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    • CommentRowNumber101.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2019
    • (edited Oct 21st 2019)

    added a graphics (here) illustarting the map from a) unstable equivariant Cohomotopy of representation spheres to b) the Burnside ring to c) the representation ring

    Now that I have done it I see that I should have made the labels “G” be explicitly “4”. Will fix later.

    diff, v41, current

    • CommentRowNumber102.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2020
    • (edited Jun 3rd 2020)

    added this historical pointer (dug out by David C. in another thread) where the term “representation group” is used to refer to a group equipped with a permutation representation:

    • William Burnside, On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

    diff, v43, current

    • CommentRowNumber103.
    • CommentAuthorUrs
    • CommentTimeJun 3rd 2020

    Sorry, wrong spot. Actually Burnside means by “permutation group” what we now call a “G-set”. Am moving the reference.

    • CommentRowNumber104.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 15th 2021
    • (edited Jun 15th 2021)

    I wonder if anything interesting happens if we shift from G acting on sets to it acting on the finite -groupoids treated in

    What plays the role of finite vector spaces there are the slices /α, where is the (,1)-category of finite -groupoids. These slices are objects of lin, whose morphisms are derived from spans of finite -groupoids, αμβ. There’s a ’cardinality’ map from lin to FinVect.

    Wild speculation: perhaps virtual permutation reps in this setting could close the gap with virtual linear reps.

    • CommentRowNumber105.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2021

    Speaking of wild speculation, I keep wondering if it’s just a coincidence that the 2-site for global equivariant homotopy theory – namely the global orbit category – is that of connected finite 1-groupoids (connected finite homotopy 1-types). This suggests that the -site for globally “higher equivariant” homotopy theory is that of all finite -groupoids.

    Moreover, the site for G-equivariant homotopy theory is the faithful slice of this site over the delooping groupoid of G, which suggests that the faithful slices (/α)faith of finite -groupoids over deloopings of finite -groups α=B𝒢 are the sites for higher 𝒢-equivariant homotopy theory.

    These ingredients are not-so-vaguely reminiscent of Goodwillie calculus, which works over the opposite and pointed site of finite -groupoids (e.g. here).

    Probably Goodwillie calculus is also a good thing to keep in mind when thinking of homotopy linear algebra.

    • CommentRowNumber106.
    • CommentAuthorDavid_Corfield
    • CommentTimeJun 15th 2021

    Homotopy linear algebra certainly fits very neatly with polynomial and analytic functors in the non-Goodwillie sense, as in

    Is there an 𝔽1 view on finite -groupoids? I see in the notes by Barwick, Perf is

    the (,1)-category of perfect H-modules – that is, of H-modules with finitely many homotopy groups, all of which are finite-dimensional.

    H𝔽1-modules would give us what? Chains of sets?

    • CommentRowNumber107.
    • CommentAuthorJohn Baez
    • CommentTimeFeb 28th 2025

    Added the second sentence here:

    The image of the comparison morphism β=K(k[]) (Def. \ref{ComparisonMapBurnsideRingRepresentationRing}) may be called the virtual linear permutation representations. Any virtual linear permutation representation is a formal difference, in the representation ring of G, of elements coming from permutation representations.

    diff, v48, current

    • CommentRowNumber108.
    • CommentAuthorJohn Baez
    • CommentTimeFeb 28th 2025

    Added more information about Serre’s counterexample.

    diff, v48, current

    • CommentRowNumber109.
    • CommentAuthorJohn Baez
    • CommentTimeMar 4th 2025

    Added proof of an exercise in Serre’s book Linear Representations of Finite Groups:

    Proposition

    (nonsurjectivity of the map from virtual permutation representations to virtual linear representations)

    Suppose G is a finite group with a linear representation ρ such that:

    1. ρ is irreducible and faithful
    2. every subgroup of G is normal
    3. ρ appears with multiplicity n2 in the regular representation of G.

    Then the map β:A(G)R(G) is not surjective.

    diff, v50, current