Dual canonical bases for the quantum general linear supergroup

Abstract

Dual canonical bases of the quantum general linear supergroup are constructed which are invariant under the multiplication of the quantum Berezinian. By setting the quantum Berezinian to identity, we obtain dual canonical bases of the quantum special linear supergroup O q (SL m|n ). We apply the dual canonical bases to study invariant subalgebras of the quantum supergroups under left and right translations. In the case n = 1, it is shown that each invariant subalgebra is spanned by a part of the dual canonical bases. This in turn leads to dual canonical bases for any Kac module constructed by using an analogue of Borel-Weil theorem.

The First Fundamental Theorem of Invariant Theory for the Orthosymplectic Supergroup

“…A brief review of early works on the theory and applications of quantum supergroups can be found in [63]. Partially successful constructions of crystal and canonical bases for quantum supergroups were given in [1,12,37,53,55,54,72,73].…”

The quantum general linear supergroup, canonical bases and Kazhdan-Lusztig polynomials

“…The present paper continues the investigation started in [2]. We shall construct the canonical basis of the tensor powers of the natural representation of U q (gl m|n ) by relating them to the Kazhdan-Lusztig theory [7] for Hecke algebras of type A.…”

“…In a recent publication [2] , we began a study of the canonical basis and dual canonical basis of quantum supergroups and their function algebras. Recall that crystal bases and canonical bases for ordinary quantum groups and the associated quantized function algebras were introduced by Kashiwara [3,4] and Lusztig [5] in the early 1990s.…”

“…In [6] the crystal bases of a class of representations of the quantum general linear supergroup was constructed. Also in [2], a dual canonical basis for the function algebra of the quantum general linear supergroup U q (gl m|n ) was obtained. However, the theory of canonical bases or global crystal bases for quantum supergroups has yet to be fully developed.…”

Back Matter

“…We have not included them in the text in Chapter 5, since they are very technical and could disrupt the exposition. We invite the reader to consult [53,110,163] for more details and the proofs of our statements, that cannot be included here.…”