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• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeOct 11th 2011
• (edited Oct 11th 2011)

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

• Claudio Procesi, Lie groups, an approach through invariants and representations, Universitext, Springer 2006, gBooks

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.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeOct 12th 2011

FWIW, I would have assumed “permutation representation” to mean in the standard representation-theory sense of vector spaces.

• CommentRowNumber3.
• CommentAuthorjim_stasheff
• CommentTimeOct 12th 2011
Is this not the usual presumption of a relevant cat?
A group in cat can act on an object in the cat
hence the group has a representation via automorphisms of that object
Is not a LINEAR representation one where the object is a vector space?
• CommentRowNumber4.
• CommentAuthorTobyBartels
• CommentTimeOct 12th 2011

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.

• CommentRowNumber5.
• CommentAuthorzskoda
• CommentTimeOct 12th 2011

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.

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeOct 12th 2011

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

$B G \to Set \stackrel{free}{\to} Vect_k.$
• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeOct 12th 2011

It makes sense to say “permutation representation” for $\cdots \to \mathrm{Set}$ and “linear permutation representation” for $\cdots \to \mathrm{Set} \to \mathrm{Vect}$.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeOct 12th 2011

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.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeOct 12th 2011

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.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeOct 12th 2011

I have added a word to the entry.

• CommentRowNumber11.
• CommentAuthorTobyBartels
• CommentTimeOct 12th 2011

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.

• CommentRowNumber12.
• CommentAuthorzskoda
• CommentTimeOct 13th 2011

they should be different, as in this case

Clear “representation by permutations” for categorical notion and cryptic “permutation representation” for linear notion are different.

• CommentRowNumber13.
• CommentAuthorTim_Porter
• CommentTimeOct 13th 2011

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).

• CommentRowNumber14.
• CommentAuthorTobyBartels
• CommentTimeOct 14th 2011
• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeSep 11th 2018
• CommentRowNumber16.
• CommentAuthorUrs
• CommentTime7 days ago

while we are at it, I (re-)wrote this entry to contain a more decent account of the plain basics

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTime7 days ago

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

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTime7 days ago
• (edited 7 days ago)

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.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTime7 days ago
• (edited 7 days ago)

made explicit the simplest non-trivial example: $A(\mathbb{Z}/2) \underoverset{\simeq}{\beta}{\longrightarrow} R(\mathbb{Z}/2)$, here

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTime6 days ago
• (edited 6 days ago)

added also pointer to Todd’s and James Montaldi’s example of $\beta$ for $G = S_4$ here.

Is this still a surjection for rational reps?

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTime6 days ago
• (edited 6 days ago)

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]

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTime4 days ago
• (edited 4 days ago)

removed the statement of surjectivity of $\beta$ for cyclic groups, since my argument had a flaw, based on a flaw in the earlier version of the lecture note mentioned above (see here). It might still be true, but I don’t know.