geometric representation theory
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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):
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
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.
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
(Schur’s lemma)
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.
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 multiple $c \cdot id$ of the identity operator.
In other words, nontrivial automorphisms of irreducible representations, a priori possible by (1), are ruled out over algebraically closed fields.
As it goes with very fundamental lemmas, the proof of Schur’s lemma follows by elementary inspection.
For the first statement:
It is immediate to see that both the kernel as well as image of a homomorphism
of any $G$-representations are $G$-invariant subspaces. 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
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
and hence
is a multiple of the identity.
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
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
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
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:
and
(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
of $G$. By the evident $\mathbb{Z}$-linear extension, the above pairing gives an actual symmetric $\mathbb{Z}$-bilinear inner product on $R(G)$:
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:
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
generally: an ortho-gonal basis;
even an ortho-normal basis if the ground field is algebraically closed field
with respect to the canonical decategorified inner product from (1).
Part (1) 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 $k$.
More generally, given an abelian category, part (1) of Schur’s lemma applies to the simple objects (see there) and the endomorphism ring of a simple object is a division ring.
For (2), if the endomorphism rings of all objects in an abelian category are finite-dimensional over an algebraically closed field $k$ (as is the case for group representations), then the endomorphism ring of a simple object is $k$ itself.
The statement of Schur’s lemma applies also to objects which are stable with respect to a Bridgeland stability condition, see there
Named after Issai Schur.
Lecture notes include
See also
Last revised on April 28, 2021 at 10:20:08. See the history of this page for a list of all contributions to it.