A finite directed category is a k-linear category with finitely many objects and an underlying poset structure, where k is an algebraically closed field. This concept unifies structures such as k-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this paper we study representations of finite directed categories, discuss their stratification properties, and show the existence of generalized APR tilting modules for triangular matrix algebras under some assumptions.

Let G be a connected almost simple algebraic group with a Dynkin automorphism . Let G be the connected almost simple algebraic group associated with G and . We prove that the dimension of the tensor invariant space of G is equal to the trace of on the corresponding tensor invariant space of G. We prove that if G has the saturation property then so does G . As a consequence, we show that the spin group Spin (2 n+ 1) has saturation factor 2, which strengthens the results of BelkaleKumar [1] and Sam [28] in the case of type B n.

We characterize the indecomposable injective objects in the category of finitely presented representations of an interval finite quiver.

The unipotent variety of a reductive algebraic group G plays an important role in the representation theory. In this paper, we will consider the closure U of the unipotent variety in the De ConciniProcesi compactification G of a connected simple algebraic group G. We will prove that U-U is a union of some G-stable pieces introduced by Lusztig in [Moscow Math. J 4 (2004) 869896]. This was first conjectured by Lusztig. We will also give an explicit description of U. It turns out that similar results hold for the closure of any Steinberg fiber in G.

In this article we study a conjecture of Geiss-Leclerc-Schröer, which is an analogue of a classical conjecture of Lusztig in the Weyl group case. It concerns the relation between canonical basis and semi-canonical basis through the characteristic cycles. We formulate an approach to this conjecture and prove it for type quiver. In the general type A case, we reduce the conjecture to show that certain nearby cycles have vanishing Euler characteristic.