The Jacobson radical

The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one [...]

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The representation theory of SU(2)

Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of (which you’ll recall we assumed way back in this previous post). As a first step, it turns out that the finite-dimensional representation theory of compact groups looks a lot like the finite-dimensional representation theory of finite groups, and [...]

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SO(3) and SU(2)

In order to study the hydrogen atom, we’ll need to know something about the representation theory of the special orthogonal group . This post consists of a few preliminaries along the way to doing this. I’ll be somewhat vague about a few things that 1) I don’t have much experience with, and 2) that would [...]

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The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation [...]

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Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials

SPUR is finally over! Instead of continuing my series of blog posts, I thought I’d just link to my paper, Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials, and the first few blog posts should more or less provide enough background to read it. My project ended up changing direction. The formula I was [...]

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Hecke algebras and the Kazhdan-Lusztig polynomials

The Hecke algebra attached to a Coxeter system is a deformation of the group algebra of defined as follows. Take the free -module with basis , and impose the multiplicative relations if , and otherwise. (For now, ignore the square root of .) Humphreys proves that these relations describe a unique associative algebra structure on [...]

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Coxeter groups

At SPUR this summer I’ll be working on the Kazhdan-Lusztig polynomials, although my mentor and I haven’t quite pinned down what problem I’m working on. I thought I’d take the chance to share some interesting mathematics and also to write up some background for my own benefit. I’ll mostly be following the second half of [...]

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The McKay correspondence I

Today we’re going to relate the representation graphs introduced in this blog post to something I blogged about in the very first and second posts in this blog! The result will be a beautiful connection between the finite subgroups of , the Platonic solids, and the ADE Dynkin diagrams. This connection has been written about [...]

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Graded representation theory

Let be a group and let be a graded representation of , i.e. a functor from to the category of graded vector spaces with each piece finite-dimensional. Thus acts on each graded piece individually, each of which is an ordinary finite-dimensional representation. We want to define a character associated to a graded representation, but if [...]

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Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on [...]

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