# Cases for which the conjecture has been verified

In what follows, A will be a block of RG with abelian defect group D, and B will be its Brauer correspondent, a block of RNG(D), so that the conjecture predicts that A and B should have equivalent derived categories.

Unless stated otherwise, in these cases the conjecture is know to be induced by a `split endomorphism two-sided tilting complex'.

## General families of blocks

• All blocks with cyclic defect group.
• All blocks of p-solvable groups.
• Dade [Dad], Harris and Linckelmann [HarLin]. In this case, the blocks A and B are Morita equivalent.
• All blocks with defect group C2×C2.
• All principal blocks with defect group C3×C3.
• Koshitani and Kunugi [pre-KosKun1]. This depends on the classification of finite simple groups, and uses results of Okuyama and others mentioned below for specific groups, and general methods due to Marcus [Mar8].

## Symmetric groups and related groups

• All blocks of symmetric groups with defect groups of order p2.
• All blocks of symmetric groups with abelian defect groups of order at most p5.
• The principal block of A4, p=2. (Defect group C2×C2.)
• The principal blocks of A6, A7 and A8, p=3. (Defect group C3×C3.)

## Sporadic groups and related groups

• The principal block of the Mathieu group M11, p=3. (Defect group C3×C3.)
• The principal block of the Mathieu group M22, p=3. (Defect group C3×C3.)
• The principal block of the Mathieu group M23, p=3. (Defect group C3×C3.)
• The principal block of the Higman-Sims group HS, p=3. (Defect group C3×C3.)
• The principal block of the Janko group J1, p=2. (Defect group C2×C2×C2.)
• The principal block of the Hall-Janko group HJ=J2, p=5. (Defect group C5×C5.)
• The non-principal block of the O'Nan group O'N with defect group C3×C3.
• The non-principal block of the Higman-Sims group HS with defect group C3×C3.
• The non-principal block of the double cover 2.J2 of the Hall-Janko group with defect group C5×C5.

## Groups of Lie type in the defining characteristic

• The principal block of PSL2(pn) in characteristic p. (Defect group Cpn.)
• Okuyama [pre-Oku2] for the general case; earlier Chuang [Chu1] for the case n=2.
• The non-principal block with full defect of SL2(p2) in characteristic p. (Defect group Cp×Cp.)

## Groups of Lie type in non-defining characteristic

• The principal block in characteristic p of the group G=GF of rational points of a connected reductive group G defined over the field Fq of q elements, where p is a prime dividing q-1 but not dividing the order of the Weyl group of G.
• This follows from a result of Puig [Pui4] ([Remark]). In this case the blocks A and B are Morita equivalent.

This general theorem includes, for p=3, some cases where G has Sylow 3-subgroup C3×C3, which were needed for the theorem of Koshitani and Kunugi. Namely: G=Sp4(q), where q=4 or 7 (mod 9); G=PSU4(q2) or PSU5(q2), where q=4 or 7 (mod 9) [see also Koshitani and Miyachi [pre-KosMiy]]. Of course, Puig's theorem also deals with the case q=1 (mod 9), but then the Sylow p-subgroup is larger than C3×C3.

• The principal blocks of the Ree groups R(32n+1)=2G2(32n+1), p=2. (Defect group C2×C2×C2.)
• It follows from work of Landrock and Michler [LanMic] that the principal blocks of R(32n+1) are all Morita equivalent (for different values of n). This reduces the conjecture to the case n=0, which is known, since R(3)=SL2(8).3.
• The principal blocks of SU3(q2), where q is a prime power, and p>3 is a prime dividing q+1. (Defect group Cr×Cr, where r=pa is the largest power of p dividing q+1.)
• The principal blocks of PSL3(q), p=3, where q=4 or 7 (mod 9). (Defect group C3×C3.)
• Kunugi [Kun3] proves that these blocks are all Morita equivalent. The conjecture then follows since Okuyama [pre-Oku1] proved the case of PSL3(4).
• The principal blocks of Sp4(q), p=3, where q=2 or 5 (mod 9). (Defect group C3×C3.)
• It follows from work of Okuyama and Waki [OkuWak] that these blocks are all Morita equivalent. The conjecture then follows, because Sp4(2) is isomorphic to the symmetric group S6.
• The principal block of Sp4(4), p=5. (Defect group C5×C5.)
• The principal blocks of PSU3(q2), p=3, where q=2 or 5 (mod 9). (Defect group C3×C3.)
• Koshitani and Kunugi [pre-KosKun2] prove that the blocks A and B are Morita equivalent.
• The principal blocks of GL4(q), p=3, where q=2 or 5 (mod 9). (Defect group C3×C3.)
• Koshitani and Miyachi [KosMiy] show that these blocks are all Morita equivalent. The conjecture then follows since Okuyama [pre-Oku1] proved the case of GL4(2), which is isomorphic to A8.
• The principal blocks of GL5(q), p=3, where q=2 or 5 (mod 9). (Defect group C3×C3.)
• Koshitani and Miyachi [KosMiy] prove that the blocks A and B are Morita equivalent.
• Unipotent blocks of weight 2 of general linear groups in non-defining characteristic p. (Defect group Cr×Cr, for r=pa some power of p.)