Re #35 and #47: I have now added something to the new renderer to handle equation referencing. In particular, the equation references at geometry of physics – perturbative quantum field theory seem to work fine now.

It does add a little more time to the rendering, and I think one will get a timeout if one tries to save this page directly now. But the timeout should I think be harmless, the edit should still go through. It remains a high priority to either make the rendering faster or else make it asynchronous (or both).

The way I have implemented the equation numbering/referencing is server side. I plan to change the theorem environment numbering/referencing to be server-side as well, in the same way (currently it is client side).

A consequence of the way I have implemented it is that each page now has the equation references within it stored in a JSON file. I will do the same for theorem environment/referencing when I get around to that. This means amongst other things that it will be fairly straightforward to implement referencing of theorems/equations across pages, and that it will be easy to give an error message if a reference does not exist, which is on the Technical TODO list (nlabmeta).

]]>Hehe, no problem. As will now be evident, I hadn’t actually clicked on the links!

]]>Sorry, I thought you were joking, or I would have answered myself.

]]>Oh, oops!

]]>I think this meaning of “lens” is a mathematical one…

]]>Thanks, Todd.

Maybe to make it more concrete:

For $M$ our multiplicities matrix,

$H = U \cdot M$its Hermite normal form, and

$\tilde H = \tilde U \cdot M$the result of deleting the zero-rows in $H$, it seems to happen in all examples checked that

$\tilde U \cdot M \cdot \tilde U^t = diag(n_i)$where $n_i \in \mathbb{N}$ is the categorified norm square of the $i$th rational irrep, hence that $\tilde U$ happens to be the basis transformation onto the rational irreps, hence to be as close to orthonormalization as is possible over the rationals.

That’s fantastic, that’s what I want to have. But this is not manifest in the way $U$ was constructed. So why does it work??

]]>There are some people who insist on $p$-groups and $p$-subgroups being nontrivial, e.g., Steven Roman in Fundamentals of Group Theory, pp. 80-81, but I guess this is just a convention.

]]>Replaced a proof that I had earlier adapted from Wikipedia with a cleaner proof that I learned from Benjamin Steinberg (Theorem 3.4). For archival purposes I will post the earlier proof here.

+– {: .proof}

(We adapt the proof from Wikipedia.) Let $P_k = \binom{G}{p^k}$ be the collection of subsets $S$ of $G$ of cardinality $p^k$, and let $G$ act on $P_k$ by taking images of left translations, $S \mapsto g S$. For $S \in P_k$, any $h \in S$ yields a monomorphism

$Stab(S) \hookrightarrow G \stackrel{g \mapsto g h}{\to} G$that (by definition of $Stab(S)$) factors through $S \hookrightarrow G$; this gives a monomorphism $Stab(S) \to S$, and so ${|Stab(S)|} \leq p^k$. Now we establish the reverse inequality for a suitable $S$. Writing $n = p^k m$, we have

${|P_k|} = \binom{p^k m}{p^k} = m \prod_{j = 1}^{p^k - 1} \frac{p^k m - j}{p^k - j}$where the product after $m$ on the right is easily seen to be prime to $p$ (any power of $p$ that divides one of the numerators $p^k m - j$ also divides the denominator $p^k - j$, so that powers of $p$ in the product are canceled). Therefore $ord_p({|P_k|}) = ord_p(m)$; let $r$ be this number. Writing out the class equation

${|P_k|} = \sum_{orbits x} \frac{{|G|}}{{|Stab(S_x)|}},$not every term on the right can have $p$-order greater than $r$, so there is at least one orbit $x$ where

$ord_p(\frac{{|G|}}{{|Stab(S_x)|}}) = ord_p({|G|}) - ord_p({|Stab(S_x)|}) \leq r = ord_p(m).$We may rearrange this inequality to say $ord_p({|G|}/m) \leq ord_p({|Stab(S_x)|})$; in other words $p^k = {|G|}/m$ divides ${|Stab(S_x)|}$. Therefore $Stab(S_x)$ has order $p^k$, which is what we wanted. =–

]]>Nothing stops $H$ from being trivial or of order prime to $p$; there the $p$-Sylow subgroup is trivial as you surmise.

]]>Oh, oh, I see your concern; thanks for spelling it out. Let me think about it some more before trying to put together a response.

]]>I’m probably being dim but shouldn’t one add a condition to $H$. I mean, what’s to stop $H$ being trivial. Or is that OK, and one can speak of the trivial group as a $p$-Sylow subgroup of itself for any $p$? Don’t we need positive powers of $p$?

]]>Thanks, I have fixed these now.

Just for emphasis: When you run into more such entries that don’t display, you can fix it by doing this:

Change URL in your browser from

`https://ncatlab.org/nlab/show/entry+name`

to

`https://ncatlab.org/nlab/edit/entry+name`

Make a trivial change in the resulting edit pane, such as adding a whitespace

hit “submit”

I am happy to admit that I haven’t thought about Diophantine linear algebra before, and need a moment of reflection on the gcd-techniques needed to force row reduction to stay integral. Once one thinks it its obvious (e.g. here), as it goes.

But is there really an integer version of Gram-Schmidt? Above I tried to convince myself, by appeal to uniqueness of Hermite normal forms, that applied to the particular case of Burnside multiplicity matrices there is a general reason why we are guarenteed to end up reading off multiplicities of irreps in permutation reps from the matrix entries. But not sure if this really works.

In the entry you say at the point where the irreps come in: “…will turn out to be…”. Is that meant to be by inspection in that particular example? Or is there a general argument that the integrally row-reduced matrix exhibits the inner products of our permutation reps against irreps?

]]>I haven’t seen a discussion of $\mathbb{F}_{1^n}$ in Connes-Consani; they have apparently proved the only finite semi-field that’s not a field is $(\{0,1\},max,+)$, and they take this to be the “prime field” of characteristic 1.

]]>Added a proof of existence of Sylow subgroups mentioned by Benjamin Steinberg at the Café.

]]>Urs, let me put it a different way. Looking at Hermite normal form, I get a sense we might have the same thing in mind, but there’s a slight communication gap.

Row reduction in linear algebra boils down to starting with a matrix $M$ and left-multiplying by elementary matrices in a certain way until we reach an echelon form. If we are working over any commutative ring $R$, then “elementary matrices” are defined to be matrices of the following form:

$1 + E$ where $E$ is a strictly lower triangular matrix with a single nonzero entry $e_{i j}$, whose left-multiplying effect is to add $e_{i j}$ times the $i^{th}$ row to the $j^{th}$ row

a permutation matrix, which we could take to be a transposition matrix whose effect is to swap two adjacent rows,

a diagonal matrix all of whose entries are invertible in $R$ (WLOG, assume but one is $1$).

In the case $R = \mathbb{Z}$, we’re never going to pop out of $\mathbb{Z}$ to $\mathbb{Q}$ entries this way (so I didn’t understand why you brought this up). The third type of elementary matrix has just $1$’s and $-1$’s down the diagonal. In general, if you start with a matrix $M$ over $R$ and left-multiply by elementary matrices defined this way, you will end with a matrix over $R$.

Now it could be that you wish to disallow the third type of elementary matrix, the diagonal ones. It’s true that whenever I’ve tried to apply the categorified Gram-Schmidt process, I’ve never had to make use of that third type. But if you want to disallow it, then I’d like to understand the theoretical grounds for why. Or, to put it more positively, why we don’t actually need it (which I have some small faith is the case, but I don’t know the reason).

Sorry if I’m being dense.

]]>Maybe there’s something in

- William Lovas, Frank Pfenning,
*A Bidirectional Refinement Type System for LF*, (doi)

Corrected the Pierce-Turner paper length from 4 to 44 pages, and included link to pdf.

]]>You’d think type refinement might crop up here, and I see a gesture in that direction at least at slide 45 of one of Noam’s talks.

I see also bidirectional typing is mentioned in the abstract of Noam’s A theory of linear typings as flows on 3-valent graphs. I guess there’s interesting combinatorics here.

]]>What does “whose degrees of freedom are a set of 9+1 large matrices” mean at BFSS model? There are 10 matrices of a large size?

]]>starting some minimum

]]>Riemann-Roch theorem over F_1

In the process of some pondering on the place of $\mathbb{F}_1$ in Sylow $p$-group theory, John Baez pointed me to this paper by Kapranov and Smirnov which speaks of Riemann-Roch over $\mathbb{F}_{1^n}$ as counting residues mod $n$ of the number of integer points of some polyhedron.

Hmm, so what’s $K \mathbb{F}_{1^n}$?

The Sylow thought, by the way, is that the $p$-Sylow subgroup of any $GL_n(Z_{p^k})$ is the maximal unipotent subgroup, and any group embeds in $S_n$ which embeds in $GL_n(Z_{p^k})$.

]]>Fixed that link.

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