# Joel Gibson

I am a postgraduate student in the School of Mathematics at Sydney University, studying under Oded Yacobi. For my contact details and teaching timetable, see my people page on the School of Maths system.

## Seminars

Currently, I am running the Student Algebra Seminar in Semester 1 of 2019. I also coordinated this seminar in Semester 2 of 2018.

In Semester 2 of 2018, I coordinated a reading group on the Geometric Satake correspondence.

## My work

I have a long-term project investigating the structure of the "product monomial crystal" in Type A. The product monomial crystal is defined in the paper Highest weights for truncated shifted Yangians and product monomial crystals, and is a subcrystal of Nakajima's crystal of monomials for any semisimple simply-laced Lie algebra, formed by the monomial-wise multiplcation of various fundamental crystals. The image on the right shows one realisation of the crystal of the sl3 adjoint representation, as a product monomial crystal.

The product monomial crystal depends on its input parameters in a rather mysterious way, interpolating between a tensor product of fundamental crystals, and the irreducible with highest weight being the sum of those fundamentals. To assist in various computations, I created a program which can compute this crystal for any input data, and list its decomposition into irreducible components. There are instructions for how to download and run the program.

## Talks

• I gave a talk on the product monomial crystal at AustMS 2018, held at the University of Adelaide, which was one of the two winners of the Bernhard Neumann prize for most outstanding student talk. Here are both the slides and annotated slides for the talk. (To view the annotated slides, open your PDF reader in "two pages per screen" mode).

## Other projects

• I've written a calculator for computing in the algebra of symmetric functions, with its basis of Schur functions. Behind the scenes, it uses crystals of GLn representations to compute the product of each pair of basis elements. It is not fast for large computations, but the code is quite simple.