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

- David Gepner, Rune Haugseng, Joachim Kock,
*∞-Operads as Analytic Monads*, (arXiv:1712.06469).

Is there an $\mathbb{F}_1$ view on finite $\infty$-groupoids? I see in the notes by Barwick, $\mathsf{Perf}_{\mathbb{C}}$ is

the $(\infty,1)$-category of perfect $H \mathbb{C}$-modules – that is, of $H \mathbb{C}$-modules with finitely many homotopy groups, all of which are finite-dimensional.

$H \mathbb{F}_1$-modules would give us what? Chains of sets?

]]>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 $\infty$-site for globally “higher equivariant” homotopy theory is that of all finite $\infty$-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 $(\mathcal{F}/\alpha)^{faith}$ of finite $\infty$-groupoids over deloopings of finite $\infty$-groups $\alpha = B \mathcal{G}$ are the sites for higher $\mathcal{G}$-equivariant homotopy theory.

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

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

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

- Imma Gálvez-Carrillo, Joachim Kock, and Andrew Tonks.
*Homotopy linear algebra*. Proc. Roy. Soc. Edinburgh Sect. A, 148(2):293–325, 2018 (arXiv:1602.05082)

What plays the role of finite vector spaces there are the slices $\mathcal{F}/\alpha$, where $\mathcal{F}$ is the $(\infty, 1)$-category of finite $\infty$-groupoids. These slices are objects of $lin$, whose morphisms are derived from spans of finite $\infty$-groupoids, $\alpha \leftarrow \mu \to \beta$. 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.

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

]]>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)

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 “$\mathbb{Z}_4$”. Will fix later.

]]>For the surjectivity of $\beta$ over $\mathbb{Q}$ on $p$-primary groups, I have added pointer also to Theorem 4.4.1 in

- Tammo tom Dieck, Section 4 of
*Transformation Groups and Representation Theory*, Lecture Notes in Mathematics 766, Springer 1979 (doi:10.1007/BFb0085965)

Just to say that the answer to Richard’s #95 is now recorded as this Prop.

]]>At table of marks *Properties* it has

The following says that the Burnside character plays the same role for finite G-sets as characters of representations play for finite-dimensional linear representations

Feel invited to expand on that!

]]>Is it worth remarking somewhere that tables of marks are the $\mathbb{F}_1$-version of character tables? I came across them (#37) by thinking that there should be such a version.

]]>I’ll send you an email on that.

]]>Ah, thanks very much! Do you know if the table of multiplicities is a standard thing considered in the literature? Just wondering for example if it can be computed from the table of marks. Sorry for the naïve questions!

]]>Thanks, Richard. Yes, I had mentioned this by email, this was a mistake I had introduced in the v2 document on the nLab. It’s been fixed in the latest version here. The two tables are just not the same (they “sit inside each other”, which had misled me.)

]]>Although now that I look at Proposition 3.10 in the paper, it looks as though the ’table of multiplicities’ on pg.16 is supposed to be exactly the table of marks, whereas this cannot be the case, because there will always be zero entries in the table of marks. Perhaps you could clarify this before we go further?

]]>Have made progress on the implementation. Checked that GAP can get the table of marks in a such a way that the cyclic subgroups can be determined almost instantaneously up to as far as I tried, which was about $Q_{1000}$, so we should have plenty of examples to be able to work with. However, GAP’s table of marks for the binary dihedral group is as follows. A full stop . or nothing means 0.

```
1: 8
2: 4 4
3: 2 2 2
4: 2 2 . 2
5: 2 2 . . 2
6: 1 1 1 1 1 1
```

Here 1-5 are cyclic and 6 is not. This is obviously not the same as the ’table of multiplicities’ in your paper/produced by Simon’s code. I imagine that things will still work out and that one should get the same characters for the image of $\beta$ in the end if one applies your algorithm, but I’ve not finished checking this yet.

]]>Found some assertions that it *is* possible to see which conjugacy classes have a cyclic representative just from the table of marks, but have not found a reference for how yet.

Also found that GAP does in some cases allow for finding representatives of the conjugacy classes. For binary dihedral groups I guess computing the subgroup lattice could be done very quickly anyhow.

]]>Of course it would not be very difficult to calculate the table of marks from scratch if one knows the conjugacy classes of subgroups. But I don’t know if there’s an easy way to compute what the latter are exactly? Edit: well, it seems that GAP can do it! So, if there is no better way, we can get GAP to compute the subgroup lattice, and get the table of marks from that. But I would imagine it would be slow if the group is of any kind of significant size, unless there are some shortcuts for the kind of groups you’re looking at.

]]>Hi Urs, thanks again! I’m getting there. I’ve now implemented the algorithm for getting $\tilde{U}$ and $\tilde{H}$ from the table of marks. But it’s still not entirely clear to me how to get the table on the top of pg.17. For how do we know which rows in the table of marks correspond to conjugacy classes of those subgroups with a single generator? (It seems, both in your paper and in GAP, that the rows of the table of marks are not labelled in such a way as to be able to immediately see this). Apologies if this is completely obvious!

]]>Right, so since the character of $k[G/H]$ at $g$ is the number $\vert (G/H)^{\langle g\rangle} \vert$ of fixed points of the cyclic group generated by $g$ on $G/H$, this should be gotten from GAP by calling the table of marks.

]]>Very helpful! Sorry for the question (am asking in passing, not looked properly), but is there is an easy way to calculate the characters of $G/H_j$? Say just with matrix manipulations of some kind?

]]>Thanks, Richard, for starting to look into this!

So for step 3 we need the invertible matrix $U$ which implements the row reduction of step 2 by left multiplication.

The rows of that matrix are the coefficients of the permutation representations $k[G/H_j]$ in the given $V_i$. So the character of $V_i$ is the sum of the characters of $k[G/H_j]$ (which in turn is combinatorially given by number of fixed points) times the corresponding entry of that matrix $U$.

Does that help?

]]>Actually step 2 is fine, at least for the simplified version. So it is just step 3 that I could do with a bit of help with, due to the notation: is it possible to compute it from the upper triangular matrix?

]]>Hi Urs, I’ve begun taking a look at doing the computations you need. Since GAP can produce the table of marks as well as the character table (I have checked this now :-)), it would be nice to do something like in #70 directly (Simon’s code has I think much more than would be needed for the remaining steps).

It would save me some time if you could spell out without the notation how steps 2 and 3 in #70 work. For example, if you could explain how to do these steps in Example 4.1 in your paper, I can probably figure out what is going on.

]]>The formulation of genuine $G$-spectra (in any $\infty$-topos) as spectral Mackey functors, hence as spectrally enriched $\infty$-functors on the Burnside category, exhibits them as motives in the sense of “sheaves with transfer” with respect to “spaces” which are just finite $G$-sets, hence a kind of discrete spaces with group actions. (Compare to pure motives or Chow motives or similar, where instead of finite $G$-sets one considers certain schemes.)

Notice how close this is to the intuition advertized in Baez-Hoffnung-Walker: A finite $G$-set may equivalently be thought of as a finite groupoid with isotropy groups contained in $G$, and hence the Burnside category is much like that category of tame spans of finite groupoids. Thus spectral Mackey functors on the Burnside category, and hence genuine $G$-spectra, are much like an $\infty$-category-theoretic realization of the intuition behind what they call the “degroupoidification” functor. (One can also get rid of the explicit dependence on $G$ here by passing further to global equivariant spectra.)

]]>