On canonical basis and path basis

RWTH Publications (RWTH Aachen)

Abstract

We show for quantum groups over classical Lie algebras that there is a unitriangular base change between the canonical basis constructed by Lusztig and Kashiwara and the path monomial basis constructed by Littelmann for every simple highest weight module. After specialization over an algebraically closed field, we prove that, in these cases, there is also a unitriangular base change between the dual canonical basis and the path vector basis constructed by Littelmann. It was conjectured by Littelmann that for any path model the associated path vector basis has a unitriangular base change to the dual canonical basis for any symmetrizable Kac-Moody Lie algebra. In addition to the above result, we provide further evidence for the conjecture by proving that an element of the dual canonical basis agrees with the corresponding path vector up to terms negligible by a partial order if it is compatible with the dual of the Frobenius splitting. This suggests that the conjecture holds for many elements of the dual canonical basis, since Qin proved that the cluster monomials belong to the dual canonical basis and Song proved that the cluster monomials are compatible with the dual of the Frobenius splitting. Based on the observed compatibility between the dual canonical basis and the path vectors, the author was posed the question, whether the path vectors form a perfect basis in the sense of Berenstein and Kazhdan, or Baumann, Kamnitzer and Knutson. We show that this is not the case already in rank 2. As part of our investigation, we define a modified version of Littelmann's root operators in the case of Lakshmibai-Seshadri paths for symmetrizable Kac-Moody Lie algebras. Although the operators are incompatible with the crystal structure, they still share interesting properties with the root operators. In particular, we conjecturally obtain a formula for the action (left multiplication) of a Chevalley generator on an element of the canonical basis similar to the one by Kashiwara. Here the leading term is with respect to a partial order defined by Littelmann; in general, the order is incompatible with the crystal structure and the leading term differs from the one in Kashiwara's formula. We prove this formula in the classical types for certain paths. Finally, we use Lusztig's tight monomial cone to study the duals of Leclerc's imaginary vectors. This allows us to show for all finite types that the canonical basis is not fully compatible with the Frobenius morphism and its splitting. Here compatible means that an element of the canonical basis gets mapped to either zero or an element of the canonical basis, in a way that is compatible with the crystal structure. Furthermore, we give a examples for semi-tight monomials arising from Lusztig's tight monomial cone for all finite types. In both cases, examples were only known in the simply-laced case before.

Authors 1

1. Felix Röhrich corresponding Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen