A Database of blocks with abelian defect group

Mary Schaps, with Ayala Bar-Ilan, Tzvia Berrebi, Mikhal Cohen, and Ruthi Leabovich

A complete description of the collection of databases and its connection to the Broue conjectures is given in the article "Explicit Tilting Complexes", which will appear in the proceedings of Groups-St. Andrews 2005, and in the meantime is on the website of Mary (Malka) Schaps. The main database is a set of hypertext indices to of data sheets for various blocks. The main databases, blocks with elementary abelian defect group, are:

SortAt_3.html, of 3-blocks
SortAt_5.html, of 5-blocks
SortAt_7.html, of 7-blocks
AbAt_11.html, of 11-blocks
AbAt_13.html.

The first three are sorted according to two important invariants of the block B, the number k(B) of ordinary characters in the blocks and the number l(B) of simple modules in the block. They are further sorted according to the numbers on the diagonal of the Cartan matrix, in ascending order. Finally, they were sorted by the degrees of the ordinary irreducible characters, reduced by division by the gcd, in ascending order. In all cases we could check, blocks with the same reduced degrees had the same decomposition matrix and thus the same sorted Cartan diagonal, since the Cartan matrix is computed from the decomposition matrix.

The decomposition matrix was not stored in GAP 4.2 for every block, and when it was not, we could not calculate the Cartan matrix, so some of those blocks occur with the heading [ ]. There were other blocks for which the decomposition matrix was not stored, but for which the degrees of the irreducible characters corresponded to the degrees for some other blocks. In this case we conjectured that the decomposition matrices were the same, up to a possible permutation of the rows. These blocks are marked with an "*" before the list of reduced degrees.

The blocks are connected by links, at the end of the block entry. Clicking on a link will bring a Morita equivalent block B', from a group G' of size less than or equal to the size of the group G of the original block B, to the top of the page. This block may also be linked to another block, and so on. The links are coded by symbols explaining the reason for the Morita equivalence:

IN: Inflation. Dividing G by a normal subgroup will give G', and the characters of the block are all trivial on the normal subgroup.
CL: Clifford theory. The block is induced from a normal subgroup.
MU: Direct Product. The group G is a direct product of G' and another group G''. The group G" has a block of defect zero whose degree is the common quotient of the degrees of the characters of B and corersponding characters of B'.
MO: Morita Theorem. G has a normal subgroup N. The block B corresponds to a conjugacy class of blocks of N. There is a subgroup H with is the stabilizer of one of these, and the block B' is Morita equivalent to a twisted group algebra over H/N.
ISO: Isomorphic.
SC : Morita equivalence related to Scopes reduction.
The links lead back to blocks for which we have tried to provide more information that on most of the data sheets, where possible.
references to proven cases, [Oku1],[Hol] etc.,can be found on Jeremy Rickard's Abelian Defect page:
http://www.maths,bris.ac.uk/~majcr/adgc/adgc.html or at the bottom of this page.
The database contains 1031blocks.

k(B) = 6, l(B) = 2

(C4X(C3XC3):C2):C2)3-block 3
(C3XC3):C23-block 1
(C3XC3):C43-block 2
[ 2, 9 ]
- 2.A6.2_1 3 -block 4, [ 2, 1, 1, 1, 1, 4, 5 ]
[ 5, 5 ]
- 2^6.A7 3 -block 8, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 4^5.A6 3 -block 22, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 4^6.A7 3 -block 18, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 4^6.A7 3 -block 20, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 4^6.A7 3 -block 22, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 4^6.A7 3 -block 51, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 4^6.A7 3 -block 52, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 4^6.A7 3 -block 53, *[ 2, 1, 1, 2, 2, 2, 2 ]
- P21/G3/L2/V1/ext2 3 -block 13, *[ 2, 1, 1, 2, 2, 2, 2 ]
- P21/G3/L5/V1/ext2 3 -block 13, *[ 2, 1, 1, 2, 2, 2, 2 ]
- 2.A11 3 -block 5, [ 2, 1, 10, 11, 11, 11, 11 ]
- 2.A13 3 -block 5, [ 2, 2, 11, 13, 13, 13, 13 ]
- (3xG2(4)).2 3 -block 5, [ 3, 1, 1, 2, 2, 2, 2 ]
- (3xL2(25)).2_2 3 -block 6, [ 3, 1, 1, 2, 2, 2, 2 ]
- (3x2^(1+6)_-.U4(2)).2 3 -block 5, *[ 3, 1, 1, 2, 2, 2, 2 ]
- (3x2^(2+8):(A5xS3)).2 3 -block 3, *[ 3, 1, 1, 2, 2, 2, 2 ]
- (3x2^(2+8):(A5xS3)).2 3 -block 4, *[ 3, 1, 1, 2, 2, 2, 2 ]
- (A4x3.L3(4).2_3).2 3 -block 2, *[ 3, 1, 1, 2, 2, 2, 2 ]
- (A4xG2(4)):2 3 -block 9, *[ 3, 1, 1, 2, 2, 2, 2 ]
- 2^(5+8):(S3xA6) 3 -block 6, *[ 3, 1, 1, 2, 2, 2, 2 ]
- 2^(6+8):(A7xS3) 3 -block 21, *[ 3, 1, 1, 2, 2, 2, 2 ]
- 2^(6+8):(A7xS3) 3 -block 9, *[ 3, 1, 1, 2, 2, 2, 2 ]
- (3xU5(2)).2 3 -block 2, [ 4, 1, 1, 2, 2, 2, 2 ]
- (A4x3):2 3 -block 1, [ 4, 1, 1, 2, 2, 2, 2 ]
- 2^(1+4)+:3^2.2 3 -block 1, [ 4, 1, 1, 2, 2, 2, 2 ]
- 2^(1+4)+:3^2.2 3 -block 6, [ 4, 1, 1, 2, 2, 2, 2 ]
- 2^(2+4):(3x3):2 3 -block 1, [ 4, 1, 1, 2, 2, 2, 2 ]
- 2^4:(3x3):2 3 -block 1, [ 4, 1, 1, 2, 2, 2, 2 ]
- 3^2:2 3 -block 1, [ 4, 1, 1, 2, 2, 2, 2 ], ?
- U72CT 3 -block 14, *[ 4, 1, 1, 2, 2, 2, 2 ]
- U72CT 3 -block 2, *[ 4, 1, 1, 2, 2, 2, 2 ]
- (3x2^(4+6):3A6).2 3 -block 10, **[ 4, 1, 1, 2, 2, 2, 2 ]
- (3x2^(4+6):3A6).2 3 -block 11, **[ 4, 1, 1, 2, 2, 2, 2 ]
- (3x2^(4+6):3A6).2 3 -block 2, **[ 4, 1, 1, 2, 2, 2, 2 ]
- (3xO8+(3):3):2 3 -block 2, **[ 4, 1, 1, 2, 2, 2, 2 ]
- (A4xO8+(2).3).2 3 -block 3, **[ 4, 1, 1, 2, 2, 2, 2 ]
- (A4xO8+(2).3).2 3 -block 4, **[ 4, 1, 1, 2, 2, 2, 2 ]
[ 5, 9 ]
- 2.A8 3 -block 5, [ 2, 1, 7, 7, 7, 7, 8 ], EQ
- 2.A7.2 3 -block 4, [ 2, 2, 5, 5, 5, 5, 7 ], ?
[ ]
- 2.SuzM4 3 -block 16, [ 2, 1, 1, 1, 1, 4, 5 ]
- 5^(1+4):4S6 3 -block 7, [ 2, 1, 1, 1, 1, 4, 5 ]
- 5^(1+4):4S6 3 -block 8, [ 2, 1, 1, 1, 1, 4, 5 ]
- 2^5.psl(5,2) 3 -block 12, [ 2, 1, 7, 7, 7, 7, 8 ]
- Isoclinic(2.A7.2) 3 -block 4, [ 2, 2, 5, 5, 5, 5, 7 ]

A Database of blocks with abelian defect group

Mary Schaps, with Ayala Bar-Ilan, Tzvia Berrebi, Mikhal Cohen, and Ruthi Leabovich

k(B) = 6, l(B) = 2

[ 2, 9 ]

[ 5, 5 ]

[ 5, 9 ]

[ ]