#representation-theory — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #representation-theory, aggregated by home.social.
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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!