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- Donkin, Stephen 3 (%)
- Robinson, Geoffrey R. 2 (%)
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- Caenepeel, S. 1 (%)
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## Submodule Structure of Generalized Verma Modules Induced from Generic Gelfand-Zetlin Modules

### Algebras and Representation (1998-03-01) 1: 3-26 , March 01, 1998

For complex Lie algebra sl(n, C) we study the submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules over some subalgebra of type sl(k, C). We obtain necessary and sufficient conditions for the existence of a submodule generalizing the Bernstein-Gelfand-Gelfand theorem for Verma modules.

## On the Existence of Auslander–Reiten Sequences of Group Representations. I

### Algebras and Representation (1998-06-01) 1: 97-127 , June 01, 1998

This is the first part of our study of the existence ofAuslander–Reiten sequences of group representations. In this part weconsider representations of group schemes in characteristic 0; in Part II weconsider representations of group schemes in characteristic p; andin Part III we give applications to representations of groups and Liealgebras.

## On Blocks with Nilpotent Coefficient Extensions

### Algebras and Representation (1998-03-01) 1: 27-73 , March 01, 1998

For modular group algebras over an arbitrary field we define new type of blocks: blocks with nilpotent extensions, and describe their source algebras. To do it, a general pattern is proposed for relations between the source algebra of a block and the source algebra of a block appearing in its decomposition in a suitable extension of the field of coefficients.

## Seminormal or t-Closed Schemes and Rees Rings

### Algebras and Representation (1998-09-01) 1: 255-309 , September 01, 1998

We define decent schemes and canonically decent projective schemes. Forsuch schemes, total quotient schemes exist, allowing to get thenormalization, seminormalization and t-closure of a decent scheme as ascheme. We exhibit the seminormalization and t-closure of a filtration on aring. If A is a decent ring and F a regular filtration onA, the associated Rees ring R is decent andProj(R) is canonically decent. The seminormalization and t-closureof R are Rees rings and the seminormalization and t-closure ofProj(R) are gotten by using projective morphisms.

## Curves on Quasi-Schemes

### Algebras and Representation (1998-12-01) 1: 311-351 , December 01, 1998

This paper concerns curves on noncommutative schemes, hereafter called quasi-schemes. Aquasi-scheme X is identified with the category
$$Mod{\text{ }}X$$
ofquasi-coherent sheaves on it. Let X be a quasi-scheme having a regularly embeddedhypersurface Y. Let C be a curve on X which is in ‘good position’ withrespect to Y (see Definition 5.1) – this definition includes a requirement that Xbe far from commutative in a certain sense. Then C is isomorphic to
$$\mathbb{V}_n^1 $$
, where n is the number of points of intersection of Cwith Y. Here
$$\mathbb{V}_n^1 $$
, or rather
$$Mod{\text{ }}\mathbb{V}_n^1 $$
, is the quotient category
$$GrModk[x_1 , \ldots ,x_n ]/\{ {\text{K}}\dim \leqslant n - 2\} {\text{ of }}\mathbb{Z}^n $$
-graded modules over the commutative polynomial ring, modulo the subcategory ofmodules having Krull dimension ≤ *n* − 2. This is a hereditary category whichbehaves rather like
$$Mod\mathbb{P}^1 $$
, the category of quasi-coherentsheaves on
$$\mathbb{P}^1 $$
.

## On the Existence of Auslander–Reiten Sequences of Group Representations. III

### Algebras and Representation (1998-12-01) 1: 399-412 , December 01, 1998

This is the third and final part of our study of the existence of Auslander–Reiten sequences ofgroup representations. In Part I we considered representations of group schemes in characteristic 0.In Part II we considered representations of group schemes in characteristic p. In this partwe give applications to representations of abstract groups and Lie algebras.

## A Result on Ext Over Kac–Moody Algebras

### Algebras and Representation (1998-06-01) 1: 161-168 , June 01, 1998

We prove the following result for a not necessarily symmetrizable Kac–Moody algebra: Let x,y ∈ W with x ≥ y, and let λ ∈ P^{+}. If *n*=*l*(*x*)-*l*(*y*), then Ext _{C(λ)}^{n}(M(*x*·*λ*),L(*y*·*λ*))=1.

## On Alperin’s Conjecture and Certain Subgroup Complexes

### Algebras and Representation (1998-12-01) 1: 383-398 , December 01, 1998

We prove a new formula about local control of the number of p-regular conjugacyclasses of a finite group. We then relate the results to Alperin’s weight conjecture to obtain newresults describing the number of simple modules for a finite group in terms of weights of solvablesubgroups. Finally, we use the results to obtain new formulations of Alperin’s weight conjecture,and to obtain restrictions on the structure of a minimal counterexample.

## Quantum Deformations of α-Stratified Modules

### Algebras and Representation (1998-06-01) 1: 135-153 , June 01, 1998

We construct quantum analogues of a class of generalized Verma modulesinduced from nonsolvable parabolic subalgebras of simple Lie algebras. Weshow that these quantum modules are true deformations of the underlyingclassical modules in the sense that the weight-space decomposition ispreserved.

## On a Projective Generalization of Alperin’s Conjecture

### Algebras and Representation (1998-06-01) 1: 129-134 , June 01, 1998

In this paper, we prove that a projective generalization of theKnörr–Robinson formulation of Alperin’s conjecture holds ifthe ‘ordinary’ form holds for a certain quotient group.