For most users, this chapter contains everything you need to calculate the basic algebra which is Morita equivalent to a block of a group algebra.
The variables in this section are given default values that should work for most users. Users should feel free to adjust these variables to suit there needs. The variables BasicWorkingSpace (2.1-1) and BasicMemAvailable (2.1-2) describe where external files will be saved and how much memory the Basic package assumes is available on your computer. Make sure that BasicWorkingSpace (2.1-1) points to a directory that has plenty of free space. This package creates a lot of temporary files, so it is a bad idea to redirect this variable to a directory containing files that will get lost among the garbage.
AutoCalcBasic (2.2-1) will change its methods depending on the amount of memory available and the amount of memory it thinks it will need. The differences between these methods are trading time for space. So increasing BasicMemAvailable (2.1-2) can help with larger groups. One should keep in mind that BasicMemAvailable (2.1-2) should not be the total memory on the machine (please leave some room for at least the operating system).
> BasicWorkingSpace | ( global variable ) |
BasicWorkingSpace is the directory where the data files for the MeatAxe will be stored. After loading the Basic package, you can change this directory by making it a readwrite variable using the function MakeReadWriteGlobal (Reference: MakeReadWriteGlobal) and then using the function Directory (Reference: Directory) as in the following example.
gap> BasicWorkingSpace;
dir("/tmp/tmp.JmXqQu/")
gap> MakeReadWriteGlobal("BasicWorkingSpace");
gap> BasicWorkingSpace:=Directory("/tmp/GapData/");
dir("/tmp/GapData/")
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> BasicMemAvailable | ( global variable ) |
BasicMemAvailable is a list where the second entry is the maximum amount of memory available to GAP in kilobytes. The basic package assumes this is also the maximum amount of memory available to the MeatAxe. If this is not the case, this number should be changed. The amount of memory available to the MeatAxe may change some of the methods used by the basic package. If you want GAP to access more memory, see Command line options (Reference: Command Line Options).
To use the automated form of the Basic package, you need three pieces of data. First, you need a group. If possible, you should use the Atlas name for your group. For a list of all the named groups see DisplayAtlasInfo (AtlasRep: DisplayAtlasInfo). If your group does not appear in this list, or you have a different description for some reason, you can use a group as long as IsGroup (Reference: IsGroup) returns true.
The argument prime needs to be an integer such that IsPrime (Reference: IsPrime) returns true and which divides the order of your group.
The integer blocknumber needs to be between 1 and the length of the list returned by BlocksInfo (Reference: BlocksInfo). While this program will return the basic algebra for any block, the blocks of defect zero are trivial and can be computed faster by hand.
> AutoCalcBasic( group, prime, blocknumber ) | ( function ) |
> AutoCalcBasic( groupname, prime, blocknumber ) | ( function ) |
Returns: a list containing the basic algebra record as the third element. The first entry in this list describes the generators of the condensation algebra, see FindAlgGens (3.1-4), and the second entry gives the words for the generators for the condensation subgroup inside its normalizer and also the words for any prime Sylow elements that might have been used, see FindCondSubgroup (3.1-2)
This function can be called either with a group group or with the Atlas name of a group, groupname. For a list of named groups, see DisplayAtlasInfo (AtlasRep: DisplayAtlasInfo). AutoCalcBasic is the automation of the Basic package. The idea is to construct the Projective Indecomposable Modules (PIMs) contained in the blocknumber block of a condensation algebra and use the relevant homomorphisms between these PIMs to construct the basic algebra. Blocks are labeled as returned by BlocksInfo (Reference: BlocksInfo). prime is the characteristic of the basic algebra and should divide the order of groupname or group.
gap> basalg:=AutoCalcBasic("M11",2,1);;
gap> basalg[1];
[ rec( genlist := [ 2, 6 ] ) ]
gap> basalg[2];
[ [ "ab^{2}a^{-1}b^{2}", "a^{-2}" ],
[ "a^{2}ba^{-2}", "aba^{-2}b^{-1}", "a^{-1}bab", "b^{2}a^{2}" ] ]
gap> RecNames(basalg[3]);
[ "group", "generators", "npims", "pimnames", "cartan", "field", "dim",
"adjmat", "1a", "2a", "4a", "1a2a1", "1a4a1", "2a1a1", "2a2a1", "4a1a1",
"4a4a1", "matrices" ]
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See also BlocksInfo (Reference: BlocksInfo), DisplayAtlasInfo (AtlasRep: DisplayAtlasInfo), FindAlgGens (3.1-4), FindCondSubgroup (3.1-2), and RecNames (Reference: RecNames).
Before using the function AutoCalcBasic with a particular group, it might be useful to run InfoKond (2.2-2). The condensation subgroup used by AutoCalcBasic is the one returned by InfoKond (2.2-2). For information on using a different condensation subgroup, see FindFaithBurn (3.1-1).
> InfoKond( group, prime, blocknumber ) | ( function ) |
> InfoKond( groupname, prime, blocknumber ) | ( function ) |
Returns: a record containing information about a particular condensation subalgebra of the group algebra. The components of this record are:
the permutation character of the conjugacy class of the condensation subgroup.
the scalar product of permchar with itself.
the number in the table of marks of the conjugacy class of the condensation subgroup.
the cartan matrix for the block of the group algebra given by blocknumber.
the dimensions of the modular characters of the group which are in the block.
the splitting field of the group.
the dimensions of the simple modules in the condensed algebra.
the dimensions of the projective indecomposable modules in the condensed algebra.
This function uses the record returned by FindFaithBurn (3.1-1) to select a condensation subgroup which will give the smallest condensation algebra for the block blocknumber of the group algebra. InfoKond can be called with either group being a group, or groupname a string name of a group. Using either method, the group being referred to must have both a character table, see CharacterTable (Reference: CharacterTable), and a table of marks, see TableOfMarks (Reference: TableOfMarks), in GAP. prime is a prime number which divides the order of the group.
gap> grpinfo:=InfoKond("M11",2,1);
rec( permchar := [ 880, 0, 16, 0, 0, 0, 0, 0, 0, 0 ], norm := [ [ 112 ] ],
numberinburn := 13, cartan := [ [ 4, 2, 2 ], [ 2, 5, 1 ], [ 2, 1, 3 ] ],
dimmodchars := [ 1, 10, 44 ], splitf := 2, dimfixmod := [ 1, 2, 4 ],
dimfixproj := [ 16, 16, 16 ] )
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