Bigrassmannian permutations and Verma modules

We show that bigrassmannian permutations determine the socle of the cokernel of an inclusion of Verma modules in type A. All such socular constituents turn out to be indexed by Weyl group elements from the penultimate two-sided cell. Combinatorially, the socular constituents in the cokernel of the inclusion of a Verma module indexed by $$w\in S_n$$ w ∈ S n into the dominant Verma module are shown to be determined by the essential set of w and their degrees in the graded picture are shown to be computable in terms of the associated rank function. As an application, we compute the first extension from a simple module to a Verma module.

A Random Walk on the Indecomposable Summands of Tensor Products of Modular Representations of SL2 $\left ({\mathbb {F}_p}\right )$

In this paper we introduce a novel family of Markov chains on the simple representations of SL2$\left ({\mathbb {F}_p}\right )$ F p in defining characteristic, defined by tensoring with a fixed simple module and choosing an indecomposable non-projective summand. We show these chains are reversible and find their connected components and their stationary distributions. We draw connections between the properties of the chain and the representation theory of SL2$\left ({\mathbb {F}_p}\right )$ F p , emphasising symmetries of the tensor product. We also provide an elementary proof of the decomposition of tensor products of simple SL2$\left ({\mathbb {F}_p}\right )$ F p -representations.

Simple Finite-Dimensional Modules and Monomial Bases from the Gelfand-Testlin Patterns

One of the most important classes of Lie algebras is sl_n, which are the n×n matrices with trace 0. The representation theory for sl_n has been an interesting research area for the past hundred years and in it, the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs and constructs monomial bases of the simple module.

δss-supplemented modules and rings

In this paper, we introduce the concept of δss-supplemented modules and provide the various properties of these modules. In particular, we prove that a ring R is δss-supplemented as a left module if and only if {R \over {Soc\left( {_RR} \right)}} is semisimple and idempotents lift to Soc(RR) if and only if every left R-module is δss-supplemented. We define projective δss-covers and prove the rings with the property that every (simple) module has a projective δss-cover are δss-supplemented. We also study on δss-supplement submodules.

Kac–Wakimoto conjecture for the periplectic Lie superalgebra

We prove an analogue of the Kac–Wakimoto conjecture for the periplectic Lie superalgebra [Formula: see text], stating that any simple module lying in a block of non-maximal atypicality has superdimension zero.

Silting Complexes and Partial Tilting Modules

It is well known that a partial tilting module may not be completed to a tilting module. However, it is still unknown whether a partial tilting module can be completed to a silting complex. The affirmative answer to this question will give an affirmative answer to the well-known rank question for tilting modules. In this paper, we prove that a partial tilting simple module can always be completed to a silting complex. More generally, we give the sufficient conditions for a partial tilting module to be completed to a silting complex.