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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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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Affine varieties and regular maps

I have to admit I’ve been using somewhat unconventional definitions. The usual definition of an affine variety is as an irreducible Zariski-closed subset of , affine -space over an algebraically closed field . A generic Zariski-closed subset is usually referred to instead as an algebraic set (although some authors also call these varieties), and the [...]

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Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when [...]

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Spectra of rings of continuous functions

An analyst thinks of the ring of polynomials as a useful tool because, on intervals, it is dense in the continuous functions in the uniform topology. If we want to understand the relationship between and polynomial rings in a more general context, it might pay off to expand our scope from polynomial rings to more [...]

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The many faces of Schur functions

The last time we talked about symmetric functions, I asked whether the vector space could be turned into an algebra, i.e. equipped with a nice product. It turns out that the induced representation allows us to construct such a product as follows: Given representations of , their tensor product is a representation of the direct [...]

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Set-multiset duality and supervector spaces

Recall that the elementary symmetric functions generate the ring of symmetric functions as a module over any commutative ring . A corollary of this result, although I didn’t state it explicitly, is that the elementary symmetric functions are algebraically independent, hence any ring homomorphism from the symmetric functions is determined freely by the images of [...]

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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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The “correct” definition of a homomorphism

(Warning: I’m trying to talk about things I don’t really understand in this post, so feel free to correct me if you see a statement that’s obviously wrong.) Why are continuous functions the “correct” notion of homomorphism between topological spaces? The “obvious” way to define homomorphisms for a large class of objects involves thinking of [...]

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Non-canonical isomorphisms

I find non-canonical isomorphisms very interesting, but I wish I knew more examples. To be vague, an isomorphism (perhaps in a category) is said to be non-canonical if it requires making an “arbitrary choice.” One of the reasons I find them interesting is that we often think of objects only up to isomorphism, but in [...]

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