ATLAS: Nomenclature of the representations (etc)

Representations and presentations

<GrpName>-<RepName>.<FormatName>

<GrpName>

• 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, . . . , g_n) 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.

<RepName>

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 finalised.

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):

• 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, Z_n = 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 without notice.

• 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.

<FormatName>

• 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 examples

M11G1-f2r44B0.m2

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.

M11G1-Ar10cB0.M

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 Magma format.

L211G1-Zr10cB0.M

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.]

M11G1-P1.M

The first presentation of M11 on its (G1) standard generators. This presentation is in Magma format.

U62G1-p1408bB0.g1

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 format.

HSd2G1-f2r22B0.m1

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 MeatAxe format.

Word programs

<InputName>-<OutputName>

Further information about word programs and their naming can be found here.

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