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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.
Classical vs Quantum Computation (Week 7)
- Nov 21, 2006
- Building a computer using the lambda calculus: the Fixed Point Theorem.
Generalising Hopf Algebras
- Mar 20, 2007
- Gizem Karaali’s paper ‘On Hopf algebras and their generalizations’, in which she describes Hopf algebras and five attempts to generalise them.
Borisov on Shapes of Cells in n-Categories
- Oct 6, 2009
- Can you comment on a paper about comparing definitions of $n$-categories?
Native Type Theory (Part 3)
- Mar 29, 2021
- The third episode of “Native Type Theory” by Christian Williams.
Geometric Representation Theory (Lecture 20)
- Jan 10, 2008
- At last: the Fundamental Theorem of Hecke Operators!

Recent Comments

Equivalence via Surjections (39)
- Tim Campion
  - Oct 19, 2025 15:36
  - Another way to look at it: Say C is a nice category equipped with a notion of equivalence and a ...
- Tim Campion
  - Oct 19, 2025 15:20
  - I guess this example (or already the example of 2-categories) illustrates another point about this: if every object is cofibrant, ...
- Timothy Campion
  - Oct 19, 2025 15:16
  - Another example in this paradigm is an n-equivalence of topological spaces, which is an isomorphism on π ≥n+1\pi_{\geq n+1} and ...
- Alexander Campbell
  - Oct 8, 2025 10:12
  - Necessarily the “coinductive” weak equivalences, because these are the weak equivalences in the unique model structure on the category of ...
- Tim Campion
  - Oct 8, 2025 02:58
  - How does this play out for strict omega-categories? “strict surjective equivalence” sounds like the same thing as a folk trivial ...

Relational Universal Algebra with String Diagrams (2)
- Matt
  - Oct 14, 2025 19:12
  - It seems to me that this but needs editing, but I’m not sure what word is perhaps missing: “This two ...
- John Baez
  - Sep 5, 2022 20:35
  - Nice! It’s not at all important, but I’m don’t remember hearing the word “inequation” — the people I know seem ...

Universe Polymorphism and Typical Ambiguity (12)
- Mike Shulman
  - Oct 14, 2025 17:02
  - The words “polymorphism” and “homomorphism” are basically unrelated.
- Matt
  - Oct 14, 2025 16:09
  - I’m interested in the relationship of this topic to classical Algebra and to universal algebra; however, I don’t see here ...
- Mike Shulman
  - Dec 11, 2012 16:43
  - Okay, I think I get what you’re saying. NF is one thing that might result if you start from a ...
- Zhen Lin
  - Dec 11, 2012 09:10
  - Hmmm. Perhaps I was not clear. Essentially, NF is the _result_ of extracting the typically ambiguous fragment of Russellian type ...
- Mike Shulman
  - Dec 11, 2012 02:09
  - In the systems I was writing about, there are many different universes, and when we write Type:TypeType:Type its meaning is ...

Group Cohomology and Homotopy Fixed Points (3)
- Dean Young
  - Oct 12, 2025 20:03
  - Wow thanks so much professor Baez! I may not have realized this without your post. I know I’ve been struggling ...
- John Baez
  - Apr 30, 2020 17:33
  - Thanks! Yes, Noether. I’ve decided to improve this paragraph: And this is called Hilbert’s Theorem 90. It’s a famous result ...
- Mathlog
  - Apr 30, 2020 17:02
  - It looks like Andreas Speiser did that in 1919. did that in 1933. There is something missing, presumably the name ...

A Complex Qutrit Inside an Octonionic One (4)
- John Baez
  - Oct 7, 2025 17:41
  - Okay, now I’m following the logic. Yes, that would be another brilliant insight-free proof. (At least for me, since I’d ...
- John Baez
  - Oct 7, 2025 16:54
  - I wrote: I know the subgroup of F 4\mathrm{F}_4 that fixes e 1,e 2,e 3e_1, e_2, e_3 contains Spin(8)\text{Spin}(8). I ...
- John Baez
  - Oct 7, 2025 16:45
  - Theo wrote: Doesn’t Part 1 follow from the statements “Yokota shows [(SU(3)×SU(3))/ℤ 3(SU(3) \times SU(3)) / \mathbb{Z}_3] preserves the standard ...
- Theo Johnson-Freyd
  - Oct 6, 2025 20:18
  - There is a result due to Kac that reads off the classification of (conjugacy classes of) torsion elements of adjoint ...

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 ...

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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 | Followups (4)

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