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\begin{document}

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\section*{Schur&#39;s lemma}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{proof}{Proof}\dotfill \pageref*{proof} \linebreak
\noindent\hyperlink{InterpretationInCategoricalAlgebra}{Interpretation in categorical algebra}\dotfill \pageref*{InterpretationInCategoricalAlgebra} \linebreak
\noindent\hyperlink{GeneralizationsAndVariants}{Generalizations and variants}\dotfill \pageref*{GeneralizationsAndVariants} \linebreak
\noindent\hyperlink{for_simple_modules}{For simple modules}\dotfill \pageref*{for_simple_modules} \linebreak
\noindent\hyperlink{for_simple_objects_in_an_abelian_category}{For simple objects in an abelian category}\dotfill \pageref*{for_simple_objects_in_an_abelian_category} \linebreak
\noindent\hyperlink{for_bridgeland_stable_objects}{For Bridgeland stable objects}\dotfill \pageref*{for_bridgeland_stable_objects} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Schur's lemma is one of the basic facts of [[representation theory]]. It concerns basic properties of the [[hom-sets]] between [[irreducible representations|irreducible]] [[linear representations]] of [[groups]].

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

\begin{enumerate}%
\item The first statement applies over \emph{every} [[ground field]]:

It says that there are no non-[[zero morphism|zero]] [[homomorphisms]] between distinct (i.e. non-[[isomorphism|isomorphic]]) [[irreducible representations]]


\item 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 vector space|finite-dimensional]]:

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



\end{enumerate}
\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

Let $G$ be a [[group]]. In the following

\begin{itemize}%
\item ``representation'' means \emph{[[linear representation]] of $G$}, linear over some [[ground field]].


\item ``finite dimensional representation'' means that the underlying [[vector space]] is a [[finite-dimensional vector space]].



\end{itemize}
An

\begin{itemize}%
\item \emph{[[irreducible representation]]} is one whose only $G$-[[invariant]] [[subspaces]] are the trivial degenerate cases: the [[zero object|zero]]-subspace and the full space itself.

\end{itemize}
\begin{prop}
\label{}\hypertarget{}{}
\textbf{(Schur's lemma)}

\begin{enumerate}%
\item 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]].


\item In the case that the [[ground field]] is an [[algebraically closed field]] of [[characteristic zero]]; [[endomorphisms]] $\phi \;\colon\; V \to V$ of a [[finite-dimensional vector space|finite dimensional]] [[irreducible representations]] $V$ are a multiple $c id$ of the [[identity morphism|identity operator]].

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



\end{enumerate}
\end{prop}
\hypertarget{proof}{}\subsection*{{Proof}}\label{proof}

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]]

\begin{displaymath}
V \overset{f}{\longrightarrow} W
\end{displaymath}
of any $G$-[[representations]] are $G$-[[invariant]] [[subspaces]]. But by the very definition of [[irreducible representation|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 object|zero]], $f$ is the [[zero morphism]]. The only case left is that the [[kernel]] is [[zero object|zero]] \emph{and} the [[image]] is all of $W$, but this means that $f$ is [[injective map|injective]] and [[surjective map|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

\begin{displaymath}
(f - c \mathrm{id}) \;\colon\; V \longrightarrow V
\end{displaymath}
is a [[homomorphism]] of $G$-[[representations]]. But then, by the first part, this must be an [[isomorphism]] or [[zero morphism|zero]]. It is not an [[isomorphism]], by construction, since now the [[eigenvectors]] with [[eigenvalue]] $c$ are in the [[kernel]]. Therefore the linear combination it must be [[zero morphism|zero]]

\begin{displaymath}
f - c id = 0
\end{displaymath}
and hence

\begin{displaymath}
f = c id
\end{displaymath}
is a multiple of the identity.

\hypertarget{InterpretationInCategoricalAlgebra}{}\subsection*{{Interpretation in categorical algebra}}\label{InterpretationInCategoricalAlgebra}

The statement of \emph{Schur's lemma} is particularly suggestive in the language of [[categorical algebra]].

Here it says that [[irreducible representations]] form a [[categorification|categorified]] \emph{[[orthogonal basis]]} for the [[2-Hilbert space]] of [[finite-dimensional vector space|finite-dimensional]] [[representations]], and even an \emph{[[orthonormal basis]]} if the [[ground field]] is [[algebraically closed field|algebraically closed]].

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

For more on this perspective see also at \emph{[[Gram-Schmidt process]]} the section \emph{\href{Gram-Schmidt+process#CategorifiedGramSchmidtProcess}{Categorified Gram-Schmidt process}}.

$\,$

We now explain this perspective of in more detail:

$\,$

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 category|enriched]] over [[Vect]].)

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

\begin{displaymath}
hom_G(-,-) \;\colon\; G Rep^{op} \times G Rep \longrightarrow Vect
\end{displaymath}
as a [[categorification|categorified]] [[inner product]] (see at \emph{[[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 vector space|finite-dimensional]] [[representations]], write

\begin{displaymath}
\langle V,W\rangle
  \;\coloneqq\;
  dim\big( hom_G(-,-) \big)
  \;\in\;
  \mathbb{N}
\end{displaymath}
for the [[dimension]] of the vector space of homomorphism between them (e.g. \hyperlink{tomDieck09}{tom Dieck 09, p. 29}).

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

\begin{displaymath}
\langle -,-\rangle 
  \;\colon\; 
  G Rep^{fin}_{/\sim} \times G Rep^{fin}_{/\sim}
  \longrightarrow
  \mathbb{N}
\end{displaymath}
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 map|linear]] with respect to the underlying [[commutative monoid|additive monoid]] structure:

\begin{displaymath}
\left\langle
    V_1 \oplus V_2
    \,,\,
    W
  \right\rangle
  \;=\;
  \left\langle
    V_1 
    \,,\,
    W
  \right\rangle
  +
  \left\langle
    V_2
    \,,\,
    W
  \right\rangle
  \phantom{AA}
  \text{and}
  \phantom{AA}
  \left\langle
    V
    \,,\,
    W_1 \oplus W_w
  \right\rangle
  \;=\;
  \left\langle
    V
    \,,\,
    W_1
  \right\rangle
  +
  \left\langle
    V
    \,,\,
    W_2
  \right\rangle
\end{displaymath}
(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]]

\begin{displaymath}
R(G) \;\coloneqq\; K\left(  G Rep^{fin}_{/\sim} \right)
\end{displaymath}
of $G$. By the evident $\mathbb{Z}$-linear extension, the above pairing gives an actual symmetric $\mathbb{Z}$-[[bilinear map|bilinear]] [[inner product]] on $R(G)$:

\begin{equation}
\langle
   -,- 
  \rangle
  \;\colon\;
  R(G) \times R(G)
  \longrightarrow 
  \mathbb{Z}
\label{DecategorifiedInnterProduct}\end{equation}
Now $R(G)$ is a [[free abelian group]] whose canonical [[generators and relations|generators]] are nothing but the [[isomorphism classes]] $V_i$ of the [[finite-dimensional vector space|finite-dimensional]] [[irreducible representations]]:

\begin{displaymath}
R(G) \;\simeq_{\mathbb{Z}}\;  \mathbb{Z}\big[ \{V_i\}_i \big]
  \,.
\end{displaymath}
In summary, in the language of [[linear algebra]], the [[irreducible representations]] $[V_i]$ constitute a canonical \emph{[[linear basis]]} of the [[representation ring]].

In terms of this language, \textbf{Schur's lemma} becomes the following statement:

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

\begin{enumerate}%
\item generally: an \emph{[[orthogonal basis|ortho-gonal basis]]};


\item even an \emph{[[orthonormal basis|ortho-normal basis]]} if the [[ground field]] is [[algebraically closed field|algebraically closed field]]



\end{enumerate}
with respect to the canonical [[decategorification|decategorified]] [[inner product]] from \eqref{DecategorifiedInnterProduct}.

\hypertarget{GeneralizationsAndVariants}{}\subsection*{{Generalizations and variants}}\label{GeneralizationsAndVariants}

\hypertarget{for_simple_modules}{}\subsubsection*{{For simple modules}}\label{for_simple_modules}

Part (1) is essentially [[category theory|category-theoretic]] and can be generalized in many ways, for example, by replacing the [[group]] $G$ by some $k$-[[associative algebra|algebra]] and taking the representations compatible with the action of $k$.

\hypertarget{for_simple_objects_in_an_abelian_category}{}\subsubsection*{{For simple objects in an abelian category}}\label{for_simple_objects_in_an_abelian_category}

More generally, given an [[abelian category]], part (1) of Schur's lemma applies to the [[simple objects]] (see \href{simple+object#SchurLemma}{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 vector space|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.

\hypertarget{for_bridgeland_stable_objects}{}\subsubsection*{{For Bridgeland stable objects}}\label{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 \href{Bridgeland+stability+condition#SchurLemma}{there}

\hypertarget{references}{}\subsection*{{References}}\label{references}

Lecture notes include

\begin{itemize}%
\item [[Tammo tom Dieck]], (1.1.2) in \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Schur%27s_lemma}{Schur's lemma}}

\end{itemize}
[[!redirects Schur's lemmas]]

[[!redirects Schur lemma]] [[!redirects Schur lemmas]]



\end{document}