Computing Ext Algebras for Finite Groups

by

Jon F. Carlson^{1} Dept. of Mathematics University of Georgia Athens, GA 30602 USA email: jfc@sloth.math.uga.edu Edward L. Green^{1} Department of Mathematics Virginia Polytechnic Institute and State University

Blacksburg, VA

USA

email: green@math.vt.edu Gerhard J. A. Schneider Rechenzentrum Universität Karlsruhe 7500 Karlsruhe 1 GERMANY email: schneider

@rz.uni-Karlsruhe.de

1 Introduction and Motivation.

The purpose of this article is to describe, in general terms, a program for computing the Ext algebras for some small simple or nearly simple groups. For notation, let G be a finite group and k a field of characteristic p > 0. We will assume that k is a splitting field for the group G. By the Ext algebra, we mean the ring,

∐_{i,j=1}^{t} Ext_{kG}^{∗}(S_{i}, S_{j}) = Ext_{kG}^{∗} (∐_{i=1}^{t} S_{i}, ∐_{j=1}^{t} S_{j}) ,

where S_{1}, …, S_{t} are the simple kG-modules. Computing the algebra means generating all of the cup product information in addition to finding dimensions and bases for the k-vector spaces Ext_{kG}^{n}(S_{i}, S_{j}). Although some of the code to implement the program is still being written, we expect to have our first calculations completed soon. In the initial stages, some parts of the process will be tailored to the particular example under study. More general methods and programs may be shaped by these early experiments.

A major aim of the program is the study of some interesting examples concerned with the basic theory of kG-modules. The building blocks for the module theory of any finite dimensional algebra are the irreducible modules. Every finitely generated module has a composition series in which all of the factors are simple modules. The construction of general modules from the simples is accomplished by homological algebra and is documented in the cohomology. Moreover there is a sense in which the entire module theory is controlled by the cohomology of the simple modules — by the Ext algebra. To begin with, any module M with only two composition factors can be expressed as an extension (exact sequence)

1Partially supported by a grant from NSF.

ζ : 0 ⟶ S_{1} ⟶ M ⟶ S_{2} ⟶ 0

where S_{1} and S_{2} are the simple factors. Such extensions are classified by the cohomology Ext_{kG}^{1}(S_{2}, S_{1}). Likewise if N is a module with three composition factors then N is expressed as an exact sequence

γ : 0 ⟶ S_{3} ⟶ N ⟶ M ⟶ 0

with S_{3} simple and M given as above. The extension class γ, which gives N, is recorded in the group Ext_{kG}^{1}(M, S_{3}). Even though this is not directly a part of the Ext algebra, we have a long exact sequence

… ⟶ Hom_{kG}(S_{1}, S_{3}) ⟶^{δ_{0}} Ext_{kG}^{1}(S_{2}, S_{3})

⟶ Ext_{kG}^{1}(M, S_{3}) ⟶

Ext_{kG}^{1}(S_{1}, S_{3}) ⟶^{δ_{1}} Ext_{kG}^{1}(S_{2}, S_{3}) ⟶ ….

Hence as a k-vector space Ext_{kG}^{1}(M, S_{3}) is the direct sum of the cokernel of δ_{0} with the kernel of δ_{1}. The important point here is that the connecting homorphisms, δ_{0} and δ_{1} are both given by cup product with the cohomology class, ζ, which presents the module M. There may be some adjustments necessary for the Massey products [2], but these can also be computed. Hence all of the information is recorded in the Ext algebra. A module with four composition factors can be analyzed by extensions and exact sequences in which Ext_{kG}^{3} of irreducible modules becomes part of the relevant data. Hence the higher cohomology has an established role in the module theory, even if it seems obscure in the complicated situations. Moreover the ring structure of the cohomology has been crucial for other studies [3].

On the other hand, any analysis of module theory through its building blocks is beset by a dilemma. The problem is that there seems to be no easy characterization of the simple modules, the building blocks, in terms of the major techniques, the homological algebra. The cohomology does not recognize irreducibility, to the point where some easily stated questions about the relationship seem nearly intractable. An example is the question of whether H^{n}(G, M) must be non-zero for some n whenever M is a simple kG-module in the principal block. The answer is not known except in some special cases of groups which are p-solvable or p-constrained [7, 8].

There is very little experimental evidence available on Ext algebras. If G is a p-group then kG has only one irreducible module and Ext_{kG}^{∗}(k, k) ≅ H^{∗}(G, k) has been calculated in many examples. Beyond that, complete calculations have been done only in cases which admit some special techniques. In characteristic p, the groups SL(2, p^{n}) have T. I. Sylow p-subgroups and the simple modules are classified by highest weights [4]. Diagrammatic methods can be used in a few cases of groups with small and well behaved Sylow p-subgroups. Examples include A_{7}, M_{11}, L_{3}(3) in characteristic 2 and A_{6} in characteristic 3 [2]. Partial results such as the cohomology with trivial coefficients, H^{∗}(G, k), have been computed in several cases.

2 The Program.

Let S_{1}, …, S_{t} denote the irreducible kG-modules. Our aim is to calculate the Ext algebra E = ∐_{i,j}Ext_{kG}^{∗}(S_{i}, S_{j}). Before describing the process we should note two things. First, such a calculation can really be made only on a block by block basis. That is, if kG = B_{1} ⊕ … ⊕ B_{s} where B_{1}, …, B_{s} are the blocks or indecomposable two-sided ideals, then each S_{i} belongs to one of the blocks. Because B_{i}B_{j} = 0 whenever i ≠ j, it is also true that Ext_{kG}^{∗}(S_{i}, S_{j}) = 0 whenever S_{i} and S_{j} belong to different blocks. Hence if we let E_{i} = ∐ Ext_{kG}^{∗}(S_{j}, S_{k}), where the sum is over all j and k such that S_{j} and S_{k} are in B_{i}, then

E = ∑_{i=1}^{s} E_{i}

where each E_{i} is a two-sided ideal in E.

It should be further noted that the techniques which we employ are probably not the ones which a person would wish to use if his object were only the calculation of cohomology with trivial coefficients. There are numerous methods from geometry and topology which can be used on H^{∗}(G, k). The most revealing exposition of these methods is the book of Adem and Milgram [1]. For the more general project which we are attempting, the only available method seems to be some form of “brute force”. Namely we compute the minimal projective resolutions for the irreducible modules and actually exhibit cocycles representing the cohomology elements. Likewise, the cup products are obtained by getting the cocycles which represent them. It is our hope that the experimental data produced by the project will lead to the development of methods with greater sophistication.

We should emphasize that “brute force”, even with large computers, requires some care. The terms of a projective resolution of a module over a group algebra can easily run into the thousands of dimensions after only a few steps. For this reason, we substitute the basic algebra, or quiver with relations, for the group algebra. This object is explained in the steps to follow, but the substitution has the effect of replacing each irreducible module by a single symbol. Thus the group algebra, which may be several thousand dimensions is replaced by something much smaller.

The project proceeds by three major steps. Each uses its own software. The steps are as follows.

\underline{Step 1}. From group algebra to basic algebra (quiver with relations). Let S_{1}, …, S_{t} be the simple kG-modules and for each i = 1, …, t let P_{i}

be the projective cover of S_{i}. Let P = ∐_{i=1}^{t}P_{i} and

E = Hom_{kG}(P, P) = ∐_{i,j=1}^{t} Hom_{kG}(P_{i}, P_{j}).

The k-algebra E is called the basic algebra of kG. It is an easy calculation that the k-dimension of E is the sum of the entries in the Cartan matrix of kG. The theorem which makes everything work is the following.

Theorem 1 The category of kG-modules is Morita equivalent to the category of E-modules.

Let e_{i} ∈ E be the idempotent which is the composition of the projection of P onto P_{i} followed by the inclusion of P_{i} into P. So 1 = ∑_{i=1}^{t}e_{i} in E, and the idempotents e_{1}, …, e_{t} are orthogonal. Under the equivalence of categories in the theorem, the projective module P_{i} of kG corresponds to the projective module Ee_{i} of E.

For each i and j let C_{ij} be a basis for the k-vector space Hom_{kG}(P_{i}, Rad P_{j}/Rad^{2}P_{j}) where Rad P_{j} denotes the radical of P_{j}. Notice here that C_{ij} corresponds to a k-linearly independent subset of Hom_{kG}(P_{i}, P_{j}).

That is, if ζ ∈ C_{ij} then we have a lifting ζ̂ such that the diagram

P_{i}
[figure: ζ̂]  ↓ ζ
Rad P_{j} ⟶ Rad P_{j}/Rad^{2}P_{j} → 0
↓ i
P_{j}

commutes. Here ζ̂ followed by the inclusion i is an element of the basic algebra E. The actual homomorphism i ∘ ζ̂ ∈ Hom_{kG}(P_{i}, P_{j}) is only defined by ζ up to the addition of an element in Rad^{2}E. However, any choice of representatives in Rad E of the elements in the C_{ij}’s will, along with the idempotents e_{1}, …, e_{t} give a set of generators for E.

The actual computational construction of the basic algebra requires several packages that deal with modules. On the computer it is practical to consider group representations rather than modules as these can be described easily via matrices. The construction relies heavily on the MEATAXE (which is available in various stand-alone implementations as well as part of Cayley or Magma [9]), the ENDOMORPHISM and HOMOMORPHISM packages developed by the third author as part of Cayley or Magma [10], and the GROEBNER package by the second author and Chuck Feustel.

In part 1 of the construction one needs to obtain explicit matrix representations of the various irreducible representations. i. e. explicit matrices for each of the group generators. Care has to be used to ensure that the underlying field is big enough to realize all absolutely irreducible representations over the same field to avoid coercion problems at a later stage. The methods have been extensively described in many papers and need not be explained again. Already at that stage we can determine the Cartan matrix for the group in question which will be necessary later. Note that the construction is not algorithmic in the strong sense, as feasible ways to construct each of the irreducible representations have to be found depending on the group in question.

In part 2 we proceed to obtain explicit matrix representations for the projective modules. Although there is a theoretical algorithm for the construction of all projective representations, it is by no means practical to construct the regular representation and to use the endomorphism package to find all the indecomposable summands. The storage requirements to manipulate a regular representation and its endomorphisms are beyond the capabilities of modern machines. Rather, theoretical insight into the case under consideration is essential to find a way to construct a projective representation in an optimal way, so that the dimensions of the matrices involved never get too big. Typical steps include induction of (projective) modules from suitable subgroups, decomposition of modules into their direct summands, using the endoring package, tensor products, etc. These constructional tools are all part of Cayley or Magma and can be invoked by simple commands. Such methods have already been used to successfully construct the projective indecomposable modules for a variety of groups, such as M_{11}, M_{12}, M_{22}, M_{23}, J_{1}, Sz(8), U_{3}(3).

Part 3 of the construction then computes all possible homomorphisms of each projective representation into any other. Such a homomorphism is again a (rectangular) matrix h, with the property that f_{1}(g) · h = h · f_{2}(g), for all group elements g and two projective representations f_{1} and f_{2}. For each pair of projective modules it suffices to find a homomorphism of largest possible rank, which can easily be read off the result of the computation. This part is in fact fully algorithmic and relies on the HOMOMORPHISM package in Cayley or Magma. At the end of part 3 we have a set of homomorphisms (matrices) which contains the essential information for the next part.

Part 4 appears to be the most complicated although in practice it is usually the fastest one. In the basic algebra, which is generated by all the homomorphisms constructed in the previous part, we first construct all possible words of length 2 by brute force. Using standard linear algebra we find all the relations between all the words of length 1 and 2 (each word is evaluated as a matrix), simply by setting up a system of equations in these matrices and by finding the solution space. In principal this approach could be continued to find all the relations between all the words of length up to 3, etc. However the number of unknowns would get out of hand quickly. The GROEBNER package yields a much faster approach to the required information, as it determines all words in the basic algebra of length 3 modulo the relations found so far. Now we compute the matrices that correspond to the words found by GROEBNER and find all the relations between these words of length up to 3. We then repeat this process several times. Because the Cartan matrix is known, we can continue until the number of elements found equals the sum of the entries in the Cartan matrix.

As a result we have constructed a basis for the basic algebra as well as the relations that hold between the basis elements, i. e. we know the quiver with relations. In particular, explicit matrices are no longer required and the computation is independent of the actual dimensions. Note however that the construction up to and including part 4 still does require the handling of matrices. Once we have the quiver it is a minor problem to give it a nice graphical presentation The feasability of the steps outlined so far has been demonstrated by completely determining the quiver with relations for U_{3}(3) over the field GF (9).

\underline{Step 2}. Compute projective resolutions for simple modules.

In the explanation of this part and the next step we will denote the irreducible modules by S_{1}, …, S_{t} and the indecomposable projective modules by P_{1}, …, P_{t}. However the reader should be aware that we are now working in the basic algebra E. Hence, among other things, the simple modules have dimension one.

The software for this step is contained in a program called GROEBNER and written by the second author and Charles Feustel [5, 6]. The initial part of the step is to create a Gröbner basis for the basic algebra E. In general terms, the ring E is a quotient of a polynomial ring R by an ideal I. The polynomial generators for R, for practical purposes can be preimages of the elements of the sets, C_{ij}, as above. Note that these are noncommuting variables. The elements of I are all polynomials in these generators, and a basis for E can be taken to be cosets modulo I of the elements e_{1}, …, e_{t} and some other monomials. If a monomial in the generators is not in the chosen basis for E, then there must be some way to rewrite it using generators for the ideal I. A Gröbner basis for E is precisely a set of generators for the ideal I that provide the rewriting rules. In order to activate the rules it is necessary to have an ordering on the monomials which give a decision process for the rewriting. That is, rewriting using a Gröbner basis means replacing a monimal by a linear combination of smaller (in the ordering) monomials. So the process can only be performed a finite number of times on any element of R. But, most importantly, it allows for the creation of a multiplication table for E in terms of the chosen monomials. Equations involving the elements of E can be reduced to linear algebra equations.

To understand the construction of the projective resolutions, we need to know how modules are expressed in this system. So the question is:

What is a module?

The answer in this system is that a module is its presentation as a homomorphic image of a projective module. That is for every module M there is a projective module Q_{0} and a surjective homomorphism ε : Q_{0} → M. The kernel of ε is also an E-module which we usually denote Ω(M). Likewise we can find a projective module Q_{1} and a surjective homomorphism ε′ : Q_{1} → Ω(M). Then composing we get an exact sequence

Q_{1} ⟶^{θ} Q_{0} ⟶ M ⟶ 0

which presents M uniquely as the quotient M ≅ Q_{0}/θ(Q_{1}). But most importantly, from the computational standpoint, the module M is characterized by the matrix of the map θ. Specifically, we can express the projective modules Q_{1} and Q_{0} as direct sums of the indecomposable projectives modules:

Q_{0} ≅ ∑_{i=1}^{t} P_{i}^{a_{i}}; Q_{1} = ∑_{i=1}^{t} P_{i}^{b_{i}}.

Then the homomorphism θ is characterized by its matrix θ = (θ_{ij}) where θ_{ij} ∈ Hom_{E}(P_{k}, P_{l}) for P_{k} and P_{l} being respectively the i^{th} and j^{th} projectives in Q_{1} and Q_{0}. That is, the entries of the matrix θ are elements E and hence can be expressed as linear combinations of the chosen monomials modulo the Gröbner basis.

Going a step further we note that a projective resolution for M,

⟶ Q_{2} ⟶^{∂_{2}} Q_{1} ⟶^{∂_{1}} Q_{0} ⟶ M ⟶ 0,

in this scheme, is a sequence of matrices representing the homomorphisms θ = ∂_{1}, ∂_{2}, etc. Hence for each i = 1, …, t we wish to compute a minimal projective resolution

… ⟶ Q_{i,1} ⟶ Q_{i,0} ⟶ S_{i} ⟶ 0

for S_{i}. Here Q_{i,0} ≅ P_{i}. We can write

Q_{i,n} = ∑_{j=1}^{t} P_{j}^{a_{inj}},

and, because of the minimality,

a_{inj} = Dim Ext_{E}^{n}(S_{i}, S_{j}).

The reason for this is that an element ζ̂ ∈ Ext_{E}^{n}(S_{i}, S_{j}) is represented by a cocycle ζ : Q_{i,n} → S_{j}. The cocycle condition is that the composition

Q_{i,n+1} ⟶^{∂} Q_{i,n} ⟶^{ζ} S_{j},

is zero. But the minimality of the resolution ensures that ∂(Q_{i,n+1}) ⊆ Rad Q_{i,n}. Hence, because S_{j} is irreducible, ζ ∘ ∂ = 0 for any element ζ of Hom_{kG}(Q_{n,i}, S_{j}). Similarly all coboundaries are zero and we have that

Hom_{E}(Q_{n,i}, S_{j}) ≅ Ext_{E}^{n}(S_{i}, S_{j}).

We see then, that by computing the minimal projective resolutions, we have determined the Ext algebra up to the point of having the k-dimensions of the Ext groups. But we also want the product structure.

\underline{Step 3}. Computing the cup products.

First we recall that the simple module S_{j} is its presentation: Q_{j,1} → Q_{j,0} → S_{j} → 0. So the cocycle ζ, given as above, really looks like:

⋯ → Q_{i,n+1} →^{∂} Q_{i,n} → Q_{i,n−1} → ⋯ → Q_{i,0} → S_{i} → 0
↓ ζ_{1}  ↓ ζ_{0} [figure: ζ]
⋯ → Q_{j,1}  →^{∂} Q_{j,0} →^{∂} S_{j} → 0 .

Here the homomorphisms ζ_{0} and ζ_{1} exist because of the projectivity of Q_{i,n} and Q_{i,n+1}. Also the diagram commutes: ∂ζ_{0} = ζ, ∂ζ_{1} = ζ_{0}∂. Furthermore the maps ζ can be extended to a complete chain map

ζ = (ζ_{∗}) : Q_{i,∗} ⟶ Q_{j,∗}

of degree n, so that for each l ≥ 0 we have a homomorphism ζ_{l} : Q_{i,l} → Q_{j,l+n} such that ∂ζ_{l+1} = ζ_{l}∂. Two chain maps represent the same cohomology class if and only if they are chain homotopic.

So the next step in the program is to extend cocycles representing cohomology classes to chain maps between projective resolutions. As in the previous step a chain map is a sequence of matrices in polynomials in the generators for E. The justification for making such a calculation is contained in the following.

Theorem 2 Suppose that ζ ∈ Ext_{E}^{m}(S_{i}, S_{j}) and γ ∈ Ext_{E}^{n}(S_{j}, S_{k}) are represented by chain maps,

ζ̂ = (ζ_{∗}) : Q_{i,∗} → Q_{j,∗} and γ = (γ_{∗}) : Q_{j,∗} → Q_{k,∗},

then the composition γ̂ ∘ ζ̂ = (γ_{∗} ∘ ζ_{∗+n}) represents the cup product γ ∪ ζ.

Consequently we compute the cup products by producing the compositions of the chain maps. In this way we can write down a multiplication table for the Ext algebra.

3 Notes.

We end the paper with a few comments and remarks on the scope and difficulties of the project.

(3.1) Capabilities of the project. In the initial stages we wish to look at the cohomology of some small simple groups. Small, here, means of order less than a million. Even some of these may be difficult in terms of computing time and capacity. The work is very experimental and it is hard to predict all of the possible difficulties. Our first experiment, which includes a computation of the projective resolutions for the simple U(3, 3)-modules in characteristic 3, indicates that most of what we intend is reasonable. However this group has 3-rank 2. Groups with larger p-rank may expose other complications. Even though most of the computations can be broken down into relatively small steps, the symbolic calculations are memory intensive.

Our intension is to look only at groups G and primes p where the prank of G is at least 2. The calculations for groups with cyclic Sylow psubgroups can be done by a general theory once the structure of the principal indecomposable modules is known. Hence from a computational standpoint this is an exercise in the use of the MEATAXE. But further, it seems likely that for the size and type of group we are considering, the cyclic Sylow psubgroup problems have already been done. For small groups many of the Brauer trees for blocks with cyclic defect have been computed.

(3.2) Nature of the problem. It is an easy check that many of the problems in steps 2 and 3 reduce to linear algebra, i.e. the solutions of systems of linear equations. For example in the construction of the chain maps, as in the following diagram, suppose that ζ_{s} has been constructed.

⋯ ⟶ Q_{i,n+s+1} ⟶^{∂_{i,n+s+1}} Q_{i,n+s} ⟶ ⋯
↓ζ_{s+1}  ↓ζ_{s}
⋯ ⟶ Q_{j,s+1}  ⟶^{∂_{j,s+1}} Q_{j,s} ⟶ · · · .

The next step would be the construction of ζ_{s+1}. But ζ_{s+1} must be represented by a matrix X such that

matrix(∂_{j,s+1}) · X = matrix(ζ_{s}) · matrix(∂_{i,n+s+1}).

The entries of all of the matrices are linear combinations of monomials in the generators of E. In the products, the Gröbner basis provides the rewriting rules to express everything uniquely as a linear combination of some distinguished set C of monomials. Then we can write X = (x_{kl}) where for each pair k, l, where x_{kl} = ∑_{σ∈C} α_{k,l,σ} · σ. The above product provides us with a system of linear equations in the indeterminants α_{k,l,σ}.

(3.3) Shortcuts. There are numerous methods for saving time and computing effort. Some of these have been built into the initial effort, while others might be considered in future efforts. For example when a chain map for a cohomology element is constructed then it is relatively easy to construct its powers and its products with other cohomology elements. Hence the chain maps of low degree should be calculated first in order to avoid duplication of effort.

An area which might yield improvements in efficiency is directly related to the nature of the problem. The fact is that the systems of equations, which arise in the computations of both the projective resolutions and the chain maps, are severely underdetermined. It is guaranteed that the systems have solutions, but usually there are many solutions. It may be possible to cut significantly the total cost of the program by looking for optimal solution. This is one of those considerations which has not been addressed in the theoretical literature, but which, for large problems, may prove very important.

(3.4) Completion. One other question of vital importance is that of when the computation is done. After all, we intend to calculate an entire Ext algebra, an infinite dimensional object, in a finite number of steps. A theoretical basis for answering the question exists in some form. As an example, suppose that there exists an element ζ ∈ Ext_{kG}^{n}(M, M) such that ζ is a regular element (nondivisor of zero) on Ext_{kG}^{∗}(N, M). If N ⊗ M has complexity two, then the cokernel of the cup product homomorphism

ζ : Ext_{kG}^{∗}(N, M) ⟶ Ext_{kG}^{∗+n}(N, M)

is the cohomology of a periodic module and hence is generated in some specific (and determinable) degree s. Then it can be concluded that Ext_{kG}^{∗}(N, M) has no new generators in degree n + s or greater (see [2]).

Similar results exist to cover other situations. However it is also true that the theoretical base for this sort of problem needs to be expanded.

References

[1] A. Adem and J. Milgram, The Cohomology of finite Groups Springer-Verlag, NewYork, 1993

[2] D. Benson and J. Carlson, Diagrammatic methods for group representations and cohomology Comm. in Algebra, 15,1987, 53-121.

[3] J. Carlson, Cohomology and modules over group algebras, Proceedings of the International Congress of Mathematicians, Kyoto 1990, Springer-Verlag, Tokyo, 1991, 317-324.

[4] –, The cohomology of irreducible modules over SL(2, p^{n}), Proc. London Math. Soc. (3) l47,1983, 480-492.

[5] D. R. Farkas, C. D. Feustel and E. L. Green, Synergy in the theories of Gröbner bases and path algebras, Canadian J. of Mathematics, 45 (4), 1993, 727-739.

[6] C. D. Feustel, E. L. Green, E. Kirkman and J. Kuzmanovich, Constructing projective resolutions, Comm. in Algebra, 21 (6), 1993, 1869-1887.

[7] P. A. Linnell, Cohomology of finite soluble groups, J. Algebra, 107,1987, 53-62.

[8] P. A. Linnell and U. Stammbach, The cohomology of p-constrained groups, J. Pure Appl. Algebra,49, 1987, 273-279.

[9] R. A. Parker, The computer calculation of modular characters (The Meat-Axe), Computational Group Theory, ed. M. Atkinson, Academic Press, London, 1984, 267-274.

[10] G. J. A. Schneider, Computing with Endomorphism Rings of Modular Representations, J. Symb. Comp., 9, 1990, 607-636.

Note: The program GROEBNER is available by anonymous ftp. Contact Ed Green at green@math.vt.edu for more information.