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Patrick Stevens

Former mathematics student at the University of Cambridge; now a software engineer.

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In which I am a wizard.

Sometimes as a student, the work piles up and I start to think “I’ll never finish this”. It becomes easy to think that there’s no point in working because the work will never be over. When that happens to me, I imagine that my course is magic/alchemy/something with flashy special effects. I’m going through the Wizardry Academy, and I’ll graduate able to manipulate the four elements. Even if I’m not the best in the year at it, I’m still able to manipulate the elements, and if I work at it, I’ll be able to manipulate them better and in flashier ways - that’s not something most people can do!

I tend not to take this analogy very far. It’s usually enough just for me to pretend I’m Kvothe for a moment, and I’m all motivated again. However, the trick kind of works for specific topics, too. At the moment, for instance, I need to know how to classify the representations of a group, per Slate Star Codex’s article.

An arcanist who is working with minerals needs to know lots of properties of those minerals, and is greatly advantaged by performing certain rituals to divine the Affinities of a metal. As you know, metals are nothing more nor less than a physical embodiment of a collection of Aspects, and you get a different kind of metal for each Aspect that has gone into its construction. All metals have an Affinity with Nothing - that’s just standard Elemental Theory. Metals only have a certain number of Affinities, too, and it turns out to be a fact that each Affinity corresponds exactly with a purity band of the metal, and you can see which purity band goes with an Affinity if you look at the Affinity through a Tracer. (On that note, recall from the first Alchemy course you ever took that there is a ritual we can perform to extract a particular Aspect already present in a metal. Purity bands are what we call the product of that ritual, and represent a distilled Aspect which is still related to the original metal.)

A mineral is an algebraic structure; a metal, a finite group. An Aspect is a group element, and so if we have different generators for the group, we get a different group. An Affinity of a group is a complex irreducible representation. All finite groups have the trivial representation, as is standard Representation Theory. Finite groups only have a certain number of irreducible complex representations, and they are in bijection with the conjugacy classes of the group. (If you apply the trace operator to a representation, you obtain a character.) From any first course in group theory, we can extract the conjugacy class of an element of a group, and it is those conjugacy classes which are in bijection with the the characters.

It’s paraphrased a bit, and my notation is a bit sloppy, but it certainly sounds more interesting than representation theory.