Joel Gibson

Symmetric algebra

This is an implementation of the Littlewood-Richardson rule for multiplying partitions, and hence computing in the symmetric function algebra with its basis of Schur functions. It can be used as a Littlewood-Richardson coefficient calculator, and more generally can multiply linear combinations of Schur functions in the ring of symmetric functions. Under the hood, it uses the tensor product rule for crystals of GL(n) representations, where n is taken sufficiently large for the partitions in question. A consequence of this approach is that it gives fairly cheap computations in the representation ring of GL(n) (the n-rowed quotient of the symmetric function algebra).

When using the symmetric function algebra, the dimensions listed are for the corresponding simple representations of the symmetric group, calculated using the hook length formula. When using the representation ring of GL(n), the dimensions listed are as representations of GL(n), calculated using a variant of the hook length formula. The evaluator can accept complex expressions such as ([1] + 3[2, 2]) * [5]. The code can be found on GitHub.

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