The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation [...]

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Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on [...]

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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 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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Young’s lattice

As we saw last time, a sequence of nested groups gives rise to a graded “poset” in which objects can be related by more than one arrow. The poset we want to construct now is given by the sequence of symmetric groups (where and are both the trivial group). Since irreducible representations of the symmetric [...]

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Introduction to symmetric functions

The theory of symmetric functions, which generalizes some ideas that came up in the previous discussion of Polya theory, can be motivated by thinking about polynomial functions of the roots of a monic polynomial . Problems on high school competitions often ask for the sum of the squares or the cubes of the roots of [...]

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Standard Young tableaux and Robinson-Schensted-Knuth

Earlier I mentioned that the theory of Young tableaux is the source of one of my favorite proofs. Today I’d like to present, again from the theory of Young tableaux, one of my favorite pairs of proofs. A standard Young tableau is a chain in Young’s lattice; equivalently, it is a sequence of Young diagrams, [...]

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