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October 2025
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October 2025's Entries

A Complex Qutrit Inside an Octonionic One
- Oct 6, 2025
- I’m trying to better characterize two maximal subgroups of the group of automorphisms of the exceptional Jordan algebra, whose intersection is the Standard Model gauge group.

Random Past Entries

TeXnical Issues
- September 2, 2006
- Sage advice on viewing this blog and posting comments thereon.
On Digital Mathematics and Drive-By Contributors
- Jan 10, 2016
- Slides from a talk about my experiences with digital mathematics.
Geometric Representation Theory (Lecture 18)
- Jan 1, 2008
- An intro to degroupoidification: the process of turning groupoids into vector spaces, and spans of groupoids into linear operators. A key prerequisite: ‘groupoid cardinality’.
Metric Coinduction
- May 28, 2009
- When does coinduction differ from induction?
Commuting Limits and Colimits over Groups
- Dec 21, 2013
- An apparently new limit-colimit commutation theorem.
Quantization and Cohomology (Week 12)
- Jan 31, 2007
- Classical mechanics, quantum mechanics and statistical mechanics as ‘matrix mechanics’ over various rigs (rings without nnegatives).

Recent Comments

Equivalence via Surjections (34)
- Tom Leinster
  - Oct 6, 2025 11:51
  - Moser, Sarazola and Verdugo’s paper has now appeared: arXiv:2509.23181.
- Mike Shulman
  - Sep 29, 2025 07:37
  - I think there’s another justification for “spans of surjective equivalences” as a notion of equivalence, that isn’t just model category ...
- Mike Shulman
  - Sep 29, 2025 07:07
  - In homotopy theory I think a more common phrase for “weakly right orthogonal to” is “having the right lifting property ...
- Tom Leinster
  - Sep 19, 2025 21:41
  - On reflection, there’s a uniform definition of “surjective equivalence” that works in many contexts—or nearly, anyway. The condition is being ...
- David Roberts
  - Sep 12, 2025 08:22
  - Ah, my mistake! I hadn’t read the comments on the post, I think I only saw it when it was ...

A Shadow of Triality? (14)
- John Baez
  - Sep 28, 2025 11:02
  - Here are some more details on how Conway and Smith get from triality to this fact: if cc is an ...
- John Baez
  - Sep 26, 2025 09:21
  - Ben wrote: They define an inverse loop (p. 83) and show that any inverse loop will satisfy xyz=yzx=zxy=1x y z=y ...
- John Baez
  - Sep 25, 2025 12:59
  - Wow, this could be the really beautiful explanation I’d been hoping for! Thanks! I’d been trying to read Conway and ...
- Ben
  - Sep 24, 2025 15:04
  - I think the answer (or something close to it) turns out to be in Conway and Smith’s On Quaternions and ...
- John Baez
  - Sep 16, 2025 18:51
  - I got some help from Vít Tuček on MathOverflow. He pointed out that Yokota gave a proof that all inner ...

Safeguarded AI Meeting (10)
- David Roberts
  - Sep 4, 2025 06:26
  - In all cases, the line-by-line proof details presented here were constructed by the author. It seems important to point out ...
- Jeffery Winkler
  - Sep 3, 2025 06:55
  - Here is a physics paper partially written using ChatGPT. https://arxiv.org/pdf/2508.21276 This will be more common in the future. Is this ...
- Jeffery Winkler
  - Sep 3, 2025 06:27
  - Here are some more links about the connections between physics and AI. Physics Today - Nobel Prize https://pubs.aip.org/physicstoday/article-abstract/77/12/12/3320663/Nobel-Prize-highlights-neural-networks-physics 2 Physics ...
- Jeffery Winkler
  - Aug 20, 2025 16:55
  - In 1954, Frederick Brown wrote a science fiction story titled “The Answer”. It’s so short I can reproduce it here. ...
- Matteo Capucci
  - Aug 18, 2025 07:15
  - I quote the comment I left on John’s blog about this: I’m part of SGAI and I believe this is ...

What Is an n-Theory? (118)
- Mike Shulman
  - Sep 1, 2025 18:56
  - Yes, I would say that.
- David Corfield
  - Sep 1, 2025 08:02
  - Ok, thanks. I was just thinking that I was now left unsure as to the principles of taxonomy of 2-theories ...
- Mike Shulman
  - Sep 1, 2025 00:49
  - It does look a little like the 3-theory of FOL, but the restrictions that each protype depends only on two ...
- David Corfield
  - Aug 30, 2025 08:37
  - Great, thanks! …we can also consider notions of “theory” that are models of an (n+1)(n+1)-theory somewhere else. I was wondering ...
- Mike Shulman
  - Aug 29, 2025 18:51
  - In the post, I argued that the semantic 1-theories in any doctrine form a 2-category — for instance, the 2-category ...

(BT) Diversity from (LC) Diversity (4)
- Mark Meckes
  - Sep 1, 2025 11:55
  - While thinking about an upcoming talk I will give about this work, I suddenly realized what I should have titled ...
- Tom Leinster
  - Aug 5, 2025 18:39
  - Thanks for the information on what’s needed of ψ\psi. Evidently the situation is different from the one I had in ...
- Mark Meckes
  - Aug 5, 2025 18:32
  - Regarding your first question, what we do in the paper is first show that D−1D-1 is a BT diversity (using ...
- Tom Leinster
  - Aug 5, 2025 17:22
  - That’s super nice! In the remarks after Theorem 1 of your post, and also in Theorem 1.7 of your paper, ...

How to Count n-Ary Trees (8)
- Lutz Warnke
  - Aug 20, 2025 22:04
  - There are many different proofs of the Lagrange inversion (not all use Bell polynomials), which might help to illuminate the ...
- John Baez
  - Jul 25, 2025 08:34
  - I keep meaning to expand this article a bit so it shows how we get from the general formula for ...
- John Baez
  - Jul 8, 2025 18:40
  - I blogged about Aguilar and Ardila’s work here: More secrets of the associahedra, January 25, 2018. But now I am ...
- Kenneth
  - Jul 8, 2025 18:32
  - Here is perhaps one way to interpret Lagrange inversion from a different (deeper?) perspective, namely generalized permutahedra: Marcelo Aguiar and ...
- Tom Copeland
  - Jul 8, 2025 00:06
  - Should the reciprocity for the rising and falling factorials (−nk)=(−1) k(n+k−1k) \binom{-n}{k} = (-1)^k \binom{n+k-1}{k} arise in your machinations and/or ...

2-Rig Conjectures Proved? (6)
- John Baez
  - Aug 10, 2025 12:55
  - Hey Mike: Kevin Coulembier seems to have found away around your problem—without agreeing that finding a way around it is ...
- John Baez
  - Jul 26, 2025 09:17
  - What we’re doing is sort of a categorification of that. A 2-rig contains some objects xx where the symmetry isomorphism ...
- John Baez
  - Jul 26, 2025 09:07
  - One way out, without changing any definitions, might be this: Show that Rep(M(n,k))\mathsf{Rep}(M(n,k)) is the free 2-rig on an object ...
- Jacques Distler
  - Jul 26, 2025 08:19
  - I know nothing about any of this. But in the more pedestrian world of ℤ 2\mathbb{Z}_2 graded-commutative algebras, there are ...
- John Baez
  - Jul 25, 2025 20:42
  - This is a great point. Unfortunately I have not yet carefully examined Coulembier’s paper, but now I have a great ...

Jack Morava (2)
- RodMcGuire
  - Aug 7, 2025 06:31
  - Aug 6: Johns Hopkins obituary
- David Corfield
  - Aug 4, 2025 15:01
  - Oh, that’s a pity! For a long time he’d comment here under the name ‘jackjohnson’ and always had highly interesting ...

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October 6, 2025

A Complex Qutrit Inside an Octonionic One

Posted by John Baez

$MathML-enabled post (click for more details).$

Dubois-Violette and Todorov noticed that the Standard Model gauge group is the intersection of two maximal subgroups of $\mathrm{F}_4$ . I’m trying to understand these subgroups better.

Continue reading “A Complex Qutrit Inside an Octonionic One” …

Posted at 5:24 PM UTC | Permalink | Post a Comment

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