The general character theory machinery is currently restricted to characters defined over the complex field. Operations on characters include the following:
The table of ordinary (complex) irreducible characters of a finite group is of critical importance. The algorithm of W. Unger (2006) is very efficient and allows Magma to compute character tables of very large groups. For example, J. Cannon used it to computer the 62 ordinary irreducible characters for the simple group J4 of order 86,775,571,046,077,562,880 given as a permutation group of degree 173,067,389. This was the response of Cannon and Unger to a talk given by J.-P. Serre at Harvard in March 2015 where he pointed out that many recent theorems etc. depend upon information taken from the Atlas character tables but these tables lack proofs of correctness. We are able to compute the character tables of all but five of the 450 groups given in the Atlas. One table was found to be incorrect. Since the Unger algorithm uses a completely different method for computing character tables this is a completely independent verification.
Let A be the group algebra K[G] for a finite group G and a finite field K. Unless otherwise stated, the A-modules discussed in this section will be defined over a finite field of characteristic p, where p divides |G|. In this section the machinery provided for investigating the irreducible A-modules for G is outlined. The issues are finding the irreducible constituents of a given A-module, finding the irreducible Brauer characters, determining all the irreducible A-modules, and finally, given an irreducible A-module N, find the projective indecomposable A-module U such that U/Rad(U)≅N, where Rad(U) is the radical of U.
A highly efficient Meataxe algorithm for modules over Fq has been developed by A. Steel over a long period. The following tools, based on the Meataxe, are available:
A Brauer character modulo p in Magma is represented as a class function (that is, element of a character ring) which is zero on p-singular group elements. In this format the standard character operations of addition, multiplication, induction, and restriction all apply directly to Brauer characters as they do to other class functions. Given the table T of ordinary characters of G and a prime p, a basic function partitions T into p-blocks.
A function is provided to construct the table of Brauer characters modulo the prime p. If G is soluble then its Brauer characters can be derived directly form the table of ordinary characters. In the case in which G is non-soluble, work is underway to produce many of the Brauer characters without first constructing the corresponding irreducible A-modules. However, at present irreducibles are first computed. The decomposition matrix D is also returned. The (i,j)-th entry of D gives the number of times the absolutely irreducible K'[G]-module j (K < = K') occurs as a constituent of the mod-p restriction of the ordinary character i.
The method for constructing the irreducible K[G]-modules proceeds as follows. Let S be the list of irreducibles that have been constructed so far. A K[G]-module N that contains an irreducible not already in the list S is identified. Then N is split using the Meataxe and the new irreducibles added to S. This is the same general approach taken by Parker and others when constructing particular irreducible modules for simple groups. However, we have made this process entirely automatic as distinct from earlier work where intervention was needed at various points (particularly in regard to choosing a suitable module N).
The indecomposable direct summands of A are projective as A-modules; they are known as the projective indecomposable A-modules. Furthermore, any finitely generated projective A-module is isomorphic to a direct sum of projective indecomposables.
The algorithm for computing projective indecomposable modules assumes that representatives of the irreducible K[G]-modules are known. The principal step of the algorithm for computing projective indecomposables is, for a given irreducible K[G]-module L∈ℒ, compute the unique projective indecomposable module U∈U with U/Rad(U)≅L.
We can successfully compute difficult examples with p small for groups G up to order about 107.
A characteristic zero Meataxe algorithm for modules was developed by A. Steel in 2009 and is used for splitting general A-modules, where A is a finite dimensional matrix algebra defined over the rational field. This yields an effective algorithm for decomposing a module into indecomposable summands. If the module is a G-module for some group G, extensive use is also made of character theory. Representations associated with characters having non-trivial Schur indices are properly handled. The difficult problem of splitting homogeneous modules (direct sums of the same indecomposable) is handled by decomposing the endomorphism ring of the module via a maximal order. Modules having dimensions in the several hundreds are routinely split into indecomposable modules.
Critical to the success of the characteristic zero Meataxe algorithm is an effective method to split a homogeneous module M into a direct sum of k isomorphic copies of a single indecomposable module S. This is achieved by first computing a maximal order O of the endomorphism ring R of M via an algorithm developed by G. Nebe and A. Steel (2009) and by then finding a singular element of O via a LLL-based heuristic search algorithm. Repeating the process gives the direct sum into the multiple copies of the indecomposable S. This also yields a practical algorithm for proving a homogeneous module to be indecomposable. Finally, the main decomposition algorithm first uses asymptotically-fast modular algorithms to split the module into homogeneous components and then splits each component by the above method.
The technique of condensation allows one to analyse modules and algebras with dimensions much larger than possible with the direct Meataxe algorithm. The condensation of an algebra A involves working with a smaller associated algebra eAe (where e is a suitable idempotent in A) which is Morita equivalent to A, which means that the algebras have exactly the same representation theory. For algebras which are group algebras, there are specific methods for producing suitable idempotents, coming from group theory, and consequently there has been much development of theory and practical algorithms for condensation and applications. Magma provides tools for the construction and analysis of the condensation of:
These tools handle modules defined either over finite fields or characteristic zero fields.
An algorithm for computing irreducible ℚ[G]-modules for a finite group G was developed by A. Steel in 2009. Given a rational character of G, the algorithm proceeds by locating a (typically reducible) module that contains the desired module; this module may arise from permutation, induced or tensor representations. Then using the Meataxe described above, the module M is split thereby, yielding the required irreducible module. A novel component of this algorithm is an automatic search for a suitable condensed module so as to reduce the dimensions of the modules that have to be split. The algorithm controls the growth of coefficients at every stage, thus returning modules whose actions are usually defined by matrices with very small integral entries. A variant of the algorithm is also provided which determines all irreducible ℚ[G]-modules for G. The machinery has been used to construct irreducible ℚ[G]-modules having dimension well over a thousand in favourable circumstances.
As an example, the irreducible degree-2024 rational representation of the Conway sporadic simple group Co2 can be computed in 4350 seconds. This involves condensing a degree-2300 permutation representation of the group down to a dimension-82 module and decomposing this via the above Meataxe. The resulting representation has integral entries with at most 2 digits.