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Let be G a finite group, let H <= G and n = [G:H]. Let ℤ Omega be the transitive permutation module on the set Omega = { Hg; g in G } of right cosets of H in G, and let 1_H^G in ℤ Irr_ℂ(G) be the corresponding permutation character. The G-set Omega is called multiplicity-free, if all constituents of 1_H^G occur with multiplicity 1.
Let Omega_i, for 1 <= i <= r, denote the orbits of H on Omega, also called the suborbits of G on Omega, where we assume Omega_1 = { H * 1 }, and where we let k_i = |Omega_i| be the index parameters. The number r of suborbits is called the rank of the permutation action of G on Omega. For 1 <= i <= r we choose omega_i in Omega_i and g_i in G such that omega_1 g_i = omega_i, where we assume g_1 = 1. The orbitals Gamma_i = { (omega_1 g, omega_i g); g in G }, for 1 <= i <= r, are precisely the orbits of the diagonal permutation action of G on Omega x Omega. For 1 <= i <= r let Gamma_{i^*} be defined by (omega_i, omega_1) in Gamma_{i^*}. Then Omega_{i^*} is called the suborbit paired to Omega_i, and we have k_{i^*} = k_i.
For 1 <= i <= r the i-th orbital graph is the simple directed graph with vertex set Omega and edge set Gamma_i. It is regular with valency k_i. Let A_i = [ a_{i, omega, omega'} ] in { 0, 1 }^{n x n}, with row index omega and column index omega', be its adjacency matrix, i. e. we have a_{i, omega, omega'} = 1 if and only if (omega, omega') in Gamma_i. The i-th orbital graph is undirected, i. e. A_i is a symmetric matrix, if and only if Omega_i is a self-paired suborbit.
By Schur's Theorem the set { A_i; 1 <= i <= r } is a ℤ-basis for the endomorphism ring E = End_{ℤG}(ℤ Omega). The algebra E^ℂ = E otimes_ℤ ℂ is split semisimple, and is commutative if and only if Omega is multiplicity-free. In this case, we have | Irr_ℂ(E^ℂ)| = r = dim_ℂ(E^ℂ), and varphi(A_1 otimes 1) = 1 for all varphi in Irr_ℂ(E^ℂ). The character table of E^ℂ is the matrix [ varphi(A_i otimes 1) ]_{varphi,i} in ℂ^{r x r}, with row index varphi in Irr_ℂ(E^ℂ) and column index 1 <= i <= r. The characters varphi in Irr_ℂ(E^ℂ) are in bijection with the constituents of 1_H^G, thus we order the rows of the character table such that the corresponding constituents of 1_H^G are ordered as in the ATLAS [CCNPW85]. In particular, this means that the index parameters k_i can be read off from the first row of the character table. We order the columns of the character table according to increasing index parameters k_i; note that this does not necessarily determine the order of the columns uniquely.
For 1 <= i <= r let P_i = [ p_{h,i,j} ] in ℤ^{r x r}, with row index 1 <= h <= r and column index 1 <= j <= r, be the representing matrix of A_i for its right regular action on E, with respect to the Schur basis of E. Hence the map E -> ℤ^{r x r}: A_i -> P_i, for 1 <= i <= r, is a faithful representation of E. The matrix P_i is called the i-th collapsed adjacency matrix or intersection matrix. . The matrix entries of P_i are given by p_{h,i,j} = |{ omega in Omega_h; (omega, omega_j) in Gamma_i }| = |Omega_h cap Omega_{i^*} g_j|. In particular, we have p_{1,i,i} = 1 and p_{1,i,j} = 0 for j <> i, as well as p_{i^*,i,1} = k_i and p_{h,i,1} = 0 for h <> i^*.
> MULTFREEINFO | ( global variable ) |
MULTFREEINFO is an immutable record. Its components are the Identifier (Reference: Identifiers) values of the GAP character tables of the sporadic simple groups and their cyclic and bicyclic extensions. The value of the component corresponding to the group G, say, is a list containing in the first position a string denoting the name of G in LaTeX format, and in each of the remaining positions a triple [ const, subgroup, ctbl ] where const is a list of positive integers, subgroup is a string that denotes the name of a subgroup H of G, in LaTeX format, and ctbl is either the matrix of irreducible characters of the endomorphism ring of the permutation module (1_H)^G or fail (indicating that the character table is not yet known). The sum of irreducible characters of G at the positions in const is a multiplicity-free permutation character of G that is induced from the trivial character of H.
The ordering of the entries corresponds to the ordering in [BL96], for the scope of the latter paper.
The complete list of faithful multiplicity-free permutation representations of the sporadic simple groups and their cyclic and bicyclic extensions is obtained as follows; there are exactly $396$ equivalence classes of these representations.
gap> infoall := MultFreeEndoRingCharacterTables( "all" );; gap> Length( infoall ); 396 gap> allnames := RecNames( MULTFREEINFO );; gap> list := List( allnames, x -> [ x, Length( MULTFREEINFO.(x) ) - 1 ] );; gap> PrintArray( list ); [ [ M11, 7 ], [ M12, 14 ], [ J1, 2 ], [ M22, 9 ], [ J2, 5 ], [ M23, 5 ], [ HS, 12 ], [ J3, 1 ], [ M24, 12 ], [ McL, 6 ], [ He, 4 ], [ Ru, 3 ], [ Suz, 6 ], [ ON, 4 ], [ Co3, 7 ], [ Co2, 10 ], [ Fi22, 9 ], [ HN, 4 ], [ Ly, 3 ], [ Th, 2 ], [ Fi23, 6 ], [ Co1, 6 ], [ J4, 2 ], [ F3+, 3 ], [ B, 4 ], [ M, 1 ], [ M12.2, 13 ], [ M22.2, 17 ], [ J2.2, 14 ], [ HS.2, 17 ], [ J3.2, 2 ], [ McL.2, 10 ], [ He.2, 4 ], [ Suz.2, 14 ], [ ON.2, 2 ], [ Fi22.2, 16 ], [ HN.2, 6 ], [ F3+.2, 5 ], [ 2.M12, 10 ], [ 2.M22, 4 ], [ 3.M22, 4 ], [ 4.M22, 0 ], [ 6.M22, 2 ], [ 12.M22, 0 ], [ 2.J2, 1 ], [ 2.HS, 5 ], [ 3.J3, 0 ], [ 3.McL, 1 ], [ 2.Ru, 1 ], [ 2.Suz, 1 ], [ 3.Suz, 4 ], [ 6.Suz, 1 ], [ 3.ON, 4 ], [ 2.Fi22, 7 ], [ 3.Fi22, 6 ], [ 6.Fi22, 6 ], [ 2.Co1, 2 ], [ 3.F3+, 2 ], [ 2.B, 1 ], [ 2.M12.2, 3 ], [ Isoclinic(2.M12.2), 1 ], [ 2.M22.2, 8 ], [ Isoclinic(2.M22.2), 2 ], [ 3.M22.2, 5 ], [ 4.M22.2, 0 ], [ Isoclinic(4.M22.2), 0 ], [ 6.M22.2, 6 ], [ Isoclinic(6.M22.2), 0 ], [ 12.M22.2, 0 ], [ Isoclinic(12.M22.2), 0 ], [ 2.J2.2, 0 ], [ Isoclinic(2.J2.2), 3 ], [ 2.HS.2, 2 ], [ Isoclinic(2.HS.2), 0 ], [ 3.J3.2, 0 ], [ 3.McL.2, 1 ], [ 2.Suz.2, 4 ], [ Isoclinic(2.Suz.2), 0 ], [ 3.Suz.2, 4 ], [ 6.Suz.2, 2 ], [ Isoclinic(6.Suz.2), 0 ], [ 3.ON.2, 0 ], [ 2.Fi22.2, 6 ], [ Isoclinic(2.Fi22.2), 5 ], [ 3.Fi22.2, 7 ], [ 6.Fi22.2, 3 ], [ Isoclinic(6.Fi22.2), 3 ], [ 3.F3+.2, 2 ] ] |
The above list shows for each group the number of pairwise inequivalent faithful multiplicity-free permutation representations. Currently there is only one case remaining where the character table of the endomorphism ring is not known. It is characterized by the value fail for the component ctbl.
gap> unknown := Filtered( infoall, x -> x.ctbl = fail );; gap> Length( unknown ); 1 gap> PrintArray( List( unknown, x -> [ x.name, x.rank, x.character[1] ] ) ); [ [ 2.B, 34, 2031941058560000 ] ] |
The multiplicity-free permutation characters of the sporadic simple groups and their cyclic and bicyclic extensions have been classified in
for the sporadic simple groups and their automorphism groups,
for the central extensions of the sporadic simple groups, and
for bicyclic extensions of the sporadic simple groups.
The following publications describe the corresponding endomorphism rings of groups G, in the sense that the collapsed adjacency matrices or the character tables of the endomorphism rings are given. (This information is contained in the sources component of the records returned by MultFreeEndoRingCharacterTables (1.4-1), it appears in the overview table shown by BrowseMultFreeActions (1.4-3), and also in the overview list and in the PDF files that are available at the package's www homepage.)
G = B, H = 2.^2E_6(2).2 and H = 2.^2E_6(2) < 2.^2E_6(2).2;
G = M, H = 2.B;
G = Fi_23, H = S_8(2) and H = 2^11.M_23; G = Fi_24'.2, H = 3^7.O_7(3).2 and H = O_10^-(2).2;
G = Co_1, H = 2_+^{1+8}.O_8^+(2);
for rank at most five and G a sporadic simple group or an automorphism group of a sporadic simple group;
G = J_4, H = 2^11: M_24;
for rank at least six and degree at most 10^7, and G a sporadic simple group or an automorphism group of a sporadic simple group;
G = HN, H = A_11 and H = U_3(8).3_1; G = HN.2, H = S_11 and H = U_3(8).6; G = Ly, H = 3.McL; G = Th, H = ^3D_4(2).3 and H = 2^5.L_5(2); G = J_4, H = 2^11: M_23; G = Fi_24', H = O_10^-(2) and H = 3^7.O_7(3); G = Fi_24'.2, H = O_10^-(2);
G = B, H = Fi_23;
G = B, H = 2^{1+22}.Co_2;
The remaining cases are assigned to [Linton] if G is perfect, and to [Mue08b] otherwise.
Besides these sources, the entry TOM means that the table of marks of the group in question is contained in the GAP library of tables of marks, and thus one can easily compute the permutation representation and the collapsed adjacency matrices, see MultFreeFromTOM (1.5-1).
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> MultFreeEndoRingCharacterTables( name ) | ( function ) |
For a string name that is the name of a sporadic simple group, or a cyclic or bicyclic extension of a sporadic simple group, MultFreeEndoRingCharacterTables returns a list of records that describe the character tables of the endomorphism rings of the faithful multiplicity-free permutation modules of this group, in a format that is similar to the classification shown in [BL96].
If name is the string "all" then MultFreeEndoRingCharacterTables returns the list of records for all sporadic simple groups and their cyclic or bicyclic extensions.
Each entry in the result list has the following components.
namethe group name name,
groupLaTeX format of name,
subgroupa string that is a name (in LaTeX format) of the subgroup from whose trivial character the permutation character is induced,
characterthe permutation character,
rankthe rank of the character,
charnmbsthe positions of the irreducible constituents of the permutation character in the ordinary character table of the group,
ATLASa string that describes (in LaTeX format) the constituents of the permutation character, relative to the perfect group involved; the format is described in the section PermCharInfoRelative (Reference: PermCharInfoRelative),
ctblthe matrix of irreducible characters of the endomorphism ring, where the rows are labelled by the irreducible constituents of the permutation character and the columns are labelled by the orbits of the subgroup on its right cosets, which are ordered according to their lengths,
matsthe collapsed adjacency matrices,
sourcesthe list of the "Label" strings of relevant entries in the package's bibliography, see Section 1.3, and
permGroupInfoa list of descriptions how the permutation representation in question can be constructed; this information is used by the function MultFreePermutationGroup (1.4-2).
In the one case where the character table is not (yet) known, the components ctbl and mats have the value fail.
For example, there are seven equivalence classes of faithful multiplicity-free permutation representations of the sporadic simple Mathieu group M_11.
gap> info := MultFreeEndoRingCharacterTables( "M11" );;
gap> Length( info );
7
gap> info[1];
rec( name := "M11",
group := "$M_{11}$",
subgroup := "$A_6.2_3$",
character := Character( CharacterTable( "M11" ),
[ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ),
rank := 2,
charnmbs := [ 1, 2 ],
ATLAS := "1a+10a",
ctbl := [ [ 1, 10 ], [ 1, -1 ] ],
mats := [ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, 1 ], [ 10, 9 ] ] ] )
gap> PrintArray( info[1].mats[2] );
[ [ 0, 1 ],
[ 10, 9 ] ]
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> MultFreePermutationGroup( record ) | ( function ) |
Let record be an entry of a list returned by MultFreeEndoRingCharacterTables (1.4-1). If the value of the permGroupInfo component of record is nonempty and if the information in this list admits the construction of a permutation group then the function MultFreePermutationGroup returns this group. Otherwise fail is returned.
> BrowseMultFreeActions( ) | ( function ) |
This function shows an overview of the database, in a browse table in visual mode (see Chapter Browse: Browsing Tables in GAP using ncurses –The User Interface).
BrowseMultFreeActions is available only if the GAP package Browse ([BL08]) is installed. The function can be also called by clicking the appropriate entry of the menu shown by BrowseGapData (Browse: BrowseGapData).
The full functionality of the function NCurses.BrowseGeneric (Browse: NCurses.BrowseGeneric) is available for the browse table. Additionally, the user inputs vb, vh, and vt can be used to open a pager showing the BibTeX, HTML, and Text format, respectively, of the values that appear in the "source" column of the selected row. Also the function BrowseBibliography (Browse: BrowseBibliography) has this feature.
> BrowseEndoRingCharacterTable( record ) | ( function ) |
Let record be an entry of a list returned by MultFreeEndoRingCharacterTables (1.4-1). The function BrowseEndoRingCharacterTable shows the character table in a format similar to the one that is printed by DisplayMultFreeEndoRingCharacterTable (1.4-5), but the table is shown in visual mode, which admits scrolling, searching, etc.; see Chapter Browse: Browsing Tables in GAP using ncurses –The User Interface for details.
BrowseEndoRingCharacterTable is available only if the GAP package Browse ([BL08]) is installed. BrowseEndoRingCharacterTable is called in BrowseMultFreeActions (1.4-3) when a row of the table shown by this function is "clicked".
> DisplayMultFreeEndoRingCharacterTable( record ) | ( function ) |
Let record be an entry of a list returned by MultFreeEndoRingCharacterTables (1.4-1). The function DisplayMultFreeEndoRingCharacterTable prints a formatted version of the character table to the screen.
The header gives the name of the group and of the point stabilizer; if the point stabilizer is not a maximal subgroup then also the name of a maximal subgroup containing the point stabilizer is given. The rows of the matrix are labelled by the irreducible constituents of the corresponding permutation character, and the columns are labelled by the orbits of the point stabilizer; note that the entries in the first row, which always corresponds to the trivial representation 1a, are exactly the lengths of these orbits. If the character table contains irrational values then these entries are displayed in a format as in the ATLAS [CCNPW85], and a legend shown below the table explains the values; ER (Reference: ER) is a GAP function that returns a square root of an integer.
See BrowseEndoRingCharacterTable (1.4-4) for a variant that uses the facilities of the GAP package Browse ([BL08]).
gap> info := MultFreeEndoRingCharacterTables( "M11" );;
gap> DisplayMultFreeEndoRingCharacterTable( info[1] );
G = M_{11}, H = A_6.2_3
| O_1 O_2
-------------
1a | 1 10
10a | 1 -1
gap> DisplayMultFreeEndoRingCharacterTable( info[4] );
G = M_{11}, H = 11:5 < L_2(11)
| O_1 O_2 O_3 O_4 O_5 O_6
-------------------------------------
1a | 1 11 11 11 55 55
11a | 1 -1 -1 11 -5 -5
16a | 1 -4-3b11 -1+3b11 -1 -5 10
16b | 1 -1+3b11 -4-3b11 -1 -5 10
45a | 1 3 3 -1 -5 -1
55a | 1 -1 -1 -1 7 -5
b11 = (-1+ER(-11))/2
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> MultFreeFromTOM( name ) | ( function ) |
For a string name that is a valid name for a table of marks in the GAP library of tables of marks, MultFreeFromTOM returns the list of pairs [ pi, G ] where pi is a multiplicity-free permutation character of the group for name, and G is the corresponding permutation group, as computed from the table of marks of this group.
If there is no table of marks for name or if the class fusion of the character table for name into the table of marks is not unique then fail is returned.
Note that the ordering of the permutation representations does not necessarily coincide with the one given by MultFreeEndoRingCharacterTables (1.4-1), and that the list returned by MultFreeFromTOM contains the trivial representation as well.
gap> multfree:= MultFreeFromTOM( "M11" );; gap> Length( multfree ); 8 gap> multfree[8]; [ [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ], Group(()) ] gap> multfree[7]; [ Character( CharacterTable( "M11" ), [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ), Group([ (1,7)(2,9)(3,11)(5,6), (1,8,5,3)(2,10,6,4) ]) ] |
> CollapsedAdjacencyMatricesInfo( G[, order][, S] ) | ( function ) |
Let G be a transitive permutations group, omega be the smallest point moved by G, and S the stabilizer of omega in G. The function CollapsedAdjacencyMatricesInfo returns a record that describes the collapsed adjacency matrices of the permutation module. The components are
matsthe collapsed adjacency matrices,
pointsa list of representatives of the orbits of S on the set of the points moved by G,
representativesa list of elements in G such that the i-th entry maps omega to the i-th entry in the points list.
The order of G can be given as the optional argument order, and the group S can be given as the optional argument S.
gap> multfree:= MultFreeFromTOM( "M11" );;
gap> CollapsedAdjacencyMatricesInfo( multfree[7][2] );
rec( mats := [ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, 1 ], [ 10, 9 ] ] ],
points := [ 1, 2 ], representatives := [ (), (1,2)(5,7)(6,9)(8,11) ] )
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> CollapsedAdjacencyMatricesFromCharacterTable( mat ) | ( function ) |
Let mat be the character table of an endomorphism algebra. The function CollapsedAdjacencyMatricesFromCharacterTable returns ...
The collapsed adjacency matrix of the orbital graph that corresponds to the j-th column of mat is given by mat^{-1} * D_j * mat, where D_j is the diagonal matrix whose diagonal is the j-th column of mat.
gap> info := MultFreeEndoRingCharacterTables( "M11" );; gap> CollapsedAdjacencyMatricesFromCharacterTable( info[1].ctbl ); [ [ [ 1, 0 ], [ 0, 1 ] ], [ [ 0, 1 ], [ 10, 9 ] ] ] |
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