nLab Schur's lemma

Idea

Schur’s lemma is one of the fundamental facts of representation theory. It concerns basic properties of the hom-sets between irreducible linear representations of groups.

The lemma consists of two parts that depend on different assumptions (a distinction often not highlighted in the literature):

1. The first statement applies over every ground field:

   It says that there are no non-zero homomorphisms between distinct (i.e. non-isomorphic) irreducible representations and any non-zero morphism among isomorphic irreducibles is an isomorphism.

2. The second statement applies only in the special case that the ground field is an algebraically closed field (such as the complex numbers) and that the representations are finite-dimensional:

   It says that, in this case, moreover the only non-trivial endomorphisms of an irreducible representation are multiples of the identity morphism.

Statement

Let $G$ be a group. In the following:

• representation means linear representation of $G$, linear over some ground field,

• finite dimensional representation means that the underlying vector space is a finite-dimensional vector space,

• an irreducible representation is one whose only $G$-invariant subspaces (fixed point spaces) are the trivial degenerate cases: the zero-subspace and the full space itself.

Proposition

(Schur’s lemma)

1. A homomorphism $\phi \;\colon\; V\to W$ between irreducible representations, is either the zero morphism or an isomorphism.

   It follows that the endomorphism ring of an irreducible representation is a division ring.

2. In the case that the ground field is an algebraically closed field, endomorphisms $\phi \;\colon\; V \to V$ of a finite dimensional irreducible representations $V$ are a multiples $c \cdot id$ of the identity operator.

   In other words, nontrivial automorphisms of irreducible representations, a priori possible by (1), are ruled out over algebraically closed fields.

Proof

As it goes with very fundamental lemmas, the proof of Schur’s lemma follows by elementary inspection.

Proof

(of Prop. )
For the first statement:

It is immediate to see that both the kernel as well as image of a homomorphism

$$V \overset{f}{\longrightarrow} W$$

of any $G$-representations are $G$-invariant subspaces (fixed point spaces). But by the very definition of irreducibility, the only such subspaces of $V$ and $W$ are the degenerate ones: their zero subspaces and the full spaces themselves.

Now if the kernel is all of $V$ or the image is zero, then $f$ is the zero morphism. The only case left is that the kernel is zero and the image is all of $W$, but this means that $f$ is injective and surjective and is hence an isomorphism.

For the second statement:

Now we use that over an algebraically closed field $k$ of every linear endomorphism of a finite dimensional vector space has an eigenvalue $c \in k$ (which is, ultimately, due to the fundamental theorem of algebra for algebraically closed fields). Now, with $f$ also the linear combination

$$(f - c \cdot \mathrm{id}) \;\colon\; V \longrightarrow V$$

is a homomorphism of $G$-representations. But then, by the first part, this must be an isomorphism or zero. But it is not an isomorphism, by construction, since now the eigenvectors with eigenvalue $c$ are in the kernel. Therefore, all of the linear combination must be zero

$$f - c \cdot id = 0$$

and hence

$$f = c \cdot id$$

is a multiple of the identity.

Interpretation in categorical algebra

The statement of Schur’s lemma is particularly suggestive in the language of categorical algebra.

Here it says that irreducible representations form a categorified orthogonal basis for the 2-Hilbert space of finite-dimensional representations, and even an orthonormal basis if the ground field is algebraically closed.

After decategorification this becomes equivalently the statement that the isomorphism classes of irreducible representations form an orthogonal basis for the representation ring, and even an orthonormal basis if the ground field is algebraically closed.

For more on this perspective see also at Gram-Schmidt process the section Categorified Gram-Schmidt process.

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We now explain this perspective of in more detail:

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Notice that the hom-sets in a category of representations $G Rep$ are canonically vector spaces: given any two homomorphisms $f,g \;\colon\; V \to W$ of $G$-representations, also the (value-wise) linear combination $c_1 f + c_2 g \;\colon\; V \to W$ is a $G$-homomorphism. ($G Rep$ is canonically enriched over Vect.)

Now it makes sense to regard this vector space-valued hom-functor on $G Rep$ as analogous

$$hom_G(-,-) \;\colon\; G Rep^{op} \times G Rep \longrightarrow Vect$$

as a categorified inner product (see at 2-Hilbert space for more on this).

This is a useful perspective even after decategorification:

For $V, W \in G Rep^{fin}$ two finite-dimensional representations, write

$$\langle V,W\rangle \;\coloneqq\; dim\big( hom_G(V,W) \big) \;\in\; \mathbb{N}$$

for the dimension of the vector space of homomorphism between them (e.g. tom Dieck 09, p. 29).

This construction only depends on the isomorphism classes of $V$ and $W$, and hence descends to a function

$$\langle -,-\rangle \;\colon\; G Rep^{fin}_{/\sim} \times G Rep^{fin}_{/\sim} \longrightarrow \mathbb{N}$$

on sets of isomorphism classes. In fact, under direct sum and tensor product of representations, $G Rep^{fin}_{/\sim}$ is a rig, and this pairing is linear with respect to the underlying additive monoid structure:

$$\left\langle V_1 \oplus V_2 \,,\, W \right\rangle \;=\; \left\langle V_1 \,,\, W \right\rangle + \left\langle V_2 \,,\, W \right\rangle$$

and

$$\left\langle V \,,\, W_1 \oplus W_w \right\rangle \;=\; \left\langle V \,,\, W_1 \right\rangle + \left\langle V \,,\, W_2 \right\rangle$$

(This is, ultimately, due to the universal property of direct sum as a biproduct.)

To further strengthen the emerging picture, we may consider the group completion of the commutative monoid $G Rep^{fin}_{/\sim}$ by passing to its Grothendieck group, in fact its Grothendieck ring if we remember also the tensor product of representations. This commutative ring is called the representation ring

$$R(G) \;\coloneqq\; K\left( G Rep^{fin}_{/\sim} \right)$$

of $G$. By the evident $\mathbb{Z}$-linear extension, the above pairing gives an actual symmetric $\mathbb{Z}$-bilinear inner product on $R(G)$:

(1)$$\langle -,- \rangle \;\colon\; R(G) \times R(G) \longrightarrow \mathbb{Z}$$

Now the underlying abelian group of $R(G)$ is a free abelian group whose canonical generators are nothing but the isomorphism classes $V_i$ of the finite-dimensional irreducible representations:

$$R(G) \;\simeq_{\mathbb{Z}}\; \mathbb{Z}\big[ \{V_i\}_i \big] \,.$$

In summary, in the language of linear algebra, the irreducible representations $[V_i]$ constitute a canonical linear basis of the representation ring.

In terms of this language, Schur’s lemma becomes the following statement:

The canonical linear basis of the representation ring given by the irreducible representations is

1. generally: an ortho-gonal basis;

2. even an ortho-normal basis if the ground field is algebraically closed field

with respect to the canonical decategorified inner product from (1).

Generalizations and variants

For simple modules

Part (1) of Schur’s lemma is essentially category-theoretic and can be generalized in many ways, for example, by replacing the group $G$ by some $k$-algebra and taking the representations compatible with the action of $k$.

For simple objects in an abelian category

More generally, part (1) of Schur’s lemma applies to simple objects in any abelian category, and the endomorphism ring of such a simple object is a division ring, as proved here.

For (2), if the endomorphism rings of all objects in an abelian category are finite-dimensional over an algebraically closed field $k$, then the endomorphism ring of a simple object is $k$ itself, as proved here. This is the case for the category of finite-dimensional complex representations of a group.

For Bridgeland stable objects

The statement of Schur’s lemma applies also to objects which are stable with respect to a Bridgeland stability condition, see there.

References

Named after Issai Schur.

Lecture notes:

• Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina: Prop. 1.16 & Cor. 1.17 in: Introduction to representation theory, Student Mathematical Library 59, AMS (2011) [arXiv:0901.0827, ams:stml-59]

• Tammo tom Dieck, §1.1.2 in: Representation theory (2009) [pdf, pdf]

See also

• Wikipedia, Schur’s lemma