[CTblLib] 3 Functions for Character Table Constructions

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# 3 Functions for Character Table Constructions

### Sections

1. Attributes for Character Table Constructions
2. Character Tables of Groups of Structure MGA
3. Character Tables of Groups of Structure GS3
4. Character Tables of Coprime Central Extensions
5. Construction Functions used in the Character Table Library

The functions in this chapter deal with the construction of character tables from other character tables. So they fit to the functions in Section Constructing Character Tables from Others in the GAP Reference Manual. But since they are used in situations that are typical for the GAP Character Table Library, they are described here.

An important ingredient of the constructions is the description of the action of a group automorphism on the classes by a permutation. In practice, these permutations are usually chosen from the group of table automorphisms of the character table in question (see AutomorphismsOfTable in the GAP Reference Manual).

Section Character Tables of Groups of Structure MGA deals with groups of structure M.G.A, where the upwards extension G.A acts suitably on the central extension M.G. Section Character Tables of Groups of Structure GS3 deals with groups that have a factor group of type S3. Section Character Tables of Coprime Central Extensions deals with special cases of the construction of character tables of central extensions from known character tables of suitable factor groups. Section Construction Functions used in the Character Table Library documents the functions used to encode certain tables in the GAP Character Table Library.

Examples can be found in Auto.

## 3.1 Attributes for Character Table Constructions

- `ConstructionInfoCharacterTable(` tbl`) A`

  If this attribute is set for an ordinary character table tbl then the value is a list that describes how this table was constructed. The first entry is a string that is the identifier of the function that was applied to the pre-table record; the remaining entries are the arguments for that functions, except that the pre-table record must be prepended to these arguments.

  ## 3.2 Character Tables of Groups of Structure MGA

- `PossibleCharacterTablesOfTypeMGA(` tblMG`,` tblG`,` tblGA`,` aut`,` identifier`) F`

  Let H be a group with normal subgroups N and M such that H/N is cyclic, M £ N holds, and such that each irreducible character of N that does not contain M in its kernel induces irreducibly to H. (This is satisfied for example if N has prime index in H and M is a group of prime order that is central in N but not in H.) Let G = N/M and A = H/N, so H has the structure M.G.A.

  Let tblMG, tblG, tblGA be the ordinary character tables of the groups M.G, G, and G.A, respectively, and aut the permutation of classes of tblMG induced by the action of H on M.G. Furthermore, let the class fusions from tblMG to tblG and from tblG to tblGA be stored on tblMG and tblG, respectively (see StoreFusion in the GAP Reference Manual).

  `PossibleCharacterTablesOfTypeMGA` returns a list of records describing all possible character tables for groups H that are compatible with the arguments. Note that in general there may be several possible groups H, and it may also be that ``character tables'' are constructed for which no group exists. Each of the records in the result has the following components.

  `table`
    the ordinary character table of a possible table for H, and

  `MGfusMGA`
    the fusion map from tblMG into the table stored in `table`.
  The possible tables differ w.r.t. some power maps, and perhaps element orders and table automorphisms; in particular, the `MGfusMGA` component is the same in all records.

  The returned tables have the `Identifier` value identifier. The classes of these tables are sorted as follows. First come the classes contained in M.G, sorted compatibly with the classes in tblMG, then the classes in H \M.G follow, in the same ordering as the classes of G.A \G.

- `PossibleActionsForTypeMGA(` tblMG`,` tblG`,` tblGA`) F`

  Let the arguments be as described for `PossibleCharacterTablesOfTypeMGA` (see PossibleCharacterTablesOfTypeMGA). `PossibleActionsForTypeMGA` returns the set of those table automorphisms (see AutomorphismsOfTable in the GAP Reference Manual) of tblMG that can be induced by the action of H on M.G.

  Information about the progress is reported if the info level of `InfoCharacterTable` is at least 1 (see SetInfoLevel in the GAP Reference Manual).

  ## 3.3 Character Tables of Groups of Structure GS3

- `CharacterTableOfTypeGS3(` tbl`,` tbl2`,` tbl3`,` aut`,` identifier`) F`
- `CharacterTableOfTypeGS3(` modtbl`,` modtbl2`,` modtbl3`,` ordtbls3`,` identifier`) F`

  Let H be a group with a normal subgroup G such that H/G @ S3, the symmetric group on three points, and let G.2 and G.3 be preimages of subgroups of order 2 and 3, respectively, under the natural projection onto this factor group.

  In the first form, let tbl, tbl2, tbl3 be the ordinary character tables of the groups G, G.2, and G.3, respectively, and aut the permutation of classes of tbl3 induced by the action of H on G.3. Furthermore assume that the class fusions from tbl to tbl2 and tbl3 are stored on tbl (see StoreFusion in the GAP Reference Manual).

  In the second form, let modtbl, modtbl2, modtbl3 be the p-modular character tables of the groups G, G.2, and G.3, respectively, and ordtbls3 the ordinary character table of H.

  `CharacterTableOfTypeGS3` returns a record with the following components.

  `table`
    the ordinary or p-modular character table of H, respectively,

  `tbl2fustbls3`
    the fusion map from tbl2 into the table of H, and

  `tbl3fustbls3`
    the fusion map from tbl3 into the table of H.

  The returned table of H has the `Identifier` value identifier. The classes of the table of H are sorted as follows. First come the classes contained in G.3, sorted compatibly with the classes in tbl3, then the classes in H \G.3 follow, in the same ordering as the classes of G.2 \G.

- `PossibleActionsForTypeGS3(` tbl`,` tbl2`,` tbl3`) F`

  Let the arguments be as described for `CharacterTableOfTypeGS3` (see CharacterTableOfTypeGS3). `PossibleActionsForTypeGS3` returns the set of those table automorphisms (see AutomorphismsOfTable in the GAP Reference Manual) of tbl3 that can be induced by the action of H on the classes of tbl3.

  Information about the progress is reported if the info level of `InfoCharacterTable` is at least 1 (see InfoCharacterTable in the GAP Reference Manual).

  ## 3.4 Character Tables of Coprime Central Extensions

- `CharacterTableOfCommonCentralExtension(` tblG`,` tblmG`,` tblnG`,` id`) F`

  Let tblG be the ordinary character table of a group G, say, and let tblmG and tblnG be the ordinary character tables of central extensions m·G and n·G of G by cyclic groups of prime orders m and n, respectively, with m ¹ n. We assume that the factor fusions from tblmG and tblnG to tblG are stored on the tables. `CharacterTableOfCommonCentralExtension` returns a record with the following components.

  `tblmnG`
    the character table t, say, of the corresponding central extension of G by a cyclic group of order m n that factors through m·G and n·G; the `Identifier` value of this table is id,

  `IsComplete`
    `true` if the `Irr` value is stored in t, and `false` otherwise,

  `irreducibles`
    the list of irreducibles of t that are known; it contains the inflated characters of the factor groups m·G and n·G, plus those irreducibles that were found in tensor products of characters of these groups.

  Note that the conjugacy classes and the power maps of t are uniquely determined by the input data. Concerning the irreducible characters, we try to extract them from the tensor products of characters of the given factor groups by reducing with known irreducibles and applying the LLL algorithm (see ReducedClassFunctions and LLL in the GAP Reference Manual).

  ## 3.5 Construction Functions used in the Character Table Library

  The following functions are used in the GAP Character Table Library, for encoding table constructions via the mechanism that is based on the attribute `ConstructionInfoCharacterTable` (see ConstructionInfoCharacterTable). All construction functions take as their first argument a record that describes the table to be constructed, and the function adds only those components that are not yet contained in this record.

- `ConstructMGA(` tbl`,` subname`,` factname`,` plan`,` perm`) F`

  `ConstructMGA` constructs the ordinary character table tbl of a group m.G.a where the automorphism a (a group of prime order) of m.G acts notrivially on the central subgroup m of m.G. subname is the name of the subgroup m.G which is a (not necessarily cyclic) central extension of the (not necessarily simple) group G, factname is the name of the factor group G.a. Then the faithful characters of tbl are induced characters of m.G.

  plan is a list, each entry being a list containing positions of characters of m.G that form an orbit under the action of a (so the induction of characters is simulated).

  perm is the permutation that must be applied to the list of characters that is obtained on appending the faithful characters to the inflated characters of the factor group. A nonidentity permutation occurs for example for groups of structure 12.G.2 that are encoded via the subgroup 12.G and the factor group 6.G.2, where the faithful characters of 4.G.2 shall precede those of 6.G.2.

  Examples where `ConstructMGA` is used to encode library tables are the tables of 3.F3+.2 (subgroup 3.F3+, factor group F3+.2) and 121.U4(3).22 (subgroup 121.U4(3), factor group 61.U4(3).22).

- `ConstructMGAInfo(` tblmGa`,` tblmG`,` tblGa`) F`

  Let tblmGa be the ordinary character table of a group of structure m.G.a where the factor group of prime order a acts nontrivially on the normal subgroup of order m that is central in m.G, tblmG the character table of m.G, and tblGa the character table of the factor group G.a.

  `ConstructMGAInfo` returns the list that is to be stored in the library version of tblmGa: the first entry is the string `"ConstructMGA"`, the remaining four entries are the last four arguments for the call to `ConstructMGA` (see ConstructMGA).

- `ConstructGS3(` tbls3`,` tbl2`,` tbl3`,` ind2`,` ind3`,` ext`,` perm`) F`
- `ConstructGS3Info(` tbl2`,` tbl3`,` tbls3`) F`

  `ConstructGS3` constructs the irreducibles of an ordinary character table tbls3 of type G.S3 from the tables with names tbl2 and tbl3, which correspond to the groups G.2 and G.3, respectively. ind2 is a list of numbers referring to irreducibles of tbl2. ind3 is a list of pairs, each referring to irreducibles of tbl3. ext is a list of pairs, each referring to one irreducible of tbl2 and one of tbl3. perm is a permutation that must be applied to the irreducibles after the construction.

  `ConstructGS3Info` returns a record with the components `ind2`, `ind3`, `ext`, `perm`, and `list`, as are needed for `ConstructGS3`.

- `ConstructV4G(` tbl`,` facttbl`,` aut`[,` ker`] ) F`

  Let tbl be the character table of a group of type 22.G where an outer automorphism of order 3 permutes the three involutions in the central 22. Let aut be the permutation of classes of tbl induced by that automorphism, and facttbl the name of the character table of the factor group 2.G. Then `ConstructV4G` constructs the irreducible characters of tbl from that information.

  The optional argument ker is an integer denoting the position of the nontrivial class of the table of 2.G that lies in the kernel of the epimorphism onto G; the default for ker is 2.

- `ConstructProj(` tbl`,` irrinfo`) F`
- `ConstructProjInfo(` tbl`,` kernel`) F`

  `ConstructProj` constructs the irreducible characters of record encoding the ordinary character table tbl from projective characters of tables of factor groups, which are stored in the `ProjectivesInfo` (see ProjectivesInfo) value of the smallest factor; the information about the name of this factor and the projectives to take is stored in irrinfo.

  `ConstructProjInfo` takes an ordinary character table tbl and a list kernel of class positions of a cyclic kernel of order dividing 12, and returns a record with the components

  `tbl`
    a character table that is permutation isomorphic with tbl, and sorted such that classes that differ only by multiplication with elements in the classes of kernel are consecutive,

  `projectives`
    a record being the entry for the `projectives` list of the table of the factor of tbl by kernel, describing this part of the irreducibles of tbl, and

  `info`
    the value of irrinfo.

  In order to encode a library table t as a ``projective table'' relative to another library table f, say, one has to do the following. First the factor fusion from t to f must be stored on the table of t, and t is written to a library file. Then the result of `ConstructProjInfo`, called for t and the kernel of the factor fusion, is used as follows. The list containing `"ConstructProj"` at its first position and the `info` component is added as last entry of the `MOT` call for this library version. The `projectives` component is added to the `ProjectivesInfo` list of f, and a new library version of f is produced (this contains the new projectives via an `ARC` call). Finally, `etc/maketbl` is called in order to store the projection for the factor fusion in the `ctprimar.tbl` data.

- `ConstructDirectProduct(` tbl`,` factors`) F`
- `ConstructDirectProduct(` tbl`,` factors`,` permclasses`,` permchars`) F`

  is a special case of a `construction` call for a library table tbl.

  The direct product of the tables described in the list factors is constructed, and all its components stored not yet in tbl are added to tbl.

  The `computedClassFusions` component of tbl is enlarged by the factor fusions from the direct product to the factors.

  If the optional arguments permclasses, permchars are given then classes and characters of the result are sorted accordingly.

  factors must have length at least two; use `ConstructPermuted` (see ConstructPermuted) in the case of only one factor.

- `ConstructSubdirect(` tbl`,` factors`,` choice`) F`

  The library table tbl is completed with help of the table obtained by taking the direct product of the tables with names in the list factors, and then taking the table consisting of the classes in the list choice.

  Note that in general, the restriction to the classes of a normal subgroup is not sufficient for describing the irreducible characters of this normal subgroup.

- `ConstructIsoclinic(` tbl`,` factors`) F`
- `ConstructIsoclinic(` tbl`,` factors`,` nsg`) F`

  constructs first the direct product of library tables as given by the list factors, and then constructs the isoclinic table of the result.

- `ConstructPermuted(` tbl`,` libnam`[,` prmclasses`,` prmchars`] ) F`

  The library table tbl is completed with help of the library table with name libnam, whose classes and characters must be permuted by the permutations prmclasses and prmchars, respectively.

- `ConstructFactor(` tbl`,` libnam`,` kernel`) F`

  The library table tbl is completed with help of the library table with name libnam, by factoring out the classes in the list kernel.

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