Principal Indecomposable Modules Research Articles

Generalized “stacked bases” theorem for modules over semiperfect rings

The history of generalized “stacked bases” theorem origins from the result of Hill and Megibben on abelian groups. We extend this theorem for modules over semiperfect rings and as a consequence we show that for a submodule H of a projective module G over a semiperfect ring, the following conditions are equivalent: there exists a decomposition into a direct sum of indecomposable modules Pi , such that G/H is a direct sum of a family of modules, isomorphic to factor modules of principal indecomposable modules.

The Loewy series of principal indecomposable modules in the regular nilpotent representations of 𝔰𝔩(3,k)

Let [Formula: see text] be a simple Lie algebra of type [Formula: see text] over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text] be the reduced enveloping algebra of [Formula: see text]. In this paper, when the [Formula: see text]-character [Formula: see text] is regular nilpotent and has standard Levi form, we precisely determine the Lowey series of principal indecomposable [Formula: see text]-modules and the dimensions for the self-extension of irreducible [Formula: see text]-modules.

Quadratic principal indecomposable modules and strongly real elements of finite groups

Let $P$ be a principal indecomposable module of a finite group $G$ in characteristic $2$ and let $\varphi$ be the Brauer character of the corresponding simple $G$-module. We show that $P$ affords a non-degenerate $G$-invariant quadratic form if and only if there are involutions $s,t\in G$ such that $st$ has odd order and $\varphi(st)/2$ is not an algebraic integer. We then show that the number of isomorphism classes of quadratic principal indecomposable $G$-modules is equal to the number of strongly real conjugacy classes of odd order elements of $G$.

The quadratic type of the 2-principal indecomposable modules of the double covers of alternating groups

The principal indecomposable modules of the double cover 2.An of the alternating group over a field of characteristic 2 are enumerated using the partitions of n into distinct parts. We determine which of these modules afford a non-degenerate 2.An-invariant quadratic form. Our criterion depends on the alternating sum and the number of odd parts of the corresponding partition.

Some remarks on the uniqueness of liftings of principal indecomposable modules for infinitesimal subgroups of semisimple groups

This note is in the nature of an appendix to the recent paper by Paul Sobaje, [5]. In that paper, Section 2.2 the question of the uniqueness of G structures on (or liftings of) principal indecomposable G1-module in the case G=SL3 is raised. We here answer this question positively as well as making some more general observations.

The principal indecomposable modules of the dilute Temperley-Lieb algebra

The Temperley-Lieb algebra \documentclass[12pt]{minimal}\begin{document}$\mathsf {TL}_{n}(\beta )$\end{document}TLn(β) can be defined as the set of rectangular diagrams with n points on each of their vertical sides, with all points joined pairwise by non-intersecting strings. The multiplication is then the concatenation of diagrams. The dilute Temperley-Lieb algebra \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}(\beta )$\end{document}dTLn(β) has a similar diagrammatic definition where, now, points on the sides may remain free of strings. Like \documentclass[12pt]{minimal}\begin{document}$\mathsf {TL}_{n}$\end{document}TLn, the dilute \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn depends on a parameter \documentclass[12pt]{minimal}\begin{document}$\beta \in \mathbb {C}$\end{document}β∈C, often given as β = q + q−1 for some \documentclass[12pt]{minimal}\begin{document}$q\in \mathbb {C}^\times$\end{document}q∈C×. In statistical physics, the algebra plays a central role in the study of dilute loop models. The paper is devoted to the construction of its principal indecomposable modules. Basic definitions and properties are first given: the dimension of \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn, its break up into even and odd subalgebras and its filtration through n + 1 ideals. The standard modules \documentclass[12pt]{minimal}\begin{document}$\mathsf {S}_{n,k}$\end{document}Sn,k are then introduced and their behaviour under restriction and induction is described. A bilinear form, the Gram product, is used to identify their (unique) maximal submodule \documentclass[12pt]{minimal}\begin{document}$\mathsf {R}_{n,k}$\end{document}Rn,k which is then shown to be irreducible or trivial. It is then noted that \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn is a cellular algebra. This fact allows for the identification of complete sets of non-isomorphic irreducible modules and projective indecomposable ones. The structure of \documentclass[12pt]{minimal}\begin{document}$\mathsf {dTL}_{n}$\end{document}dTLn as a left module over itself is then given for all values of the parameter q, that is, for both q generic and a root of unity.

Standard modules, induction and the structure of the Temperley-Lieb algebra

The basic properties of the Temperley-Lieb algebra TLn with parameter β = q+ q −1 , q ∈ C\{0}, are reviewed in a pedagogical way. The link and standard (cell) modules that appear in numerous physical applications are defined and a natural bilinear form on the standard module s is used to characterise their maximal submodules. When this bilinear form has a non-trivial radical, some of the standard modules are reducible and TLn is non- semisimple. This happens only when q is a root of unity. Use of restriction and induction allows fo r a finer description of the structure of the standard modules. Finally, a particu lar central element Fn ∈ TLn is studied; its action is shown to be non-diagonalisable on certain indecomposable modules and this leads to a proof that the radicals of the standard modules are irreducible. Moreover, the space of homomorphisms between standard modules is completely determined. The principal indecomposable modules are then computed concretely in terms of standard modules and their inductions. Examples are provided throughout and the delicate case β = 0, that plays an important role in physical models, is studied systematically.

Derived Functors Related to Wall-Crossing

The setting is the representation theory of a simply connected, semisimple algebraic group over a field of positive characteristic. There is a natural transformation from the wall-crossing functor to the identity functor. The kernel of this transformation is a left exact functor. This functor and its first derived functor are evaluated on the global sections of a line bundle on the flag variety. It is conjectured that the derived functors of order greater than one annihilate the global sections. Also, the principal indecomposable modules for the Frobenius subgroups are shown to be acyclic.

The trivial intersection problem for characters of principal indecomposable modules

For a prime p and a finite group G let Φ p ( G ) denote the complex character associated to the projective indecomposable module in characteristic p with trivial head. Let Irr ( Φ p ( G ) ) denote the set of irreducible characters occurring as constituents in Φ p ( G ) . We characterize all finite simple groups which satisfy Irr ( Φ p ( G ) ) ∩ Irr ( Φ q ( G ) ) = { 1 G } for all primes p ≠ q .

A Swan length theorem and a Fong dimension theorem for Mackey algebras

We first present some results about Mackey algebras of p-groups over fields of characteristic p, including their primitive idempotents and decompositions of their simple and principal indecomposable modules under restriction. We then apply these results together with a Green's indecomposability theorem for Mackey algebras to obtain Mackey algebra versions of some classical results of group algebras which are mostly related to restriction, induction and dimensions of modules. Our results about dimensions include Mackey algebra analogues of Dickson's theorem, Swan's theorem and Fong's dimension formula.

The module structure of the Solomon–Tits algebra of the symmetric group

Let ( W , S ) be a finite Coxeter system. Tits defined an associative product on the set Σ of simplices of the associated Coxeter complex. The corresponding semigroup algebra is the Solomon–Tits algebra of W. It contains the Solomon algebra of W as the algebra of invariants with respect to the natural action of W on Σ. For the symmetric group S n , there is a 1–1 correspondence between Σ and the set of all set compositions (or ordered set partitions) of { 1 , … , n } . The product on Σ has a simple combinatorial description in terms of set compositions. We study in detail the representation theory of the Solomon–Tits algebra of S n over an arbitrary field, and show how our results relate to the corresponding results on the Solomon algebra of S n . This includes the construction of irreducible and principal indecomposable modules, a description of the Cartan invariants, of the Ext-quiver, and of the descending Loewy series. Our approach builds on a (twisted) Hopf algebra structure on the direct sum of all Solomon–Tits algebras.

Formula omitted]-pairs of symmetric group algebras and their intermediate defect 4 blocks

In this paper we study [ 3 : 2 ] -pairs of symmetric group algebras and their ‘intermediate’ block in detail. The aim is to understand how one block of a [ 3 : 2 ] -pair can inherit two properties—the Ext-quiver being bipartite and the principal indecomposable modules having a common Loewy length 7—from the other. We establish some sufficient conditions for this inheritance, and verify these conditions for some special blocks.

Filtrations in Rouquier Blocks of Symmetric Groups and Schur Algebras

We study Rouquier blocks of symmetric groups and Schur algebras in detail, and obtain explicit descriptions for the radical layers of the principal indecomposable, Weyl, Young and Specht modules of these blocks. At the same time, the Jantzen filtrations of the Weyl modules are shown to coincide with their radical filtrations. We also address the conjectures of Martin, Lascoux–Leclerc–Thibon–Rouquier and James for these blocks.

Cartan matrix over the 0-Hecke algebra of type F4

Let (W,S) be the finite Weyl group with S as its Coxeter generating set. For w∈W, let R(w)={si∈S∣l(wsi)<l(w)} and L(w)={si∈S∣l(siw)<l(w)}, where we denote by l(w) the minimal length of an expression of w as a product of simple reflections. To any Weyl group one can associate a corresponding finite-dimensional algebra called 0-Hecke algebra H where K is any field. Norton [J. Austral. Math. Soc. Ser. A 27 (1979) 337–357] pointed out that the principal indecomposable modules and the irreducible modules over the 0-Hecke algebra H parametrized by a subset J of S. We denote by U(Ĵ) and M(Ĵ) respectively the principal indecomposable module and the irreducible module parametrized by J. For two subset J, L of S, let CJL= the number of times M(L) is a composition factor of U(Ĵ). Norton [J. Austral. Math. Soc. Ser. A 27 (1979) 337–357] shown that CJL=|YL∩(YJ)−1| where YL={w∈W∣R(w)=L} and (YJ)−1={w∈W∣L(w)=J}. In this article, we describe explicitly CJL for the 0-Hecke algebra of type F4 by applying the canonical expression of every element in the Weyl group of type F4. Thus we determine the Cartan matrix over the 0-Hecke algebra of type F4.

On Certain Blocks of Schur Algebras

In this paper, what is already known about defect 2 blocks of symmetric groups is used to deduce information about the corresponding blocks of Schur algebras. This information includes Ext-quivers and decomposition numbers, as well as Loewy structures of the Weyl modules, principal indecomposable modules and tilting modules.

On the principal blocks of [formula omitted] and [formula omitted] over a field of characteristic 3

In this paper, we construct the Ext-quivers of the principal blocks of F S 9 and F S 10 , where F is an algebraically closed field of characteristic 3, and obtain the Loewy structures of their principal indecomposable modules.

On the principal blocks of F[Sfr ]11 and F[Sfr ]12 over a field of characteristic 3

In this paper, we construct the Ext-quivers of the principal blocks of F[Sfr ]11 and F[Sfr ]12, where F is an algebraically closed field of characteristic 3. We also obtain the Loewy structures of three principal indecomposable modules of the principal block of F[Sfr ]11.

Translation cokernels of rational modules

A fundamental approach to the homological algebra of rational modules in positive characteristic p should begin with the translation properties of injective modules. This point of view leads to the study of functors related to coherent translation and their effect on the injective indecomposable modules indexed by p-regular dominant weights. The main result shows that the calculation can often be reduced to a corresponding calculation on the principal indecomposable modules of the Frobenius kernel.1

On the structure of nonsemisimple Hopf algebras

The left module structure of finite-dimensional quantum algebras is analysed using the theory of primitive idempotents. In particular, a complete structural result in terms of principal (projective) indecomposable modules (p.i.m.) is given in the case of (q a root of 1) by finding a complete set of primitive idempotents. The structure of p.i.m. is analysed in detail. The Jacobson radical of the algebra is investigated and its significance in the study of nonsemisimple symmetries in physical systems is discussed.

Representations of Finite-Dimensional Hopf Algebras

LetHdenote a finite-dimensional Hopf algebra with antipodeSover a field k. We give a new proof of the fact, due toOS, thatHis a symmetric algebra if and only ifHis unimodular andS2is inner. IfHis involutory and not semisimple, then the dimensions of all projectiveH-modules are shown to be divisible by chark. In the case where k is a splitting field forH, we give a formula for the rank of the Cartan matrix ofH, reduced modchark, in terms of an integral forH. Explicit computations of the Cartan matrix, the ring structure ofG0(H), and the structure of the principal indecomposable modules are carried out for certain specific Hopf algebras, in particular for the restricted enveloping algebras of completely solvablep-Lie algebras and ofsl(2,k).