The IN-refinement of the inductive AM-condition from [Spä13a, Definitions 6.2 and 6.4] has been checked for the simple groups that are listed in the following table.
The list covers all sporadic simple groups and some simple groups with exceptional Schur multiplier.
| A5 | L3(2) | A6 | A7 | U3(3) |
| M11 | A8 | U4(2) | Sz(8) | M12 |
| J1 | M22 | J2 | S6(2) | G2(3) |
| M23 | 2F4(2)' | HS | J3 | M24 |
| G2(4) | McL | He | Ru | Suz |
| ON | Co3 | Co2 | Fi22 | HN |
| Ly | Th | Fi23 | Co1 | J4 |
| Fi24' | B | M |
For the Schur covers of each of these groups and each prime divisor p of the group order, the character tables of the normalizers of the defect groups of all p-blocks are available as GAP tables, except the Sylow 2- and 3-normalizer in 3.Fi24', 2.B, and M.
The structure of these Sylow normalizers are known, see [Wil98]. This information suffices for showing that the condition in question holds in these cases.
In some cases, the class fusion between the (character tables of the) subgroup and the group is not (yet) known. The condition in question holds for all possible class fusions in these cases.
The Inductive BAW condition from [Spä13b, Definition 4.1] has been checked for the simple groups in the above table, except for the groups J4, F24', B, and M; in these cases, the available information for p = 2 and p = 3 is incomplete.
Note that [AD12] shows the AWC-goodness of all sporadic simple groups, which is related to the non-blockwise version of the Alperin weight conjecture.
[AD12] An, J. and Dietrich, H., The AWC-goodness and essential rank of sporadic simple groups, J. Algebra, 356 (2012), 325–354.
[Spä13a] Späth, B., A reduction theorem for the Alperin–McKay conjecture, J. reine angew. Math., 680 (2013), 153–189.
[Spä13b] Späth, B., A reduction theorem for the blockwise Alperin weight conjecture, J. Group Theory, 16 (2) (2013), 159–220.
[Wil98] Wilson, R. A., The McKay conjecture is true for the sporadic simple groups, J. Algebra, 207 (1) (1998), 294–305.