Arkady Berenstein (University of Oregon)
Title: Canonical bases for sl_2-modules Abstract: The notion canonical bases comes back to Kazhdan-Lusztig's and Lusztig's pioneering works and now became an indispensible tool in representation theory. Even though the rigorous definition is not always available, for a finite-dimensional sl_2(C)-module V such basis B should satisfy a number of natural properties:
It has to be compatible with the weight decomposition of V.
A part of B has to span the subspace V^+ of highest weight vectors and the subspace V^- of the lowest weight ones.
Under the action of the nilpotents e and f there must be a leading term, which condition essentially implies a crystal structure on B.Â
If we call a basis B satisfying conditions 1.-3. ``perfect", then one can easily show that any finite-dimensional V has at least one perfect basis because the most natural basis in a simple module V_\lambda is perfect and perfectness holds under direct sums. However, this observation implicitly assumes that we already know multiplicity of each V_\lambda in V, which is not always possible. For instance if V is a symmetric power of V_\mu, \mu>2, the multiplicities are in general unknown, which means an absence of a natural perfect bases which would count them. Situation is a much better when V is a tensor product of simple modules -- a natural perfect basis exists and satisfies some remarkable properties which make it canonical:
All structure constants of the action of e and f on B are nonnegative integers.B has a cluster structure under a natural embedding of all V=V_{\lambda_1}\otimes … \otimes V_{\lambda_n} into the polynomial algebra of 2n variables.
If V has a structure of a commutative algebra so that sl_2(C) acts on it by derivations then canonicicty of B also requires positivity of structure constants under multiplication and the cluster structure as well.
I will explain these for the canonical basis in tensor product and, if time permits, outline possible generalizations.
Villa Battelle, Wednesday 4th December, 14:30