Dade's Conjecture for Special Linear Groups in the Defining Characteristic

✍ Scribed by Hideki Sukizaki

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 171 KB
• Volume: 220
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1999.7926

✦ Synopsis

For such a p-chain C we denote m j=0 N G U j by N G C , and by C we denote the length m of C. Moreover, for a given p-block B of G and a non-negative integer d, let Irr N G C B d denote the set of irreducible characters χ of N G C , such that χ belongs to a block of N G C inducing B and such that p d is the highest power of p dividing N G C /χ 1 , that is, χ has defect d. Note that if two radical p-chains C and C are G-conjugate then Irr N G C B d = Irr N G C B d for all B and d. In [1] Dade gives the following conjecture.

Conjecture 1 (Dade's ordinary conjecture). Suppose that a finite group G satisfies O p G = 1 and that a p-block B of G is not of defect 0. Then

for all d > 0, where denotes the sum over a set of representatives of the G-conjugacy classes of radical p-chains of G.