Structures on hom-sets

Suppose I hand you a commutative ring . I stipulate that you are only allowed to work in the language of the category of commutative rings; you can only refer to objects and morphisms. (That means you can’t refer directly to elements of , and you also can’t refer directly to the multiplication or addition [...]

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Boolean rings, ultrafilters, and Stone’s representation theorem

Recently, I have begun to appreciate the use of ultrafilters to clean up proofs in certain areas of mathematics. I’d like to talk a little about how this works, but first I’d like to give a hefty amount of motivation for the definition of an ultrafilter. Terence Tao has already written a great introduction to [...]

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Zeta functions, statistical mechanics and Haar measure

An interesting result that demonstrates, among other things, the ubiquity of in mathematics is that the probability that two random positive integers are relatively prime is . A more revealing way to write this number is , where is the Riemann zeta function. A few weeks ago this result came up on math.SE in the [...]

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The adjoint functor theorem for posets

Recently in Measure Theory we needed the following lemma. Lemma: Let be non-constant, right-continuous and non-decreasing, and let . Define by . Then is left-continuous and non-decreasing. Moreover, for and , . If you’re categorically minded, this last condition should remind you of the definition of a pair of adjoint functors. In fact it is [...]

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The induced representation

Charles Siegel over at Rigorous Trivialities suggested a NaNoWriMo for math bloggers: instead of writing a 50,000-word novel, just write a blog post every day. I have to admit I rather like the idea, so we’ll see if I can keep it up. Continuing the previous post, what we want to do now is to [...]

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Some adjoint functors

Note: as usual, I will be playing free and loose with category theory in this post. Apologies to those who know better. One way to define a subgroup of a group is as the image of a homomorphism into . Given the inclusion map , the functor in the category of groups acts contravariantly to [...]

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The orthogonality relations for representations of finite groups

In order to continue our discussion of symmetric functions it will be useful to have some group representation theory prerequisites, although I will use many of the results in the representation theory of the symmetric groups as black boxes. I had planned on using this post to discuss Frobenius reciprocity, but got so carried away [...]

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