Abstract
In O. Brunat and J. Gramain (2014) recently proved that any two blocks of double covers of symmetric groups are Broue perfectly isometric provided they have the same weight and sign. They also proved a corresponding statement for double covers of alternating groups and Broue perfect isometries between double covers of symmetric and alternating groups when the blocks have opposite signs. Using both the results and methods of O. Brunat and J. Gramain in this paper we prove that when the weight of a block of a double cover of a symmetric or alternating group is less than p then the block is Broue perfectly isometric to its Brauer correspondent. This means that Broue’s perfect isometry conjecture holds for the double covers of the symmetric and alternating groups.We also explicitly construct the characters of these Brauer correspondents which may be of independent interest to the reader.
Highlights
Broue’s abelian defect group conjecture postulates that every block with abelian defect is derived equivalent to its Brauer correspondent
Gramain proved an analogue of [5, Theorem 7.2] at the level of characters for the double covers. In other words they proved that any two blocks of double covers of symmetric groups with the same weight and sign are Broueperfectly isometric
They proved a corresponding statement for double covers of alternating groups and Broueperfect isometries between double covers of symmetric and alternating groups when the blocks have opposite signs
Summary
Broue’s abelian defect group conjecture postulates that every block with abelian defect is derived equivalent to its Brauer correspondent. Gramain proved an analogue of [5, Theorem 7.2] at the level of characters for the double covers (see [2, Theroem 4.21]) In other words they proved that any two blocks of double covers of symmetric groups with the same weight and sign are Broueperfectly isometric. They proved a corresponding statement for double covers of alternating groups and Broueperfect isometries between double covers of symmetric and alternating groups when the blocks have opposite signs The main notion they employed in their proof was that of an MN-structure. In this paper we develop an MN-structure for the Brauer correspondent of a block of a double cover This involves very explicitly constructing the characters of such a group.
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