Abstract
Lusztig's theory of PBW bases gives a way to realize the crystal B(∞) for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different 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. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except E8, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.
Highlights
We consider the crystal B(∞) for a simple Lie algebra over C of -laced type. This is a combinatorial object that contains a great deal of information about the algebra and its finite-dimensional representations. It is usually defined by a complicated algebraic construction, but it can often be realized in quite simple ways
Lusztig explicitly describes how the realizations are related for reduced expressions that differ by a braid move
We explicitly describe the unique crystal isomorphism between marginally large tableaux and Kostant partitions in these types
Summary
We consider the crystal B(∞) for a simple Lie algebra over C of -laced type This is a combinatorial object that contains a great deal of information about the algebra and its finite-dimensional representations. Lusztig’s early algebraic construction of the canonical basis of the quantum group Uq(g) (see [13, Chapters 41 and 42] and references therein) can be interpreted as giving a number of parameterizations of B(∞), one for each reduced expression for the longest word w0 in the Weyl group For each of these realizations, at least one of the crystal operators is very simple, but others may be complicated. We give a set of conditions on a reduced expression that ensure Lusztig’s crystal structure on Kostant partitions is given by a simple bracketing procedure, similar to the type An structure on multisegments. See [17] for the general results and [18] for the connection to marginally large tableaux in type Dn
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