From 1f0d304a52fbf0d5fd8e2da6e2c088cd8e39f44c Mon Sep 17 00:00:00 2001 From: Zhiming Wang <zmwangx@gmail.com> Date: Wed, 19 Nov 2014 23:27:12 -0800 Subject: 20141119 Convolution of irreducible characters --- .../2014-11-19-convolution-of-irreducible-characters.md | 16 ++++++++++++++++ 1 file changed, 16 insertions(+) create mode 100644 source/_posts/2014-11-19-convolution-of-irreducible-characters.md diff --git a/source/_posts/2014-11-19-convolution-of-irreducible-characters.md b/source/_posts/2014-11-19-convolution-of-irreducible-characters.md new file mode 100644 index 00000000..1176091a --- /dev/null +++ b/source/_posts/2014-11-19-convolution-of-irreducible-characters.md @@ -0,0 +1,16 @@ +--- +layout: post +title: "Convolution of irreducible characters" +date: 2014-11-19 20:40:37 -0800 +comments: true +categories: [math, expository, 'representation theory', algebra] +--- +__*TL; DR:* The actual PDF write-up is [here](/pdf/20141119-convolution-of-irreducible-characters.pdf).__ + +--- + +Yesterday I was trying to establish the formula for orthogonal primitive central idempotents of a group ring. It is possible to establish the result through the convolution of irreducible characters. However, I stuck quite a while on trying to work out the convolutions themselves. For a formidable and unenlightening proof using "matrix entry functions" (i.e., fix a basis, induce a matrix representation, and explicitly expand everything in matrix elements), see [this post](http://drexel28.wordpress.com/2011/03/02/representation-theory-using-orthogonality-relations-to-compute-convolutions-of-characters-and-matrix-entry-functions/) (in fact, this is just one in a series of posts that lead up to the result). That's a really sad proof. + +It turns out that I really should have been working the other way round --- first establish the orthogonal idempotents (the proof of which is really simple and elegant, I was just trapped in a single thread of thought), then use that to compute the convolution of irreducible characters. + +I feel like this is worth presenting (as the only proof I saw online is the really sad one above), so I TeX'ed it up. I tried to convert to MathJax HTML but eventually gave up (that's the story for another post). So, the write-up is in good ol' PDF, available [here](/pdf/20141119-convolution-of-irreducible-characters.pdf). -- cgit v1.2.1