ATLAS: Nomenclature of the representations (etc)
Representations and presentations
All our representations and presentations are stored in files whose names have the following form:
where <GrpName>, <RepName> and <FormatName> consist solely of alphanumeric characters. For the purposes of discussion, we shall denote by G the group we have a representation or presentation of.
The <GrpName> part of the identifier is a concatenation of strings <GrpN> and <Gens>, say. The <GrpN> part gives the structure of G - preferrably well enough to determine the isomorphism type of G. The naming procedure does not deviate too much from what follows below:
There are various other ambiguities such as isoclinism, eg for 2.S7, and `flaps', eg for 32.U4(3).23, and worse things in the covers of L3(4). We have not yet developed a comprehensive system to deal with this, but I have used 2S7 and 2S7i to denote the two double covers of S7, with 2S7 being the version whose character table appears in the ATLAS.
Nonabelian simple groups have the same notation as in the ATLAS, but with subscripts and superscripts written without any indication that they are sub/superscripts. Also, remove any brackets, and replace + and - by p and m respectively. Ambiguities and exceptions are dealt with in the next few points.
Where there is more than one description of a simple group, we prefer the description of the group according to the preferences: Alternating > Linear > Unitary, Symplectic > Orthogonal > Exceptional group of Lie type. Thus we prefer A5 to L24 or L25.
We have used L27 instead of L32, and U42 instead of S43.
The groups O10+(2) and O10-(2) are denoted O10p2 and O10m2 respeectively. All orthogonal groups (of large enough rank) are denoted similarly.
The twisted groups 3D4(q), 2E6(q), 2B2(q), 2F4(q), 2G2(q) are denoted TD4q, TE6q, Szq, TF4q and Rq respectively. The exception is that we use TF42 to denote the group 2F4(2)' - we use TF42d2 to denote 2F4(2).
We do not consider L213 to be ambiguous - it represents L2(13). This ATLAS is unlikely ever to contain an L21(3) page!
Adding a d2 to the right of a group name denotes adding a .2 to the group. Thus PGL2(11) = L2(11):2 is denoted L211d2. Ambiguities are resolved with the use of letters, with d2a corresponding to an extension .21, d2b corresponding to .22 and so on. Thus we use L34d2c for what the ATLAS calls L3(4):23.
Exceptions to the above rule are: the use of Sn instead of And2, and the use of S6, PGL29 and M10 to denote various extensions of A6.
d3, d4, d5 operate with rules similar to that of d2.
Extensions .S3, .22, .D8, .D12, .S4 are denoted using S3, V4, D8, D12, S4. As with d2, lower case letters are used to resolve ambiguities. For example, A6V4 denotes Aut(A6) = A6.22, and L34S3a denotes L3(4):3:22 [since L3(4):3:21 = L3(4):6].
The <Gens> part consists of the letter G, followed by a nonnegative integer. If <Gens> is G0, this means that we are representing/presenting G on an arbitrary (unspecified) generating set. If <Gens> is Gi, where i > 0, then this means that we are representing G on its ith tuple of standard generators. A tuple (g1, g1, . . . , gn) of generators of G are said to be standard generators if they also come with a set of conditions that specify the tuple up to conjugation by elements of Aut(G). We have defined more than one tuple of standard generators for some groups. The generating tuples G1, G2, etc will be fixed. We may vary the tuple G0, and even have two nonautomorphic tuples called G0 concurrently.
For example, if we are considering a [re]presentation of M12:2 on its (2C, 3A, 12A)generators, currently the only standard generators defined for M12:2, this part of the identifier would be M12d2G1.
The <RepName> part of the identifier tells you whether the file contains a presentation, permutation representation, monomial representation, or a matrix representation of G. It also gives information on the degree of the representation and, for matrix representations, the ring over which it is taken.
If the file contains a presentation of G [on generators Gi], then <RepName> is Pj, where j is a [strictly] positive integer, and <RepName> is usually P1. Theoretically, we could have two or more presentations of G on the same generating set, but a group would satisfy one presentation if and only if it satisfied the other.
For permutation, monomial and matrix representations, the
<RepName> identifier consists of an element giving (some
indication of) the isomorphism class of the representation, followed
by Bi to indicate that this is the ith base to which
we have written the representation. These representations should be FAITHFUL
for the group named in <GrpName>
For permutations, <RepName> consists of p followed
by a degree, possibly with a distinguishing `letter', followed by
Bi, indicating that this is the ith base to which we
have written the representation. (With the intention that B1,
B2, etc. are definitely fixed, and that B0 is also fixed
but may be altered in `exceptional' circumstances.
For monomials, <RepName> might possibly consist of m followed by a degree (the degree of the corresponding permutation
group) possibly with a distinguishing `letter', possibly followed by an
indication of which representation was induced up to obtain the monomials,
and finally Bi, indicating that this is the ith base
to which we have written the representation. Actually, we have no
representations that are explicitly of this type as we have not designed a
format in which we would like to express them, though some of the matrix
representations happen to be monomial. Nothing in this paragraph has been
For matrix representations, <RepName> consists of a ring
identifier, followed by r, followed by the degree (possibly with a
distinguishing letter), followed by Bi (the ith base
for this isomorphism class of representations).
The following rings have been used in this ATLAS (and in
addition Q was used in version 1):
Of these, the notation fq is fixed, and Z and
Zn are likely to remain fixed. The notation A
is not fixed and is liable to change without notice.
- fq denotes the finite field of order q [and not written over a proper subfield].
- Z denotes the ring, Z, of integers.
- Zn denotes the ring, Zn = Z/nZ, of integers modulo n [when n > 1 is not prime - currently [26/11/99] only Z4 occurs].
- A denotes a representation written over a ring of algebraic integers, but not over the integers themselves.
The following are not currently [04/02/02] used and are liable to change
- Q denotes a representation written over the rationals, but not over the integers.
- R denotes a representation written over the reals, but not over the rationals or algebraic integers.
- C denotes a representation written over the complexes, but not over the reals or algebraic integers.
- Fp denotes an arbitrary field of characteristic p.
- H [or D] denotes a representation written over a
non-commutative division ring (`the' quaternions for example).
- q denotes a polynomial ring, or the field of fractions of such a thing. Anything falling into one of the above categories is excluded from this section.
The <FormatName> part of the identifier consists of a single letter, possibly followed a number. This number will be a positive [nonzero] integer, but there is no (theoretical) bound on how big this number may be. The letter gives the format of the representation/presentation. The possibilities are as follows:
If no number is present, then all generators of G are in a single file. If a number i is present, then this means that the file contains the ith generator[s] of G.
- m denotes a representation in ASCII format that is intended for input to the MeatAxe.
- g denotes a representation/presentation in GAP format.
- b denotes a representation in MeatAxe binary format.
- M denotes a representation/presentation in Magma format.
Some information on the various formats has been provided on
- The M11G1-f2r44 indicates that we have representation of M11 on its (G1)
standard generators as 44 × 44 matrices over GF(2). Since the r44 comes
without a distinguishing letter, and there is a unique absolutely irreducible
representation of M11 of degree 44 in characteristic 2, the representation we
have is taken to be isomorphic to that one. Finally, the B0 indicates that this
is the 0th basis to which we have written such a representation, and the .m2
indicates that this is the 2nd generator in MeatAxe format.
- The M11G1-Ar10 indicates that we have representation of M11 on its (G1)
standard generators as 10 × 10 matrices over a ring of algebraic
integers other than Z. [But this does not indicate that it is
impossible to write an equivalent representation over Z.] Looking in
the ATLAS indicates that M11 has 3 absolutely irreducible representations in
characteristic 0, so the r10c indicates that we have the 3rd one of these.
This representation is written with respect to its B0-basis and is in
- The L211G1-Zr10 indicates that we have representation of L2(11) on its (G1)
standard generators as 10 × 10 matrices over Z. In characteristic
0, L2(11) has two absolutely irreducible and one other irreducible characters
of degree 10, so the r10c indicates that this is a copy the degree 10
representation with character irreducible over Q but not over
Q(b11). [There are no troublesome representations with non-trivial
Schur index for this group; these throw more spanners into the works.]
- The first presentation of M11 on its (G1) standard generators. This
presentation is in Magma format.
- A permutation representation U6(2) on its (G1) standard generators. This is
the second of the 3 primitive permutation representations of U6(2) on 1408
points. It is the first generator in GAP
- HS:2 on its (G1) standard generators as 22 × 22 matrices over
GF(2). There is no irreducible representation of degree 22 in characteristic
2 for HS:2 so there is a certain amount of ambiguity over what the precise
isomorphism class of this representation is (we have not invented naming
conventions to resolve such a difficulty). In this case, the representation
is the unique uniserial module of shape 20.1.1, but the notation does not even
imply indecomposability. It is the first generator in
Our word programs are stored in files whose names have the following form:
where <InputName> and <OutputName> consist solely of
alphanumeric characters and describe the desired inputs and outputs. Typically,
<InputName> is a valid <GrpName> and <OutputName> consists of a part indicating what the program does
(eg, cyc, ccls, max3,
a[ut]2p2) and a part Wi which
indicates that it is the ith version of such a program. An example is
U62G1-max4W2 which is the second word program to calculate a
representative of the 4th class of maximal subgroups of U6(2), starting from
its standard generators. (In this case, this subgroup is one of the 3 classes
of subgroups isomorphic to U4(3):2b, and we must make sure
that out choice of which one constitutes a representative of the 4th class
is compatible with infomation given in the ATLAS and elsewhere
on these pages.)
Further information about word programs and their naming can be
Go to main ATLAS (version 2.0) page.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 25th November 1999.
Last updated 05.02.02 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.