20141119 Convolution of irreducible characters

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+---
+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.
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+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.
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+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).