#representationtheory — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #representationtheory, aggregated by home.social.
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🧮 New preprint, Part III: *A Centre-Charge Selection Rule for the Wilson-Line Potential* — the fundamental domain of gauge–Higgs unification is representation-dependent.
If you compute a Hosotani potential on \(T^2/\mathbb{Z}_2\), you probably halve the search region in \(\alpha_2\). For every representation able to host a Standard-Model quark generation, that halving is invalid — and nothing warns you.
Why: advancing the Wilson line by one period is, up to a Weyl reflection, multiplication by the central element \(-\mathbf 1\in Z(SU(4))\). A rep answers a central element with \(\zeta^{c(R)}\) — that is what \(n\)-ality *is*. So the character has no choice:
\[D(-t)=(-1)^{a+2b+3c}D(t),\quad D(t)=s_\lambda(1,-1,t,t^{-1}).\]That alphabet has determinant \(-1\): it sits on the **improper** component of \(O(4)\), where an irrep and \(\mu\otimes\det\) cancel. So this is not a delicate cancellation — it *empties a sublattice*. Every mode with \(k_2\) odd and \(m\) even is identically absent. Not small. Absent.
The centre charge, read for fifty years as a restriction on which representations carry which charges, also restricts **what the potential may contain**.
Needs \(N\) even, so \(SU(4)\) is the smallest group that can carry the rule. Classification certified in Lean 4, sorry-free.
📄 https://zenodo.org/records/21438928
💻 https://github.com/karlesmarin/centre-parity-selection -
🧮 New preprint — Part II of a series: *Three Gates to a Quark Generation*.
The question. In 6D gauge–Higgs unification, which representations of \(SU(4)\) hold a Standard-Model quark generation as chiral zero modes on \(T^2/\mathbb{Z}_2\)?
The answer, exactly. An irreducible rep with Dynkin labels \((a,b,c)\) works iff
\[(a+2b+3c)\ \text{odd}\ \ \wedge\ \ b\ge 1\ \ \wedge\ \ a+b+c\ge 3.\]
Three gates: arithmetic (the \(\mathbb{Z}_4\) centre charge), geometric (the middle Dynkin node excited), and size. Only 8 reps qualify below dim 400.For the curious. Strip a generation to its skeleton, the “\(\pm\tfrac12\) cell”: \(Q(\mathbf 2,\tfrac16),\, u(\mathbf 1,\tfrac23),\, d(\mathbf 1,-\tfrac13)\), where \(\tfrac16=\tfrac12\bigl(\tfrac23+(-\tfrac13)\bigr)\) puts the doublet at the hypercharge midpoint of its singlets — Pati–Salam's \(Y=T^3_R+\tfrac{B-L}{2}\). Now delete the middle node of \(\circ\!-\!\circ\!-\!\circ\): out comes a Levi \(SU(2)_L\times SU(2)_R\times U(1)\). Its \(SU(2)_L\)-singlet sector is \((b{+}1)\) copies of the Clebsch–Gordan tower \([\tfrac a2]\otimes[\tfrac c2]\); the orbifold's chirality keeps half of each, giving a closed zero-mode count
\[N(a,b,c)=(b{+}1)\,\tfrac{a+c+1}{2}.\]The twist: why \(SU(4)\)? The cell is three charge constraints on a hypercharge with \(\operatorname{rank}(G){-}1\) free parameters — over-determined exactly at rank 3. So \(SU(4)\) is the unique critical rank where the cell is a rigid gate; for \(SU(5),SU(6)\) it dissolves and generic reps admit. Machine-checked exactly to dimension 900.
📄 https://doi.org/10.5281/zenodo.21432628
💻 https://github.com/karlesmarin/su4-sm-cell-criterion#RepresentationTheory #Physics #GaugeTheory #GrandUnification
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A computer found one solution to a hard model-building problem. Then we proved you will *always* find one — and machine-checked the proof in Lean 4. 🧮
The setting: which fermions complete a quark block in a 6D \(SU(4)\) gauge–Higgs model on an orbifold, so that every local consistency condition (anomalies + "tadpoles") cancels? These are exact integer / representation-theory conditions over \(SU(4)\) weights.
One witness is easy to distrust — a fluke? The structural answer is no. Writing the anomaly map \(A\) and the tadpole map \(\Theta\) as linear functionals of the added matter, two exact facts settle it:
• \(\operatorname{rank}[A;\Theta]=8+2=10\): the tadpole is *independent* of the anomalies — no conserved invariant traps it;
• anomaly-neutral additions realise *every* tadpole direction (Farkas certificates), so their cone is all of \(\mathbb{R}^2\).Hence *every* anomaly-free completion is tadpole-compatible: an Existence theorem, not luck. The certificate is checked by the Lean 4 kernel, depending only on propext. The same rank+cone test ships as a reusable tool for any orbifold model. Honest scope: one infrared step stays open.
📄 https://zenodo.org/records/21432626
💻 https://github.com/karlesmarin/ghu-su4-completion#Lean4 #FormalMath #ProofAssistant #RepresentationTheory #Maths #Physics
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For any prime p there are exactly two non-abelian groups of order p³:
1. The Heisenberg group of the field with the p elements, which consists of matrices of the form
[1 a c]
[0 1 b]
[0 0 1]
with a, b, c in Fₚ.2. The semidirect product Z/p² ⋊ Z/p with Z/p acting on Z/p² via a·x := (1+ap)x.
Those two non-isomorphic groups have isomorphic character tables!
What are some other nice examples of infinite families of groups with isomorphic character tables?
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Sat Dec 7, 2024 on zoom & in-person
Session in memory of Richard Parker at the annual Nikolaus conference at Aachen (on group & representation theory). Main speakers:
Gerhard Hiß
Gabriele Nebe
Colva Roney-Dougal -
Ah, it is Schur’s Theorem!
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To paraphrase the question, why is a sum of all operators in a given matrix representation is equal to identity?
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What was the name of the property that makes sum of all operators of a given representation kind of like a delta function when multiplying another operator?
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My fifth Math Research Livestream is now available on YouTube:
In this one, I started reworking my preprint on an almost-elementary formula for the partition numbers. After spending a few more hours messing around with this during the stream, I'm still not sure how I feel about the result. I've had a lot of positive feedback, but I don't know if I will benefit much from putting more energy into this paper. If you give it a watch please let me know what you think!
#math #livestream #Twitch #algebra #AbstractAlgebra #RepresentationTheory #combinatorics
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My next Math Research Livestream starts in about 40 minutes on Twitch! Check it out at https://www.twitch.tv/charlotteaten. I'll be reimagining my preprint (https://arxiv.org/abs/2308.10177) on an almost-elementary formula for the partition numbers.
#math #livestream #Twitch #algebra #AbstractAlgebra #RepresentationTheory #combinatorics
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I've posted my talk on a relatively elementary formula involving the partition numbers to YouTube! You can find it at https://youtu.be/KGwsHqIH970 and you can see the slides (now with fewer typos) at https://aten.cool/documents/aten_du_algebra_logic_2023.pdf. The preprint itself can be found on the arXiv at https://arxiv.org/abs/2308.10177.
#partitions #combinatorics #algebra #AbstractAlgebra #RepresentationTheory
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Quiver algebras have the property that, in any fixed dim, rep(Q) is a vector space. Is there some characterization of which algebras have this property?
(Unital algebras seem to be ruled out. Except kQ *is* unital, but for those you can just "throw away" the 1, and all the primitive idempotents in fact, and it works. I guess it's because they have an k-linear decomposition \(kQ = E \oplus I\) where E is the subalgebra of idempotents and I is an ideal, and any representation of Q is uniquely determined by what it does on I (which is a non-unital subalgebra), since it has no choice of what to do on E. Not quite sure what the right general principle is here though...)
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Anyone know the tame/wild classification for finite *cyclic* quivers? The oft-quoted one is for acyclic.
I can see any quiver w/ two cycles is wild, and any graph that is just one cycle is tame. Having trouble finding anything written about classifying other unicyclic quivers.
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Quantum mechanics anyone? Dozens have been disappointed by UCLA’s administration ineptly standing in the way of Dr. Mike Miller being able to offer his perennial Winter UCLA math class (Ring Theory this quarter), so a few friends and I are putting our informal math and physics group back together.
We’re mounting a study group on quantum mechanics based on Peter Woit‘s Introduction to Quantum Mechanics course from 2022. We’ll be using his textbook Quantum Theory, Groups and Representations:An Introduction (free, downloadable .pdf) and his lectures from YouTube.
Shortly, we’ll arrange a schedule and some zoom video calls to discuss the material. If you’d like to join us, send me your email or leave a comment so we can arrange meetings (likely via Zoom or similar video conferencing).
Our goal is to be informal, have some fun, but learn something along the way. The suggested mathematical background is some multi-variable calculus and linear algebra. Many of us already have some background in Lie groups, algebras, and representation theory and can hopefully provide some help for those who are interested in expanding their math and physics backgrounds.
Everyone is welcome!
#group-theory #lie-groups #peter-woit #physics #quantum-mechanics #representation-theory
https://boffosocko.com/2023/01/26/quantum-mechanics-study-group-for-peter-woit/
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While it may look ugly (or scary) to you, I think this is one of the most beautiful formulas in #RepresentationTheory. It is an explicit formula for any #representation of a 2 x 2 matrix. It lets you increase a 2 x 2 matrix to any size you want, while still miraculously preserving matrix multiplication!
It was first derived by Wigner and is known in physics as Wigner D-matrix.
https://en.wikipedia.org/wiki/Wigner_D-matrix -
Appendix C of our paper has a great introduction to #RepresentationTheory of 2 x 2 matrices. If you ever wanted to learn this stuff, I recommend you have a look!
Here is how the first few representations look like. While it's not obvious, these maps are homomorphisms from 2 x 2 matrices to d x d matrices, and there exists one such map for every dimension d!