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We study convex order and a cuspidal system for the Khovanov-Lauda-Rouquier (KLR) algebras of twisted affine type. This allows us to classify the irreducible modules over KLR algebras of twisted affine type. Particularly, we are interested in the imaginary modules. They can be constructed from the colored minuscule imaginary modules. We describe explicitly minuscule imaginary modules of certain colors for the Cartan matrices . Moreover, we discuss the the relation between the dimension of minuscule imaginary modules of color and Catalan numbers. For untwisted affine types, Kleshchev and Muth showed that the square of a certain permutation on the imaginary tensor space of a fixed color is a nonzero map. However, it is not the case for twisted affine types. We present a new result that these maps of the imaginary tensor spaces of certain color for some twisted affine types are equal to zero.
In 1898 Frobenius discovered a construction which, in present terminology, associates with every module of a subgroup the induced module of a group. This construction proved to be of fundamental importance and is one of the basic tools in the entire theory of group representations.This monograph is designed for research mathematicians and advanced graduate students and gives a picture of the general theory of induced modules as it exists at present. Much of the material has until now been available only in research articles. The approach is not intended to be encyclopedic, rather each topic is considered in sufficient depth that the reader may obtain a clear idea of the major results in the area.After establishing algebraic preliminaries, the general facts about induced modules are provided, as well as some of their formal properties, annihilators and applications. The remaining chapters include detailed information on the process of induction from normal subgroups, projective summands of induced modules, some basic results of the Green theory with refinements and extensions, simple induction and restriction pairs and permutation modules. The final chapter is based exclusively on the work of Weiss, presenting a number of applications to the isomorphism problem for group rings.
The module categories of Khovanov-Lauda-Rouquier algebras categorify the integral form of the negative half of the quantum group U_q(g) coming from any symmetrizable Kac-Moody algebra g. We construct a family of simple modules over KLR algebras and show how they can be used to obtain the building blocks of existing classifications of simple finite-dimensional modules in finite types. The construction extends to infinite types, where we obtain simple modules whose structures are easy to describe. We give many explicit examples of this construction in rank 2 cases.
There are two algebras associated to a reductive Lie algebra g: the De Concini- Kac quantum algebra and the Kac-Moody Lie algebra. Recent results show that the principle block of De Concini -Kac quantum algebra at an odd root of unity with (some) fixed central character is equivalent to the core of a certain t-structure on the derived category of coherent sheaves on certain Springer Fiber. Meanwhile, a certain category of representation of Kac-Moody Lie algebra at critical level with (some) fixed central character is also equivalent to a core of certain t-structure on the same triangulated category. Based on several geometric results developed by Bezurkvanikov et al. these two abelian categories turn out to be equivalent. i.e. the two t-structures coincide.
The authors study imaginary representations of the Khovanov-Lauda-Rouquier algebras of affine Lie type. Irreducible modules for such algebras arise as simple heads of standard modules. In order to define standard modules one needs to have a cuspidal system for a fixed convex preorder. A cuspidal system consists of irreducible cuspidal modules—one for each real positive root for the corresponding affine root system X , as well as irreducible imaginary modules—one for each -multiplication. The authors study imaginary modules by means of “imaginary Schur-Weyl duality” and introduce an imaginary analogue of tensor space and the imaginary Schur algebra. They construct a projective generator for the imaginary Schur algebra, which yields a Morita equivalence between the imaginary and the classical Schur algebra, and construct imaginary analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy formula.
This reprint of a 1983 Yale graduate course makes results in modular representation theory accessible to an audience ranging from second-year graduate students to established mathematicians. Following a review of background material, the lectures examine three closely connected topics in modular representation theory of finite groups: representations rings; almost split sequences and the Auslander-Reiten quiver; and complexity and cohomology varieties, which has become a major theme in representation theory.