The ideal-variety correspondence

I guess I didn’t plan this very well! Instead of completing one series I ended one and am right in the middle of another. Well, I’d really like to continue this series, but seeing as how finals are coming up I probably won’t be able to maintain the one-a-day pace. So I’ll just stop tagging [...]

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Irreducible components

If it wasn’t clear, in this discussion all rings are assumed commutative. Given a variety like we’d like to know if there’s a natural way to decompose it into its “components” . These aren’t its connected components in the topological sense, but in any reasonable sense the two parts are unrelated except possibly where they [...]

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The Noetherian condition as compactness

Let’s think more about what an abstract theory of unique factorization of primes has to look like. One fundamental property it has to satisfy is that factorizations should be finite. Another way of saying this is that the process of writing elements as products of other elements (up to units) should end in a finite [...]

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The weak Nullstellensatz and affine varieties

Hilbert’s Nullstellensatz is a basic but foundational theorem in commutative algebra that has been discussed on the blogosphere repeatedly, but thematically now is the appropriate time to say something about it. The idea of the weak Nullstellensatz is quite simple: the polynomial ring has evaluation homomorphisms sending for some point , so we can think [...]

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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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Primes and ideals

Probably the first important result in algebraic number theory is the following. Let be a finite field extension of . Let be the ring of algebraic integers in . Theorem: The ideals of factor uniquely into prime ideals. This is the “correct” generalization of the fact that , as well as some small extensions of [...]

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The Jacobi-Trudi identities

Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let be a partition. A semistandard Young tableau of shape is a filling of the Young diagram of with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau is defined as where [...]

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The Lindstrom-Gessel-Viennot lemma

One of my favorite results in algebraic combinatorics is a surprisingly useful lemma which allows a combinatorial interpretation of the determinant of certain integer matrices. One of its more popular uses is to prove an equivalence between three other definitions of the Schur functions (none of which I have given yet), but I find its [...]

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Groups vs. abelian groups

A few weeks ago on MathOverflow Greg Muller asked, “why do groups and abelian groups feel so different?” The answers were very interesting and came from several different perspectives, but I still don’t feel as if the question was resolved. So I’ll try to synthesize and summarize some of the answers and hopefully something will [...]

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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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