Algebraic Modules
Here we consider the topics of algebraic modules and sources of simple modules, mostly in blocks with abelian defect group.
In [1], Alperin defined a module to be algebraic if it satisfies a polynomial with integer coefficients, where addition and multiplication are thought of as the direct sum and the tensor product. The correct place to think of this is the Green ring of all virtual modules, where 'minus' a module makes sense. An alternative definition of an algebraic module is that a module M is algebraic if and only if there is a finite list of indecomposable modules M1,M2,...,Mn such that any indecomposable summand of any tensor power of M is isomorphic to one of the Mi.
In [2], Alperin proved that if G is one of the groups SL2(2n), then all of the simple modules are algebraic. This is extended in one direction in my D.Phil. thesis, and in another direction in as-yet unpublished work.
Theorem (Craven, [4]). Let G be a finite group with abelian Sylow 2-subgroups, and let K be a field of characteristic 2. Then all simple KG-modules are algebraic.
Theorem (Kovacs, unpublished). Let p be a prime, and let K be a field of characteristic p. Let G be one of the groups SL2(pn). Then all simple KG-modules are algebraic.
A theorem that in some sense is orthogonal to the first theorem listed is the following, proved by Charles Eaton, Radha Kessar, Markus Linckelmann, and me, when we were at MSRI in March 2008, building on previous work of the first three authors.
Theorem (Craven–Eaton–Kessar–Linckelmann, [7]). Let G be a finite group, and suppose that G possesses a 2-block B whose defect group is elementary abelian of order 4. Then all simple B-modules are algebraic.
This result is equivalent to a conjecture of Erdmann [8] on the Green correspondence in blocks whose defect group is Klein four, and implies that there are only three Puig equivalence classes of block with Klein four defect group, settling the last open problem for these blocks. Like cyclic blocks, these blocks are now completely understood.
The two theorems above seem like they have a common generalization, namely the following conjecture.
Conjecture (Craven, [4]). Let G be a finite group, and let B be a 2-block of G with abelian defect group. Then all simple B-modules are algebraic.
It is unclear at the moment how this fits into the fabric of the current conjectures and results. Although algebraic modules have been defined for over thirty years now, their use has been limited. Apart from Alperin's result alluded to earlier, and Berger's result [3] that all simple modules for soluble groups are algebraic (extended by Feit [9] to p-soluble groups, where of course the field has characteristic p), are the only results to be found in the literature explicitly dealing with algebraic modules.
In the case of endo-trivial and endo-permutation modules, the property of being algebraic is equivalent to being torsion in the Dade group, and this has been of interest over the past few years. Puig's conjecture that the source of the simple module in a nilpotent block is torsion in the Dade group was one of the difficult parts of the proof of the theorem on blocks with Klein four defect groups given earlier. It is not clear whether the methods in algebraic modules can be used to a great extent in proving this conjecture.
It should be noted that for odd primes, not every simple module in a block with abelian defect group is algebraic. Indeed, there are simple modules in the principal block of the smallest Mathieu group that are not algebraic, along with simple modules in the principal block of M23, some faithful simple modules of 4.M22 and of 42.PSL3(4), and potentially simple modules in other non-principal blocks. An article on the problem for 3-blocks with abelian defect group is in the pipeline.
The above paragraphs dealt only with simple modules. While these are interesting, most modules are not simple.
Theorem (Craven, [5]). Let M be a non-periodic module, and suppose that M is algebraic. Then none of the non-trivial Heller translates Ωi(M) is algebraic.
This result shows that non-periodic algebraic modules are in some sense rare. In fact, if we increase the complexity by 1, we get a much stronger result.
Theorem (Craven, [5]). Let M be an indecomposable module of complexity 3, and let Γ denote the component of the stable Auslander–Reiten quiver containing M. (Note that Γ is of type A∞.) Then M lies on the end of Γ and is the only algebraic module on Γ.
It is not known whether there is an A∞ component of the Auslander–Reiten quiver for any group that had more than one algebraic module on it. If G is a semidihedral group of order 16, then the component of the Auslander–quiver containing the trivial module also contains the non-trivial, torsion endo-trivial module for G, and so that is an example of a component containing two algebraic modules.
It should be noted that this is merely an introduction to algebraic modules, and there is much more known, but this should give you a flavour as to the directions of my research in the area.
References
[1] Jonathan Alperin, On Modules for the Linear Fractional Groups, International Symposium on the Theory of FInite Groups, 1974, Tokyo (1976), pp. 157–163.
[2] Jonathan Alperin, Projective Modules for SL2(2n), J. Pure Appl. Algebra 15 (1979), no. 3, 219–234. MathSciNet
[3] Thomas Berger, Solvable Groups and Algebraic Modules, J. Algebra 57 (1979), no. 2, 387–406. MathSciNet
[4] David A. Craven, Simple Modules for Groups with Abelian Sylow 2-subgroup Are Algebraic, J. Algebra 231 (2009), no. 5, 1473–1479. MathSciNet, DOI, PDF
[5] David A. Craven, Algebraic Modules and the Auslander–Reiten Quiver, J. Pure Appl. Algebra 215 (2011), 221–231. DOI, PDF
[6] David A. Craven, Tensor Products of Modules for Groups of Lie Type, Algebr. Represent. Theory 126 (2013), 377-404. DOI, PDF.
[7] David A. Craven, Charles Eaton, Radha Kessar, and Markus Linckelmann, Blocks with a Klein Four Defect Group, Math. Z. 268 (2011), 441-476. DOI, PDF
[8] Karin Erdmann, Blocks whose Defect Groups are Klein Four Groups: A Correction, J. Algebra 76 (1982), no. 2, 505–518. MathSciNet
[9] Walter Feit, Irreducible Modules of p-Solvable Groups, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif. 1979), pp. 405–411. MathSciNet