Geometric Representation Theory (Lecture 14)
Posted by John Baez
This time in the Geometric Representation Theory seminar I tried to state the Fundamental Theorem of Hecke Operators… and I screwed up. Luckily, I screwed up in an instructive way!
Pick a group $G$. We’ve seen that every $G$invariant relation between finite $G$sets gives an intertwining operator between the resulting permutation representations of $G$. These operators are called Hecke operators.
Any category theorist worth their salt should want to make this into a functor. Since there’s a category $FinRel^G$ with
 finite $G$sets as objects
 $G$invariant relations as morphisms
and a category $FinVect^G$ with
 finitedimensional representations of $G$ on complex vector spaces as objects
 $G$invariant operators (= intertwining operators) as morphisms
one might hope the Hecke operator trick gave a functor
$F: FinRel^G \to FinVect^G$
But, it doesn’t!
That wasn’t my mistake. I can be dumb… but I’m not that dumb.
In fact, I began my lecture explaining this problem and its origin. Fundamentally, the problem is that a relation is a matrix taking values in the rig of truth values, $\{0,1\}$, while a linear operator between vector spaces equipped with bases is a matrix taking values in the rig $\mathbb{C}$. There’s a tempting inclusion
$\{0,1\} \hookrightarrow \mathbb{C}$
and this is lets us turn relations into linear operators. By functorial abstract nonsense, it turns $G$invariant relations into $G$invariant linear operators… and this is the Hecke operator trick.
But, this tempting inclusion is not a rig homomorphism, because “true or true does not mean twice as true”! So the Hecke operator trick does not give a functor
$F: FinRel^G \to FinVect^G$
To get around this problem I introduced spans of finite sets, which you can think of (in a slightly wimpy way) as matrices of natural numbers. The inclusion
$\mathbb{N} \hookrightarrow \mathbb{C}$
is a rig homomorphism, so we do get a functor
$F: FinSpan^G \to FinVect^G$
where $FinSpan^G$ is the category with
 finite $G$sets as objects
 spans of finite $G$sets as morphisms
My problem came when I wanted to state the really cool fact about Hecke operators using this functor. The really cool fact is that Hecke operators coming from atomic $G$invariant relations — those that can’t be broken down further using ‘or’ — form a basis of intertwining operators between the resulting permutation representations. I tried to state this as follows.
First of all, I noted that for any objects $X,Y \in FinSpan^G$, $hom(X,Y)$ is a module of the rig $\mathbb{N}$, while $hom(F X, F Y)$ is a complex vector space — that is, a module of the rig $\mathbb{C}$. That’s all true.
Then I noted that
$F: hom(X,Y) \to hom(F X , F Y)$
gives rise to a linear operator
$hom(X,Y) \otimes_{\mathbb{N}} \mathbb{C} \to hom(F X , F Y)$
That’s true too.
And then I claimed this operator was onetoone and onto! And that’s false. As Jim pointed out right after class — damn him! — it’s onto but not onetoone.
But, all is not lost. In later lectures, Jim used my error as a springboard to study various forms of decategorification, and to give a really deep account of the form we need in this course— namely, degroupoidification. Degroupoidification is something we wanted to talk about anyway, but now we see how it allows us to state the Fundamental Theorem of Hecke Operators in a really beautiful, conceptual way.

Lecture 14 (Nov. 13)  John Baez on matrix mechanics and Hecke operators. Any rig $R$ gives a category $Mat(R)$ whose objects are finite sets and whose morphisms are $R$valued matrices. Any rig homomorphism from $R$ to $R'$ gives a functor from $Mat(R)$ to $Mat(R)$. The homomorphism from $\mathbb{N}$ to $\mathbb{C}$ lets us turn spans of finite sets into linear operators between finitedimensional vector spaces. We can thus turn $G$invariant spans between $G$sets into intertwining operators between finitedimensional representations of $G$. These are Hecke operators. A flawed attempt to formally state the “Fundamental Theorem of Hecke Operators” in terms of this functor.

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http://mainstream.ucr.edu/baez_11_13_stream.mov  Downloadable video
 Lecture notes by Alex Hoffnung
 Lecture notes by Apoorva Khare

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Re: Geometric Representation Theory (Lecture 14)
So what’s the final answer? The suspense is killing me!