COMBINATORIAL DESCRIPTIONS OF THE CRYSTAL STRUCTURE ON CERTAIN PBW BASES

Ben Salisbury, Adam Schultze, Peter Tingley

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

Using the theory of PBW bases, one can realize the crystal B(∞) for any semisimple Lie algebra over C using Kostant partitions as the underlying set. In fact there are many such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. Here we show that, for certain reduced expressions, the crystal operators can also be described by a much simpler bracketing rule. We give conditions describing these reduced expressions, and show that there is at least one example in every type except possibly E8, F4 and G2. We then discuss some examples.

Original languageEnglish
Pages (from-to)501-525
Number of pages25
JournalTransformation Groups
Volume23
Issue number2
DOIs
StatePublished - Jun 1 2018

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