Walter BridgesAffiliation: University of North Texas Title Of Talk: Integer partitions and statistics for random representations of Lie algebras
URLS: Abstract: Choose a partition of a large integer uniformly at random. How big do we expect the largest part to be? How many 1’s are there? What does the Young diagram look like? For that matter, how can we generate large partitions efficiently, to collect data and make conjectures? In 1993, Fristedt introduced a statistical mechanics inspired approach to these sorts of questions that has proved widely useful in analytic combinatorics. Adapting Fristedt’s conditioning device to our setting, I will describe how a typical, large-dimensional representation looks for the family of complex Lie algebras, $\mathfrak{sl}_{r+1}(\mathbb{C})$. (The case $r = 1$ corresponds to integer partitions.) In particular, we give asymptotic probability distributions for the multiplicity of small irreducible representations, as well as the largest dimension, the largest height, and the total number of irreducible resentations appearing in the decomposition of a representation sampled uniformly from all representations with the same dimension. This is joint work with Kathrin Bringmann and Caner Nazaroglu.
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