@Zoran #41: More or less. You need some cocompleteness assumptions on $C$ to be assured that colimit exists. Or else you need to put some restriction on what sorts of $M$’s are allowed (for example, 0 for all but finitely many $n$, if it deserves to be called polynomial). But basically it’s the right idea, and what is being spelled out over at Schur functor. The plethystic tensor product goes by a number of names, for example “substitution product” of (polynomial) linear species. Cf. also the more general notion of club.

]]>Todd wrote:

One other “complaint” (not sure that’s the right word) is the sentence

Conceptually, the importance of $\mathbb{P}$ is that it is a skeleton of the groupoid of finite sets and bijections.

That’s one importance, but perhaps to me the chief conceptual importance is that $\mathbb{P}$ (or finite sets and bijections) is the free symmetric monoidal category on one generator: that’s the main thing we’re using about $\mathbb{P}$ throughout the analysis.

I guess I meant to say something like “Hey! Glomming all the symmetric groups to form a groupoid is not just some *ad hoc* technical trick! This is a groupoid you know and love! Finite sets! And that means we’re talking about linear structure types!”

Whenever I say something like “*the* importance of…” or “the *true* meaning of…”, it’s not that I really think there’s just one true significance to anything. I really just mean “hey, this is a really interesting shift of viewpoint, that feels deep!” Indeed I sometimes say the real reason the groupoid of finite sets is important is that it’s the free symmetric monoidal category on one object.

So, feel free to change the wording there. I just wanted to get $core(FinSet)$ into the picture.

]]>It’s funny that you don’t like the emphasis on Young symmetrizers, Todd, because that’s precisely the emphasis that I want to avoid in our actual paper. But the classical treatment of Schur functors relies on Young symmetrizers very heavily. In fact, some people seem to think a Schur functor is a functor coming from a Young symmetrizer! Of course these are just what we’d call the irreducible Schur functors. But I figure the $n$Lab article should explain the whole story. So I wanted to explain Schur functors coming from Young symmetrizers in the introductory section, and then show how direct sums of these ’irreducible’ Schur functors are precisely functors of this form:

$X \mapsto S_R(X) = \sum_{n \ge 0} R_n \otimes_{\mathbb{C}[S_n]} X^{\otimes n}$where $R = \{R_n\}$ is any list of finite-dimensional reps of $S_n$ with $R_n = 0$ for large enough $n$. Then I wanted to note that such a list $R$ can be thought of as a ’polynomial species’ $R: core(FinSet) \to Vect$. And then we can say “Bye-bye, Young symmetrizers!”

I did all this, more or less, in the first 2 sections of the $n$Lab article - but I guess I didn’t sufficently emphasize the “disposable” nature of the Young symmetrizers.

There may also be a problem - as Dan suggests - with explaining the concept of Schur functor in a way where the definition keeps expanding in scope. I want it to be clear the definitive concept for us was the category $Schur$ (defined over an arbitrary but fixed field of characteristic zero). But it could probably stand to be a lot clearer.

I guess Todd will take a crack at improving everything…

]]>Does $\Lambda$ have a universal characterisation in terms of the category of plethories? Otherwise, why is it described as the “most famous example”? (Then again, I suppose the most famous finite simple group doesn’t have a universal characterisation.) If it did, that might give a way to characterise $Schur$ in its bicategory.

]]>One other “complaint” (not sure that’s the right word) is the sentence

Conceptually, the importance of $\mathbb{P}$ is that it is a skeleton of the groupoid of finite sets and bijections.

That’s one importance, but perhaps to me the chief conceptual importance is that $\mathbb{P}$ (or finite sets and bijections) is the free symmetric monoidal category on one generator: that’s the main thing we’re using about $\mathbb{P}$ throughout the analysis.

]]>As partway toward the answer of characterizing Schur functors, the more general analytic functors (where the $R(n)$ may be nonzero for infinitely many $n$) were characterized by Joyal in an appendix of his paper on the Lie species, Lecture Notes in Mathematics 1234: necessary and sufficient conditions are that they preserve filtered colimits and weak pullbacks, and there may be one other condition that I’m forgetting.

But what is notable about his result is that he shows how to extract the coefficients $R[n]$ of the species from the analytic functor $R(x) = \sum_{n \geq 0} R[n] \otimes_{S_n} X^{\otimes n}$. This is by no means obvious!

This is related to a worry I have: John wrote that there is a category $Schur$ consisting of Schur functors $FinVect \to FinVect$ and natural transformations between them. Then later there is a claimed equivalence between this category and the category of polynomial species. Apparently the claim is that $Schur$ is by definition the essential image of a full embedding

$PolySpecies \to [FinVect, FinVect]$Now (1) it’s true, but I don’t see it as quite obvious, that this is a full embedding – I suspect one has to grapple with the theory laid out in Joyal’s appendix to prove this result, unless I’m missing something, and (2) as jdc asks, there is still the question of what are the objects of the essential image, *intrinsically*?

See, this is the beauty of going beyond the conception of Schur functors as certain endofunctors on $FinVect$ to the much wider playing field of endofunctors on general symmetric monoidal Cauchy complete $Vect$-categories (or in fancier but more precise language, pseudonatural transformations $U \to U$ developed later in the paper). There are two important points:

(1) Instead of trying to characterize classical Schur endofunctors just on $FinVect$ (as functors satisfying certain conditions like preserving filtered colimits, weak pullbacks, etc.), we instead concentrate the essence of “Schurness” by saying that they are precisely those gadgets that can be defined on all symmetric monoidal linearly Cauchy complete categories in such a way as to respect “change of base”: the precise way to express it is to say that Schur functors are, seen in the proper light, pseudonatural transformations $U \to U$ (as you, John, had suggested from the get-go). And also,

(2) By expanding to that bigger playing field, we can then invoke the representability result and get the theorem that the category of pseudonatural transformations and modifications really is the category of polynomial species! This is much softer and easier (less technical) than Joyal’s analysis!

I happen to think that’s quite elegant and conceptual, and I think this is what we should be gently hinting at in the introduction. Ultimately we are trying to say that Schur functors are utterly simple and natural and conceptually inevitable, and so are things like plethysm. But somehow I fear that basic message is now getting lost or masked by saying things like

This description of irreducible representations of $S_n$ [in terms of Young symmetrizers] paves the way toward an important generalization of Schur functors…

because it seems to place the Young symmetrizer approach at the forefront of the development, which is not the direction we’re really taking.

]]>Those first two sections look good to me. My only quibble is that you don’t explicitly define what you mean by a Schur functor, but instead say things like “may also be called”. This gets a bit confusing near the end where you say that the composite of two Schur functors is a Schur functor, and it’s not completely clear whether this is a theorem or is a further extension of the definition. (I know you mean it to be a theorem. As a hint of the proof, maybe it would be worth saying that a composite of polynomials is a polynomial?)

Incidentally, can Schur functors be characterized as the functors $FinVect \to FinVect$ such that some property holds?

[I’ll be away from Friday afternoon until Sunday night and won’t be able to post during that time. And I think John is leaving tomorrow for Singapore! Have a great time, John!]

]]>I have tried to fix the mistake Dan noticed in the Schur functor article - at least in the first two sections, namely the idea and definition section, and the section called the category of Schur functors. These two sections are supposed to be nice and readable now - and correct, too.

Please, everyone, see if these two sections look okay!

I have *not* yet fixed the later sections.

Dan wrote:

I’m using the induction product throughout, so when I talk about spin networks in the category of $\mathbb{P}$-representations, parallel lines mean induction product!

John wrote:

I guess you’re thinking of them as ’generic’ spin networks that don’t care which representation of which group is the fundamental object $X$ that you’re using to build up fancier objects via Schur functors. And then you’re trying to go as far as you can in this ’generic’ context before specializing to a particular group. Right?

Right! And amazingly, the correction factor in spin network evaluations is just a polynomial in $N$, where $N$ is the $N$ in $SU(N)$!

John wrote:

This reminds me of a paper I wrote about 2d Yang-Mills theory with gauge group $SU(N)$ versus 2d Yang-Mills theory with gauge group $SU(\infty)$.

I’m also keeping in mind that it might be relevant to computations in the $N \to \infty$ limit of lattice gauge theory.

Dan wrote:

But I think a lot of the interest in the representation theory comes from the explicit way the various irreps of $S_n$ embed in $C[S_n]$, and how they interact (which is revealed by the non-commuting of the Young symmetrizers).

John wrote:

I guess so, but this has always struck me as confusing rather than interesting. I wish someone would say some of the key ideas in something resembling plain English, but people always seem to take a highly computational approach.

Well, I don’t think I have enough of an overview to answer this to
your satisfaction. All I can say is that many people put large
amounts of effort into understanding the properties of the minimal
idempotents $p_\lambda$, even though for abstract reasons it’s clear
that there exists a set $g_i$ of minimal *central* idempotents.
I’m hoping my work helps bridge the gap here, by revealing that
the $g_i$’s can be easily expressed in terms of the $p_\lambda$’s.

Why care about the $p_\lambda$’s at all? Well, they have some very nice properties. For example, as I mentioned earlier, if $\sigma$ is a permutation then $\sigma^{-1} p_S \, \sigma$ is $p_T$, where the tableaux $T$ is obtained from $S$ by permuting the entries using $\sigma$. This doesn’t work for the $g_i$’s. Moreover, by staring at the wiring diagram (maybe for a long time!) you can convince yourself that for any tableaux $S$ and $T$ of the same shape, and any $\sigma$,

$p_S \sigma p_T = m p_S \tau$where $m$ is in $\{ 0, 1, -1 \}$ and $\tau$ is the permutation sending $T$ to $S$. This is extremely useful for computations involving composites of the Young symmetrizers. So, except for the fact that they aren’t orthogonal, the Young symmetrizers are very nicely behaved.

Another reason why studying $p_S$ for all tableaux $S$ rather than just for one of each shape is that there is no canonical ordering to the boxes, so why should you pick one arbitrarily?

And a third reason why the interrelationships are important is that one is often really working with $V^{\otimes n}$, where $V$ is the defining rep of $SU(N)$, and you have a specific tensor that you want to express as a sum of others with nice symmetry properties. This is a vague way of describing what happens in lattice gauge theory, but I think it’s pretty general.

So I’ve been trying and (as you can see) not quite succeeding in doing what I want to do while avoiding a lot of these computations. So, thanks for straighteing me out. I think something like Todd’s original approach, but really downplaying Young tableaux, will make me happiest.

I’m also all for downplaying the Young tableaux and making the
story more functorial. I’m just pointing out that it might not
explain the *whole* story, just a portion of it. (But hopefully
I’m wrong!)

Dan wrote:

And when I said that the category of $\mathbb{P}$-representations doesn’t have duals, I meant with respect to the

induction product.

Yeah, sorry, I realized that in the gym while thinking more about your post. My first comment about that was fired off before I had absorbed what you were really saying in that post. I eventually caught on:

It’s with respect to the induction product that $Schur$ is the free symmetric monoidal $k$-linear Cauchy complete category on one object. And that one object can’t have a dual because if it did, there would be an isomorphism

$X^* \cong F(X)$that held for some fixed Schur functor $F$ and *every* object $X$ in *every* symmetric monoidal $k$-linear Cauchy complete category $C$. And this fails already when $C$ is the category of finite-dimensional reps of $U(1)$.

I’m using the induction product throughout, so when I talk about spin networks in the category of $\mathbb{P}$-representations, parallel lines mean induction product! I haven’t seen these sorts of spin networks discussed explicitly before, but they are very natural from a certain point of view.

I guess you’re thinking of them as ’generic’ spin networks that don’t care which representation of which group is the fundamental object $X$ that you’re using to build up fancier objects via Schur functors. And then you’re trying to go as far as you can in this ’generic’ context before specializing to a particular group. Right?

This reminds me of a paper I wrote about 2d Yang-Mills theory with gauge group $SU(N)$ versus 2d Yang-Mills theory with gauge group $SU(\infty)$. This was an attempt to formalize some ideas physicists have about the $N \to \infty$ limit of $SU(N)$ gauge theory. The Hilbert space for the circle in 2d $SU(\infty)$ Yang-Mills theory has a basis given by Young diagrams. To get the Hilbert space for $SU(N)$ Yang-Mills theory you can impose some relations called ’Mandelstam relations’ which cut the basis down to Young diagrams with fewer than $N$ rows. These correspond to the usual basis of class functions on $SU(N)$ - the usual basis of states for $SU(N)$ Yang-Mills on the circle.

At the time I didn’t fully understand how this was related to the functor $Schur \to Rep(SU(N))$, but it was one of the things that got me interested in this…

You asked about choosing a minimal idempotent $q_\lambda$. Yes, if you do, I think you’ll get the usual Schur functor.

Okay, good. I’ve been thinking about that and it’s starting to seem obvious.

But I think a lot of the interest in the representation theory comes from the explicit way the various irreps of $S_n$ embed in $C[S_n]$, and how they interact (which is revealed by the non-commuting of the Young symmetrizers).

I guess so, but this has always struck me as confusing rather than interesting. I wish someone would say some of the key ideas in something resembling plain English, but people always seem to take a highly computational approach. So I’ve been trying and (as you can see) not quite succeeding in doing what I want to do while avoiding a lot of these computations. So, thanks for straighteing me out. I think something like Todd’s original approach, but really downplaying Young tableaux, will make me happiest.

]]>When I wrote that $p_S p_T = 0$ when $S \gt T$, that was just background that is well known. That was just leading up to what I haven’t seen before, which is the formula for the $g_i$’s.

And when I said that the category of $\mathbb{P}$-representations doesn’t have duals, I meant with respect to the *induction product*. If $W$ was a dual for a rep $V$ of $S_n$, then it would have to have a non-trivial component of degree $-n$, but these things are graded by the natural numbers. I’m using the induction product throughout, so when I talk about spin networks in the category of $\mathbb{P}$-representations, parallel lines mean induction product! I haven’t seen these sorts of spin networks discussed explicitly before, but they are very natural from a certain point of view. For example, if $V_i$ is the image of an idempotent $e_i$ in $C[S_{n_i}]$, for $i = 1, 2$, then the induction product of $V_1$ with $V_2$ is the image of an idempotent $e_1 . e_2$ which is the element of $C[S_{n_1 + n_2}]$ formed by placing $e_1$ and $e_2$ “side-by-side”.

You asked about choosing a minimal idempotent $q_\lambda$. Yes, if you do, I think you’ll get the usual Schur functor. But I think a lot of the interest in the representation theory comes from the explicit way the various irreps of $S_n$ embed in $C[S_n]$, and how they interact (which is revealed by the non-commuting of the Young symmetrizers).

]]>Dan wrote:

If you fix a vector space $X$ (or an object in a category $C$ like you’ve described), you get a functor $Ch$ sending a representation $V$ of $S_n$ to $Hom(V, X^{\otimes n})$, and this extends linearly to the category of representations of $\mathbb{P}$. It’s an interesting and important fact that $Ch$ is monoidal.

This is part of the point of our approach to Schur functors. You can think of $Schur$ as the category of representations of $\mathbb{P}$ that are finite direct sums of irreducible representations. In our $n$Lab paper, Todd shows that $Schur$ is the ’free symmetric monoidal $k$-linear Cauchy complete category on one object’. Call this object $x$. So, if you fix any object in any category $C$ of this sort, you get an essentially unique $k$-linear symmetric monoidal functor

$Ch: Schur \to C$sending $x$ to $X$.

But it probably should be called $Tr$ rather than $Ch$.

I’ve been dreaming of things like this for years, starting in my paper on 2-Hilbert spaces, where I described universal properties for the categories of finite-dimensional unitary representations of $U(k)$, and $SU(k)$, and $O(k)$, and $SO(k)$, and $Sp(k)$. But Todd figured out the right class of symmetric monoidal categories for which $Schur$ is the free guy on one object.

]]>Dan wrote:

The category of representations of $\mathbb{P}$ does

nothave duals,

A representation of $\mathbb{P}$ has duals if it’s just a finite direct sum of finite-dimensional reps of $S_n$’s, no?

]]>Dan Christensen (aka “jdc”) wrote:

That’s the other thing I’ve figured out recently: if you order the standard tableaux with $n$ boxes using last-letter order (or maybe the reverse, depending exactly on your conventions), you find that $p_S p_T = 0$ for $S \gt T$.

I think the fact you just mentioned is Lemma 4.23 of Fulton and Harris’ *Representation Theory*. I didn’t know your formula for orthogonal idempotents $g_i$. But clearly I’m not the person who would know this sort of thing.

I’m still a bit confused - so let me ask a question to see if I’m finally catching on. Todd has described a Schur functor for each irrep $V_\lambda$ of $S_n$ as follows:

$X \mapsto V_\lambda \otimes_{k[S_n]} X^{\otimes n}$Is the following functor naturally isomorphic to his? Take $k[S_n]$ and write it as a sum of matrix algebras, one for each $\lambda$. Take the element “1” in the $\lambda$th matrix algebra and write it as a sum of commuting minimal idempotents - there’s a lot of arbitrariness in how to do this, but the sum will have $dim(V_\lambda)$ terms. Take any one of these minimal idempotents and call it $q_\lambda$. Then consider the functor

$X \mapsto q_\lambda X^{\otimes n}$Is this naturally isomorphic to Todd’s?

]]>I wrote a new section **Plethysm product and symmetric operads** within the entry plethysm. Please check.

Yes, John, that sounds right. The idempotents coming from the decomposition into matrix algebras are just the projections onto the isotypic components, not the projections onto a decomposition into irreps. (When you view $C[S_n]$ as an $S_n \times S_n$-rep using left and right multiplication, then this is the decomposition into irreps; but not when you view it as an $S_n$-rep with just one of the actions.)

And while the usual Young symmetrizers (indexed by all standard tableaux) do have images that form a direct sum decomposition into irreps, they are *not* the projections associated to that decomposition because they are not orthogonal.

That’s the other thing I’ve figured out recently: if you order the standard tableaux with $n$ boxes using last-letter order (or maybe the reverse, depending exactly on your conventions), you find that $p_S p_T = 0$ for $S \gt T$. Write the list of all standard tableaux as $T_1, T_2, \ldots, T_r$ in last letter order and write $p_i$ for $p_{T_i}$. For each $i$, define $g_i = p_i (1-p_{i+1}) \cdots (1-p_r)$. Then the $g_i$’s have the same images as the $p_i$’s and *are* the natural projections onto these images (so, in particular, they are orthogonal).

As I mentioned earlier, it’s a common mistake to assume that the $p_i$’s are orthogonal, but it turns out that some of the work done using the $p_i$’s can be redone with the $g_i$’s (with additional complications due to the extra factors).

So my next question: has anyone seen these $g_i$’s anywhere? I couldn’t find them written down.

]]>I always get confused about central idempotents versus non-central idempotents in group algebras - I worry about this constantly. So it’s quite possible I’m getting mixed up again in the $n$Lab entry. Let me try to explain:

If $G$ is a finite group, the group algebra $\mathbb{C}[G]$ is a direct sum of matrix algebras, one for each irrep $\lambda$ of $G$. The element “1” in each of these matrix algebras gives a central idempotent $p_\lambda$ in $\mathbb{C}(G)$, and these idempotents add up to $1 \in \mathbb{C}(G)$. If $X$ is any rep of $G$ it follows that $p_\lambda X$ is a subrep of $X$, and the direct sum of these subreps is all of $X$.

Now specialize to the case where $G = S_n$ and $X = V^{\otimes n}$. This is what I was trying to talk about. I think this construction gives one Schur functor

$V \mapsto p_\lambda V^{\otimes n}$for each Young diagram.

(None of this depends on having a specific formula for the central idempotents $p_\lambda$. I actually think my formula for the central idempotents is right, but that’s a separate, detachable question.)

But now I think I see the problem that’s been nagging at me all the time. These Schur functors of mine coming from Young diagrams are direct sums of copies of some other, irreducible Schur functors! And those are the ones that Todd is talking about.

Does that sound right?

]]>Now that I have a bit more time, let me say something about what I’ve been thinking about on this topic.

First, I really like the definition of a Schur functor as $Hom(V, X^{\otimes n})$ (or the dual version), with $V$ an $S_n$-rep, since it avoids using Young tableaux as an intermediate step. (On the other hand, while concise, it doesn’t give the same intuition as the definition using Young symmetrizers.)

But rather than focusing on this as a functor of $X$, I’ve been
thinking about it as a functor of $V$. Write $\mathbb{P}$ for the
groupoid of finite sets and bijections (or the obvious skeleton,
namely the coproduct of the groupoids $S_n$). Then consider at the
category of representations of $\mathbb{P}$. As you note, this category
has a monoidal structure, which takes representations $V$ and $W$
of $S_p$ and $S_q$ to $V \otimes W$ induced up from $S_p \times S_q$
to $S_{p+q}$, and then extended linearly to general representations of
$\mathbb{P}$. (This is called the *induction product* in the literature,
and it puts a ring structure on the direct sum over $n$ of the spaces
of class functions on $S_n$.)

If you fix a vector space $X$ (or an object in a category $C$ like you’ve described), you get a functor $Ch$ sending a representation $V$ of $S_n$ to $Hom(V, X^{\otimes n})$, and this extends linearly to the category of representations of $\mathbb{P}$. It’s an interesting and important fact that $Ch$ is monoidal.

In my applications, $X = \mathbb{C}^k$ is the defining representation
of the unitary group $U(k)$, so $Ch$ sends representations of
$\mathbb{P}$ to representations of $U(k)$. (Decategorified, the
character of $Ch(X)$ is called the *$U(k)$-characteristic* of the
character of $V$. That’s where the notation $Ch$ comes from. I’m not
trying to name something after myself! :-)

Now, the category of representations of $U(k)$ has duals, so it has a
natural trace structure (which just takes the usual trace of a linear
map). The category of representations of $\mathbb{P}$ does *not* have
duals, but it nevertheless has an obvious trace structure. (See
Selinger’s paper, A survey of graphical languages for monoidal
categories, Chapter
5 for what I
mean by a trace structure.)

It’s clear that the functor $Ch$ doesn’t preserve traces, since the
dimensions of $V$ and $Hom(V, X^{\otimes n})$ usually don’t agree.
However, it really makes sense to think of an irrep as being
1-dimensional, and if you adjust the two trace structures to take this
into account, you find that $Ch$ does preserve traces. And this leads
to an important computational consequence: if you want to evaluate a
$U(k)$ spin network which is $Ch$ applied to a $\mathbb{P}$ spin
network, then as long as the diagram factors through one irrep (which
controls the normalization factors), it suffices to evaluate the
$\mathbb{P}$ spin network you started with. The amazing thing is that
the value of the $\mathbb{P}$ spin network is independent of $k$!
This explains and generalizes some techniques that Cvitanovic
describes in his book *Group Theory*, and turns out to be useful in
lattice gauge theory.

One reason I’m writing this to find out if anyone has seen this functor $Ch$ written down before. My searches have turned up nothing.

]]>While I haven’t checked every detail, I think everything that you wrote is correct.

By the way, instead of $V_\lambda \otimes_{S_n} X^{\otimes n}$, I have seen $Hom_{S_n}(V_\lambda, X^{\otimes n})$ used to describe the Schur functor. This should correspond to dualizing $V_\lambda$, so it would just give a different mapping between irreps and Schur functors.

]]>Thanks Dan (glad we got that squared away!). I certainly recognize your name, and it’s nice to hear from you!

Funny – I had had a worry that I suspect corresponds to one of the points that you brought up, and then for some reason thought there was nothing to worry about after all (and retracted my worry), and now I’m worried again! Well, not actually worried – I think the original approach to the Schur functors that I had written down works fine. John had done some rewriting, and the original approach was to be moved to another section, but I didn’t get around to doing that yet.

Here was my original approach. The group algebra $k[S_n]$ is a bimonoid in the symmetric monoidal category of finite-dimensional vector spaces over $k$, $FinVect_k$. Now for any symmetric monoidal Cauchy complete $Vect_k$-enriched category $C$, there is a symmetric monoidal enriched functor $FinVect_k \to C$, uniquely so up to isomorphism. Being a symmetric monoidal functor, it carries bimonoids to bimonoids. Thus, in any such $C$, we have a bimonoid which by abuse of notation we again call $k[S_n]$. Also, if $V_\lambda$ is an irreducible module over $k[S_n]$ in $FinVect_k$, the same symmetric monoidal functor carries $V_\lambda$ to a module over the bimonoid $k[S_n]$ in $C$. Again, by abuse of notation, we call this object $V_\lambda$.

Now, the category of modules $Mod(B)$ over a bimonoid $B$ in a symmetric monoidal category $C$ carries a monoidal category structure. The tensor product on $Mod(B)$ is constructed with help from the comultiplication on $B$, and the monoidal unit with help from the counit, and the underlying functor $U: Mod(B) \to C$ is a strong (symmetric) monoidal functor. Now for any object $X$ of $C$, the $n$-fold tensor $X^{\otimes n}$ is naturally a module over $k[S_n]$. Take the tensor product of $V_\lambda$ and $X^{\otimes n}$ in $Mod(k[S_n])$.

Finally, the average of all the group elements,

$e = \frac1{n!} \sum_{\sigma \in S_n} \sigma,$defines an idempotent operator on the module $V_\lambda \otimes X^{\otimes n}$. The splitting of this idempotent in $C$ (which we assumed Cauchy complete) gives the object of coinvariants $V_\lambda \otimes_{S_n} X^{\otimes n}$. This gives $S_\lambda(X)$, hence defines the Schur functor $S_\lambda$.

Do you agree?

Okay, now I think the description of $S_\lambda$ that is in the current draft doesn’t work. Here we start with an idempotent $e_\lambda$ corresponding to the identity of the matrix algebra (on the underlying vector space of the isotype $V_\lambda$), and form a corresponding idempotent operator on $X^{\otimes n}$. This operator

$E_\lambda: X^{\otimes n} \to X^{\otimes n}$could also be written

$X^{\otimes n} \cong k[S_n] \otimes_{S_n} X^{\otimes n} \stackrel{e_\lambda \otimes_{S_n} 1}{\to} k[S_n] \otimes_{S_n} X^{\otimes n} \cong X^{\otimes n}$and the identity on $X^{\otimes n}$ would be $\sum_\lambda E_\lambda$ where $\lambda$ ranges over Young diagrams. But, left multiplication by the idempotent,

$e_\lambda: k[S_n] \to k[S_n],$splits through $\hom_{Vect}(V_\lambda, V_\lambda)$ which as a module is isomorphic to $V_{\lambda}^{d(\lambda)}$, where $d(\lambda) \coloneqq \dim(V_\lambda)$. So we’re *not* getting

as we want, but rather

$V_{\lambda}^{d(\lambda)} \otimes_{S_n} X^{\otimes n}$which is too big! So I seem to agree: the current description we have is flawed, and maybe we should revert to the original approach.

John, is this making sense to you?

]]>By the way, I didn’t mean to post semi-anonymously above. It was my first time posting to the n-forum and I was in a rush. Hopefully this comment shows up correctly.

Also, in my rush, my comments weren’t phrased very carefully. I should have said that I think your approach is very cool, and I’m looking forward to reading it more carefully to understand what’s really going on with Schur functors. Moreover, even if I’m right that your approach produces reducible Schur functors, I suspect that it can be adapted to reproduce the usual story.

… Hmm, it still just shows “jdc” instead of “Dan Christensen”. I ticked the box to let people know my real name, but it still doesn’t show up.

]]>Thanks for explaining why having $p_\lambda$ central would imply that it is idempotent. Unfortunately (as I said), the usual Young symmetrizers are not idempotent, so this proof doesn’t apply to them. And the later $p_\lambda$’s, being projectors onto isotypic components, are clearly idempotent.

I haven’t read the rest of the Schur functor entry in detail, but it could be that you are right that one could ignore the first section and let the rest stand on its own. However, then you are in the situation that what you write as $S_\lambda$ wouldn’t be the usual Schur functor which (for example) produces irreps of the unitary groups. It would be a much larger gadget which might be isomorphic to a sum of the usual $S_\lambda$’s.

Moreover, I think the true richness of the representation theory of the symmetric group lies precisely in how the isotypic components break up into irreps in a way that has beautiful combinatorics underlying it. If the only ingredient you use is the decomposition into isotypic components (which is canoncial, of course), then I’d be surprised if you can reproduce the theory of Schur functors. But I’m no expert on Schur functors, so maybe I’m wrong.

]]>Thanks jdc –

John put in the stuff on the explicit description of Young symmetrizers, so I’ll let him bear most of the brunt in addressing these questions (while I run off scot-free (-: ). But in some sense I think it’s no big deal, because the whole approach we’re taking in this paper is based on general abstract theory which doesn’t actually need the explicit combinatorial descriptions. I suspect John put that stuff in just to round out the discussion for those accustomed to the more combinatorial approaches.

One small note about the idempotence of $p_\lambda = p^A p^S$: when I first read that, I puzzled about it too. But the equation

$p^A p^S p^A p^S = p^A p^A p^S p^S$is clearer if we expand it to say (using centrality of $p_\lambda$):

$p^A p^S p^A p^S = p_\lambda p^A p^S = p^A p_\lambda p^S = p^A p^A p^S p^S$which reduces to $p^A p^S = p_\lambda$ if $p^A$ and $p^S$ are idempotent. Does this answer to your question there, or was it something else?

As for your penultimate paragraph: what you’re finding fault with again harks back to the descriptions of the first section. But suppose we ignore the first section! Instead, start all over again, by invoking Maschke’s theorem (which implies that the group algebra is a finite product of matrix algebras), and defining elements $p_\lambda$ to be the primitive central idempotent elements of $k[S_n]$ that correspond to identity elements of the matrix algebras. (Don’t worry for now about the Young diagram/tableaux meaning of $\lambda$; just take it to be a symbol that indexes isomorphism classes of irreducible representations, one for each matrix algebra.) Then, in any symmetric monoidal linear category in which idempotents split, you can interpret how $p_\lambda$ acts as an idempotent operator $X^{\otimes n} \to X^{\otimes n}$ for any object $X$. The images of these projections give the $S_\lambda(X)$ that we are defining en route to the Schur functor $S_\lambda$.

This is not to dismiss your criticisms (which are appreciated!), but rather to emphasize that I think they don’t much affect the approach we’re really taking. We could just erase the stuff about Young symmetrizers in the beginning, and I think we’d be okay.

]]>I just looked through the n-lab entry on Schur functors and I’m confused about some of the statements there. Maybe I’m just interpreting things incorrectly.

In the first section, the Young symmetrizer $p_\lambda$ is defined, and then it says that $p_\lambda$ lies in the center of the group algebra. I don’t think this is true. In general, $p_\lambda$ really depends on a tableaux $T$ of shape $\lambda$ (to map the boxes to the factors in the tensor product) and if $\sigma$ is a permutation, $\sigma p_T \sigma^{-1}$ is equal to $p_S$, where $S$ is the tableaux that has the entries of $T$ permutated by $\sigma$. For example, take the Young diagram corresponding to the partition (2,1) labelled with 1 and 2 in the first row and 3 in the second, and take $\sigma$ to be the transposition (2,3). You can work this case out explicitly.

As another example, is $S$ and $T$ are standard tableaux of the same shape but with different entries, then one of $p_S p_T$ and $p_T p_S$ is automatically zero, but the other is sometimes (but rarely) non-zero. (The failure to notice this is a mistake in both the book by James and Kerber and Cvitanovic’s book.)

Also, even if $p_\lambda$ were central, I don’t follow the proof given that claims to show that it is idempotent. (It is, of course, but the proof is a bit harder.)

Later in that section, it says that the image $S_\lambda(V)$ of $p_\lambda : V^{\otimes n} \to V^{\otimes n}$ is invariant under $S_n$, but this is also false. Taking the same partition (2,1) as above, $S_\lambda(V)$ consists of tensors which are symmetric in the first 2 variables and antisymmetric in the first and third. This condition isn’t invariant under exchange of factors.

All of those comments were about the first section. I also have a
comment about the section on the action of Young
symmetrizers.
There it defines $p_\lambda$ as the projection onto a certain
matrix algebra inside the group algebra.

I just wanted to point out that this definition
doesn’t reduce to the one in the first section. Each $p_\lambda$ from the first section projects onto a
single irrep inside the group algebra, while the $p_\lambda$’s
here project onto sums of irreps (viewing $C[S_n]$ as an $S_n$-rep
under multiplication from one side).
You might hope that the $p_\lambda$ from this section is just
a sum of the $p_\lambda$’s from the first section indexed by
standard tableaux of shape $\lambda$ (and others have thought this too),
but the non-orthogonality of the Young symmetrizers makes this false
in general. While the images of the $p_\lambda$’s (indexed by all
standard tableaux) *do* give a direct sum decomposition of $C[S_n]$,
the projections $p_\lambda$ are *not* the natural projection maps
of that decomposition!

I’ve recently found a way to describe those projections, but I have to rush off now, so I’ll write more later.

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