The group of endotrivial modules in the normal case

The group of endotrivial modules has recently been determined for a finite group having a normal Sylow p-subgroup. In this paper, we give and compare three different presentations of a torsion-free subgroup of maximal rank of the group of endotrivial modules. Finally, we illustrate the constructions in an example.     

Gluing representations via idempotent modules and constructing endotrivial modules

Let G be a finite group and k be a field of characteristic p. We show how to glue Rickard idempotent modules for a pair of open subsets of the cohomology variety along an automorphism for their intersection. The result is an endotrivial module. An interesting aspect of the construction is that we end up constructing finite dimensional endotrivial modules using infinite dimensional Rickard idempotent modules. We prove that this construction produces a subgroup of finite index in the group of endotrivial modules. More generally, we also show how to glue any pair of kG-modules.     

Algebraic modules and the Auslander-Reiten quiver

Recall that an algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by the direct sum and tensor product. In this article we prove that non-periodic algebraic modules are very rare, and that if the complexity of an algebraic module is at least 3, then it is the only algebraic module on its component of the (stable) Auslander-Reiten quiver. For dihedral 2-groups, we also show that there is at most one algebraic module on each component of the (stable) Auslander-Reiten quiver. We include a strong conjecture on the relationship between periodicity and algebraicity.     

Two theorems on the vertices of Specht modules

Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module $S^\lambda$ has non-trivial cyclic vertex if and only if $\lambda$ has p-weight 1. 2. If p does not divide n and $S^{(n-r, 1^r)}$ is indecomposable then its vertex is a p-Sylow subgroup of $S_{n-r-1} \times S_r$.Received: 15 August 2002     

On the number of anisotropic simple submodules in modules with a form

Let G be a finite group, and V a finite-dimensional semisimple G-module over a finite field. Assume that V is endowed with a nonsingular bilinear form which is symmetric or symplectic, and which is invariant under the action of G. In this setting, we compute the number of anisotropic simple submodules of V.Received: 25 May 2004     

Vertices of simple modules and normal subgroups ofp-solvable groups

Let π be a set of prime numbers andG a finite π-separable group. Let θ be an irreducible π′-partial character of a normal subgroupN ofG and denote by I_π′ (G‖θ), the set of all irreducible π′-partial characters Φ ofG such that θ is a constituent of Φ_N. In this paper, we obtain some information about the vertices of the elements in I_π′ (G‖θ). As a consequence, we establish an analogue of Fong's theorem on defect groups of covering blocks, for the vertices of the simple modules (in characteristicsp) of a finitep-solvable group lying over a fixed simple module of a normal subgroup.     

On simple modules and normal subgroups of p-solvable groups

Let N be a normal subgroup of a p-solvable group G and let M be a simple FN-module, where F is an algebraically closed field of characteristic p. Next, denote by IRR⁰(FG|M) the set of all simple FG-modules V lying over M such that the p-part of dim_F V is as small as possible. In this paper, |IRR⁰(FG|M)| and the vertices of modules in IRR⁰(FG|M) are determined. The p-blocks of G to which modules in IRR⁰(FG|M) belong are also determined.Received: 5 December 2003     

Green vertices and sources of simple modules of the symmetric group labelled by hook partitions

We determine the Green vertices and sources of many of the simple modular representations of the finite symmetric group being parametrized by hook partitions. Received: 20 August 2006     

Endotrivial modules for nilpotent restricted Lie algebras

Let $$mathfrak {g}$$ be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For $$pgeqslant 5$$, we show that every endotrivial $$mathfrak {g}$$-module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for $$p geqslant 5$$ has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.     

Endotrivial modules for the general linear group in a nondefining characteristic

Suppose that \(G\) is a finite group such that \(\mathrm{SL }(n,q)\subseteq G \subseteq \mathrm{GL }(n,q)\), and that \(Z\) is a central subgroup of \(G\). Let \(T(G/Z)\) be the abelian group of equivalence classes of endotrivial \(k(G/Z)\)-modules, where \(k\) is an algebraically closed field of characteristic \(p\) not dividing \(q\). We show that the torsion free rank of \(T(G/Z)\) is at most one, and we determine \(T(G/Z)\) in the case that the Sylow \(p\)-subgroup of \(G\) is abelian and nontrivial. The proofs for the torsion subgroup of \(T(G/Z)\) use the theory of Young modules for \(\mathrm{GL }(n,q)\) and a new method due to Balmer for computing the kernel of restrictions in the group of endotrivial modules.     

Computations in differential and difference modules

Constructive methods based on the Gröbner bases theory have been used many times in commutative algebra over the past 20 years, in particular, they allow the computation of such important invariants of manifolds given by systems of algebraic equations as their Hilbert polynomials. In differential and difference algebra, the analogous roles play characteristic sets.In this paper, algorithms for computations in differential and difference modules, which allow for the computation of characteristic sets (Gröbner bases) in differential, difference, and polynomial modules and differential (difference) dimension polynomials, are described. The algorithms are implemented in the algorithmic language REFAL.     

Large dimension homomorphism spaces between Specht modules for symmetric groups

Let F be a field of characteristic p. We show that Hom_FΣn(S^λ,S^μ) can have arbitrarily large dimension as n and p grow, where S^λ and S^μ are Specht modules for the symmetric group Σ_n. Similar results hold for the Weyl modules of the general linear group. Every previously computed example has been at most one-dimensional, with the exception of Specht modules over a field of characteristic two. The proof uses the work of Chuang and Tan, providing detailed information about the radical series of Weyl modules in Rouquier blocks.     

A characterization of endotrivial modules over p-groups

Let P be a finite p-group, and let k a field of characteristic p>0. We prove that a finietly generated kG-module M is endotrivial if and only if

Group algebra series and coboundary modules

The shift action on the 2-cocycle group Z²(G,C) of a finite group G with coefficients in a finitely generated abelian group C has several useful applications in combinatorics and digital communications, arising from the invariance of a uniform distribution property of cocycles under the action. In this article, we study the shift orbit structure of the coboundary subgroup B²(G,C) of Z²(G,C). The study is placed within a well-known setting involving the Loewy and socle series of a group algebra over G. We prove new bounds on the dimensions of terms in such series. Asymptotic results on the size of shift orbits are also derived; for example, if C is an elementary abelian p-group, then almost all shift orbits in B²(G,C) are maximal-sized for large enough finite p-groups G of certain classes.     

On fixed point sets and Lefschetz modules for sporadic simple groups

We consider 2-local geometries and other subgroup complexes for sporadic simple groups. For six groups, the fixed point set of a noncentral involution is shown to be equivariantly homotopy equivalent to a standard geometry for the component of the centralizer. For odd primes, fixed point sets are computed for sporadic groups having an extraspecial Sylow p-subgroup of order p³, acting on the complex of those p-radical subgroups containing a p-central element in their centers. Vertices for summands of the associated reduced Lefschetz modules are described.     

Finding coherent cyclic orders in strong digraphs

A cyclic order in the vertex set of a digraph is said to be coherent if any arc is contained in a directed cycle whose winding number is one. This notion plays a key role in the proof by Bessy and Thomassé (2004) of a conjecture of Gallai (1964) on covering the vertex set by directed cycles. This paper presents an efficient algorithm for finding a coherent cyclic order in a strongly connected digraph, based on a theorem of Knuth (1974). With the aid of ear decomposition, the algorithm runs in O(nm) time, where n is the number of vertices and m is the number of arcs. This is as fast as testing if a given cyclic order is coherent.     

Infinite dimensional modules for Frobenius kernels

We prove that the projectivity of an arbitrary (possibly infinite dimensional) module for a Frobenius kernel can be detected by restrictions to one-parameter subgroups. Building upon this result, we introduce the support cone of such a module, extending the construction of support variety for a finite dimensional module, and show that such support cones satisfy most of the familiar properties of support varieties. We also verify that our representation-theoretic definition of support cones admits an interpretation in terms of Rickard idempotent modules associated to thick subcategories of the stable category of finite dimensional modules.     

The Loewy length of a tensor product of modules of a dihedral two-group

While the finite-dimensional modules of the dihedral 2-groups over fields of characteristic 2 were classified over 30 years ago, very little is known about the tensor products of such modules. In this article, we compute the Loewy length of the tensor product of two modules of a dihedral 2-group in characteristic 2. As an immediate consequence, we determine when such a tensor product has a projective direct summand.     

A theorem of Jon F. Carlson on filtrations of modules

We give an alternative proof to a theorem of Carlson [J.F. Carlson, Cohomology and induction from elementary abelian subgroups, Quart. J. Math. 51 (2000) 169-181] which states that if G is a finite group and k is a field of characteristic p, then any kG-module is a direct summand of a module which has a filtration whose sections are induced from elementary abelian p-subgroups of G. We also prove two new theorems which are closely related to Carlson’s theorem.