48.3 Brauer Table Records

Brauer table records are similar to the records which represent ordinary character tables. They contain many of the well--known record components, like identifier, centralizers, irreducibles etc.; but there are two kinds of differences:

First, the operations record is BrauerTableOps instead of CharTableOps (see Operations Records for Character Tables). Second, there are two extra components, namely

ordinary, which contains the ordinary character table corresponding to the Brauer table, and

blocks, which reflects the block information; it is a list of records with components

defect:

the defect of the block,

ordchars:

a list of integers indexing the ordinary irreducibles in the block,

modchars:

a list of integers indexing the Brauer characters in the block,

basicset:

a list of integers indexing the ordinary irreducibles of a basic set; note that the indices refer to the positions in the whole irreducibles list of the ordinary table, not to the positions in the block,

decinv:

the inverse of the restriction of the decomposition matrix of the block to the basic set given by the basicset component, and possibly

brauertree:

if exists, a list that represents the decomposition matrix which in this case is viewed as incidence matrix of a tree (the so--called Brauer tree); the entries of the list correspond to the edges of the tree, they refer to positions in the block, not in the whole irreducibles list of the tables. Brauer trees are mainly used to store the information in a more compact way than by decomposition matrices, planar embeddings etc. are not (or not yet) included.

Note that Brauer tables in the library have different format (see Organization of the Table Libraries).

We give an example:

    gap> PrintCharTable( CharTable( "M11" ) mod 11 );
    rec( identifier := "M11mod11", text := "origin: modular ATLAS of finit\
    e groups, tests: DEC, TENS", prime := 11, size :=
    7920, centralizers := [ 7920, 48, 18, 8, 5, 6, 8, 8 ], orders :=
    [ 1, 2, 3, 4, 5, 6, 8, 8 ], classes :=
    [ 1, 165, 440, 990, 1584, 1320, 990, 990 ], powermap :=
    [ , [ 1, 1, 3, 2, 5, 3, 4, 4 ], [ 1, 2, 1, 4, 5, 2, 7, 8 ],,
      [ 1, 2, 3, 4, 1, 6, 8, 7 ],,,,,, [ 1, 2, 3, 4, 5, 6, 7, 8 ]
     ], fusions := [ rec(
          name := "M11",
          map := [ 1, 2, 3, 4, 5, 6, 7, 8 ],
          type := "choice" ) ], irreducibles :=
    [ [ 1, 1, 1, 1, 1, 1, 1, 1 ], [ 9, 1, 0, 1, -1, -2, -1, -1 ],
      [ 10, -2, 1, 0, 0, 1, E(8)+E(8)^3, -E(8)-E(8)^3 ],
      [ 10, -2, 1, 0, 0, 1, -E(8)-E(8)^3, E(8)+E(8)^3 ],
      [ 11, 3, 2, -1, 1, 0, -1, -1 ], [ 16, 0, -2, 0, 1, 0, 0, 0 ],
      [ 44, 4, -1, 0, -1, 1, 0, 0 ], [ 55, -1, 1, -1, 0, -1, 1, 1 ]
     ], irredinfo := [ rec(
           ), rec(
           ), rec(
           ), rec(
           ), rec(
           ), rec(
           ), rec(
           ), rec(
           ) ], blocks := [ rec(
          defect := 1,
          ordchars := [ 1, 2, 3, 4, 6, 7, 9 ],
          modchars := [ 1, 2, 3, 4, 6 ],
          decinv :=
           [ [ 1, 0, 0, 0, 0 ], [ -1, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ],
              [ 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 1 ] ],
          basicset := [ 1, 2, 3, 4, 6 ],
          brauertree :=
           [ [ 1, 2 ], [ 2, 7 ], [ 3, 7 ], [ 4, 7 ], [ 5 .. 7 ] ] ), rec(
          defect := 0,
          ordchars := [ 5 ],
          modchars := [ 5 ],
          decinv := [ [ 1 ] ],
          basicset := [ 5 ] ), rec(
          defect := 0,
          ordchars := [ 8 ],
          modchars := [ 7 ],
          decinv := [ [ 1 ] ],
          basicset := [ 8 ] ), rec(
          defect := 0,
          ordchars := [ 10 ],
          modchars := [ 8 ],
          decinv := [ [ 1 ] ],
          basicset := [ 10 ] )
     ], ordinary := CharTable( "M11" ), operations := BrauerTableOps, orde\
    r := 7920, name := "M11mod11", automorphisms := Group( (7,8) ) ) 

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