context

Representation theory

Mathematics

Faithful representations

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 GG on some VV is thought of as a functor BGBAut(V)\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 (𝒞,)(\mathcal{C}, \otimes) be a closed monoidal category, AA a monoid object in 𝒞\mathcal{C} and

ρ:AVV \rho \;\colon\; A \otimes V \longrightarrow V

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

ρ˜:A[V,V] \tilde \rho \colon A \longrightarrow [V,V]

from AA to the object of endomorphism of VV (the internal hom).

Definition

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

For \infty-Actions

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

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

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

V V/G V/Aut(V) Obj^ κ BG BAut(V) Obj κ. \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)

ρ˜:GAut(V). \tilde \rho \colon G \longrightarrow \mathbf{Aut}(V) \,.

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

Properties

Existence

Proposition

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

E.g. Kowalski 14, proof of theorem 6.1.2

Proposition

For GG 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 VV is a finite dimensional representation of an affine algebraic group GG over a field kk, then every finite dimensional representation of GG is isomorphic to a subquotient of n(VV *)\otimes^n(V \oplus V^\ast), where V *V^\ast is the dual representation?.

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

Examples

Example

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

ρ˜:GEnd(V) \tilde \rho \colon G \longrightarrow End(V)

from GG to the set of endofunctions of VV.

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

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

So, equivalently the action is faithful if

ρ˜:GAut(V) \tilde \rho \colon G \longrightarrow Aut(V)

is injective, hence is a monomorphism of groups.

More specifically, if VV 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 ρ˜\tilde \rho factors through the general linear group GL(V)GL(V) of VV through a group homomorphism

ρ˜:GGL(V). \tilde \rho \colon G \longrightarrow GL(V) \,.

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

The same applies when GG 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

Last revised on November 9, 2018 at 06:00:43. See the history of this page for a list of all contributions to it.