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Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the permutation representation in the standard representation theory books like in
I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.
Edit: new (related) entries for Claudio Procesi and Arun Ram.
FWIW, I would have assumed “permutation representation” to mean in the standard representation-theory sense of vector spaces.
I’d be inclined to call the current topic of the page a representation by permutations. I note here that $C$ should be a groupoid; otherwise “permutation” is completely out.
Right I do call it “representation by permutations” (see 1), while the permutation representation in the rep. theory sense entails the linear extension of the action from a vector space basis to the entire linear (=vector) space.
Zoran #5: I am in agreement with this terminology. When I talk privately with certain category theorists (e.g., Jim Dolan), sometimes we’ll refer to a permutation representation when we mean a representation in $Set$ (which latter by the way is also acceptable), but this is not widely recognized outside a certain community.
Jim #3: true, but the rest of the world has apparently not caught up yet! When people say ’representation’ without qualifications, the default assumption is that it’s a linear representation, and a permutation representation is generally taken to mean a composite functor
$B G \to Set \stackrel{free}{\to} Vect_k.$It makes sense to say “permutation representation” for $\cdots \to \mathrm{Set}$ and “linear permutation representation” for $\cdots \to \mathrm{Set} \to \mathrm{Vect}$.
Sure, it makes sense to say that, but if you only said “permutation representation” by itself to the mathematician-in-the-street, he is likely to interpret that as what you just called “linear permutation representation”. I thought that’s what this discussion was about: what the conventional standard is (as opposed to what the convention should be), as reported in Zoran’s #1.
I thought that’s what this discussion was about: what the conventional standard is
Okay. I thought the discussion is about what to write in the $n$Lab entry.
Of course we should always explain different use of terminology, for transparency. But I don’t think that we have to enslave ourselves to conventional standards if they should be different, as in this case.
I have added a word to the entry.
One difference between a representation in $Set$ and a permutation representation in $Vect$ is the morphisms (the intertwiners). Even the isomorphisms differ, so we can’t pretend that they’re the same thing.
they should be different, as in this case
Clear “representation by permutations” for categorical notion and cryptic “permutation representation” for linear notion are different.
I should point out that people who work on representations of symmetric groups and applications in combinatorics refer to permutation representations although there is often not a linear representation in sight. This continues the tradition of D.E. Littlewood. (You will see how I know this if you look at http://en.wikipedia.org/wiki/Dudley_E._Littlewood).
added a remark Comparison map from Burnside ring to representation ring
starting a section Examples – Virtual permutation representations. But we happen to be discussing this not here in this thread, but in the thread on the Burnside ring, see there
In the proof of the proposition about $\beta$ being surjective for some classes of groups (this prop.), I have expanded out the argument for cyclic groups, as a corollary of that for $p$-primary groups. Like so:
To see surjectivity for cyclic groups: By the previous statement we have surjectivity already for those cyclic groups whose order is a prime power. But by the fundamental theorem of cyclic groups, every cyclic group is a direct product group of cyclic groups of prime power order. Moreover, every irreducible representation of a direct product group is an external tensor product of irreps of the group factors (this prop.). But $\beta$ sends “external Cartesian products” to external tensor products, by the same elementary argument which shows that $\beta$ sends plain Cartesian products to tensor products. This way the statement reduces to that for $p$-primary cyclic groups.
Hm, maybe my argument for the cyclic groups (#18) is wrong after all:
That the irrep of a direct product group is a tensor product of irreps, is that true also over $\mathbb{Q}$?
[ edit: hm, should be okay, as in [prop. 2.3.17 here]
Ah, the surjectivity of beta for cyclic groups over the rationals is example 4.4.4 in tom Dieck’s notes http://www.uni-math.gwdg.de/tammo/rep.pdf
added pointer to Dress 86, section 3 for the statement that $\beta$ is surjective for symmetric groups in characteristic zero.
Re #24: well, that is indeed wonderful since I didn’t know where to find it in the literature. Now I do.
It’s a citation, but still not a proof, since it just says
it follows easily from the classical representation theory of $\Sigma_n$…
What is indeed easy to see is that in the discussion of Specht modules all the ingredients enter that one expects to see for this to be true. But if it’s so easy, what is the explicit general formula that expresses the Specht module corresponding to a given Young tabloid in terms of virtual permutation reps?
I might ask on the group-pub mailing list (which, for some weird reason, is private) to see if this is treated explicitly anywhere.
Didn’t know of this group. If you can, please do!
So David kindly checked on that group theory list. In reply there is so far a sketch of the idea of the proof:
if you look in the representation theory of $S_n$, the situation is the following:
you have an ordering on the set of partitions of $n$, which is, if I am not mistaken, the lexicographical ordering.
Then for every partition $\lambda$, you define $M_\lambda = Ind_{S_{\lambda}}^S_n 1$.
Inside $M_{\lambda}$ you find a direct sum of a new irreducible module $V_{\lambda}$ and modules of the form $V_{\mu}$, for $\mu \lt \lambda$.
This already shows that if you consider the representing matrix of the above transformation, you get an upper diagonal matrix, with 1s on the diagonal.
(In $A(S_n)$ I have restricted here to subgroups of the form $S_{\lambda} = \prod_{i=1}^r S_{\lambda_i}$)
and, upon further request for a citable reference, a pointer to Sagan 01 (but it seems that’s meant as a general pointer to the rep theory of the symmetric group, not to a citable proof of the statement in question).
Okay, thanks Urs and David – I’ll check through that later. One of those things that is absolutely routine to experts, it seems.
Well, thinking about what was written for about 90 seconds, this
Inside $M_{\lambda}$ you find a direct sum of a new irreducible module $V_{\lambda}$ and modules of the form $V_{\mu}$, for $\mu \lt \lambda$
looks pretty hand-wavy: about as hand-wavy as the Gram-Schmidt page. But maybe it will become clearer in time. If I get desperate, maybe I’ll put it to MO.
Thanks, Todd, for looking into this.
(The situation reminds me of what Langlands said (somewhere here) about specialists hurting their field by being so specialist.)
Here is a first list of our results (using computer algebra by Simon Burton) on cokernels of $A(G) \overset{\beta}{\to} R^{(fin)}_k(G)$ for nonabelian finite subgroups of $SU(2)$: jpg.
In all cases, beta is surjective over $\mathbb{R}$ onto the ring of integer-valued (i.e. not irrational-valued) characters.
Interesting how many items in that right-most column are essentially $\mathbb{Z}[\rho]/\mathbb{Z}[2\rho]$. Is that due to the central extensions by $\mathbb{Z}/2$?
Don’t know if this is related to any central extension. Here one such contribution appears for every complex irrep with integer characters but of quaternionic/symplectic type.
I’ll show the details of the computation later when the file has taken shape a bit more.
For what it’s worth, here is an overview of the physics picture behind that result in #33: jpg
If we see group characters appear in for fractional D-branes, should we not expect marks to play a role in M-theory?
Much like character theory simplifies working with group representations, marks simplify working with permutation representations and the Burnside ring. (Wikipedia)
According to Pfeiffer here:
The table of marks of $G$ arises from a characterization of the permutation representations of $G$ by certain numbers of fixed points. It provides a compact description of the subgroup lattice of $G$ and enables explicit calculations in the Burnside ring of $G$.
Plausibly, but I remain uncertain what exactly there is to be said.
A special kind of mark, namely that of cyclic groups $C_g$ generated by elements $g \in G$, already does play a central role, because that is equivalently just the character of a permutation representation $k[G/H]$ (for any ground field $k$):
$m(H, C_g ) \;=\; \chi_{k[G/H]}(g) \,.$1 to 38 of 38