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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 SetSet. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group GG on a set XX, and looks what happens in the vector space of functions into a field KK. As we know, for a group element gg the definition is, (gf)(x)=f(g 1x)(g f)(x) = f(g^{-1} x), for f:XKf: X\to K is the way to induce a representation on the function space K XK^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 CC 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 SetSet (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

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

    It makes sense to say “permutation representation” for Set\cdots \to \mathrm{Set} and “linear permutation representation” for SetVect\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 nnLab 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 SetSet and a permutation representation in VectVect 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

    diff, v15, current

    • 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

    diff, v16, current

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

    In the proof of the proposition about β\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 pp-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 β\beta sends “external Cartesian products” to external tensor products, by the same elementary argument which shows that β\beta sends plain Cartesian products to tensor products. This way the statement reduces to that for pp-primary cyclic groups.

    diff, v19, current

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

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

    diff, v20, current

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

    added also pointer to Todd’s and James Montaldi’s example of β\beta for G=S 4G = S_4 here.

    Is this still a surjection for rational reps?

    diff, v20, current

    • 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 β\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.

    diff, v24, current