The proposed research covers the representation theory of Lie superalgebras, Hecke algebras and affine Lie algebras over fields of characteristic zero as well as over fields with positive characteristics. In recent work, the PI and collaborators have obtained a conceptual solution to the irreducible character problem in various parabolic categories of classical Lie superalgebras. The PI proposes a new approach to attack the well-known open problem on the irreducible characters for the FULL category O for Lie superalgebras. The PI also proposes to develop a spin analogue of the invariant theory for Weyl groups and a spin analogue of Hecke algebras. These formulations call for the computation and comparison of spin fake degrees and spin generic degrees, as a striking spin analogue of a deep classical theory (which is intimately related to finite groups of Lie type). The PI further proposes to initiate a systematic study of the modular representations of infinite-dimensional Lie algebras in positive characteristic.
The mathematical language used to describe symmetries in nature often involves the concepts of groups, algebras, or their variants. Wang's research has helped to solve long standing old problems for Lie superalgebras (part of the language used in describing the supersymmetry), and also to raise many exciting research problems. The PI's research has attracted students to the active area of representation theory of groups and algebras. He is a popular speaker for summer and winter schools aiming at training graduate students and young mathematicians at the beginning of their careers.
The mathematical language used to describe the nature involves the concept of groups and algebras; in particular Lie superalgebras are indispensable in describing the supersymmetry. Finding the characters of simple modules (that is, the basic building blocks of modules) of a Lie algebra or group is a fundamental problem in a branch of mathematics called representation theory. While the Lie algebra character problem has afforded a beautiful solution thanks to Kazhdan-Lusztig (KL) theory (using the KL basis of a Hecke algebra) developed since 1979, a formulation of KL theory for the analogous problem for Lie superalgebras is much more challenging, due to the fact that the Weyl groups/Hecke algebras no longer control the superalgebra linkage. The simple Lie superalgebras are divided into different types, such as ABCD, etc. In this project, we solved the type-A superalgebra simple character problem completely, settling a decade old conjecture fromulated by Brundan in terms of Lusztig's canonical basis for quantum groups. Moreoever we formulated and solved the type-B superalgebra character problem, which has been open for more than 3 decades. To that end, we initiated a new theory of canonical basis arising from quantum symmetric pairs (different from the original one for quantum groups), which in a distinguished case recovers the type-B KL basis and gives us a new formulation of the type-B KL theory. The new canonical basis is shown to be intimately related to geometry of flag varieties. In another part of the project, we also constructed the canonical basis associated to a class of quantum superalgebras and their modules for the first time. This is a super generalization of the canonical basis theory developed by Lusztig and Kashiwara since 1990. Several PhD students and postdoc have been actively involved in various aspects of the project above, and they have rapidly moved to the research frontier upon carrying out successfully these work. The results of our project are expected to lead to new higher categorical structures and open up some new research directions to be explored in the coming years.
