TY - JOUR

T1 - COMBINATORIAL DESCRIPTIONS OF THE CRYSTAL STRUCTURE ON CERTAIN PBW BASES

AU - Salisbury, Ben

AU - Schultze, Adam

AU - Tingley, Peter

N1 - Publisher Copyright:
© 2017, Springer Science+Business Media, LLC.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85024491181&partnerID=8YFLogxK

U2 - 10.1007/s00031-017-9434-9

DO - 10.1007/s00031-017-9434-9

M3 - Article

AN - SCOPUS:85024491181

SN - 1083-4362

VL - 23

SP - 501

EP - 525

JO - Transformation Groups

JF - Transformation Groups

IS - 2

ER -