Cases for which the conjecture has been verified

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

Thanks to Naoko Kunugi for help in preparing this page.

General families of blocks

• All blocks with cyclic defect group.
  • Rickard [Ric1], Linckelmann [Lin2] and Rouquier [Rou3], [Rou1].
• 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.
  • Chuang [Chu2].
• All blocks of symmetric groups with abelian defect groups of order at most p⁵.
  • Chuang and Kessar [pre-ChuKes].
• The principal block of A₄, p=2. (Defect group C₂×C₂.)
  • Rickard [Ric6].
• The principal blocks of A₆, A₇ and A₈, p=3. (Defect group C₃×C₃.)
  • Okuyama [pre-Oku1].

Sporadic groups and related groups

• The principal block of the Mathieu group M₁₁, p=3. (Defect group C₃×C₃.)
  • Okuyama [pre-Oku1].
• The principal block of the Mathieu group M₂₂, p=3. (Defect group C₃×C₃.)
  • Okuyama [pre-Oku1].
• The principal block of the Mathieu group M₂₃, p=3. (Defect group C₃×C₃.)
  • Okuyama [pre-Oku1].
• The principal block of the Higman-Sims group HS, p=3. (Defect group C₃×C₃.)
  • Okuyama [pre-Oku1].
• The principal block of the Janko group J₁, p=2. (Defect group C₂×C₂×C₂.)
  • Gollan and Okuyama [pre-GolOku].
• The principal block of the Hall-Janko group HJ=J₂, p=5. (Defect group C₅×C₅.)
  • Holloway [pre-Hol].
• The non-principal block of the O'Nan group O'N with defect group C₃×C₃.
  • Koshitani, Kunugi and Waki [pre-KosKunWak].
• The non-principal block of the Higman-Sims group HS with defect group C₃×C₃.
  • Holm [Hol1], Koshitani, Kunugi and Waki [pre-KosKunWak].
• The non-principal block of the double cover 2.J₂ of the Hall-Janko group with defect group C₅×C₅.
  • Holloway [pre-Hol].

Groups of Lie type in the defining characteristic

• The principal block of PSL₂(pⁿ) in characteristic p. (Defect group C_pⁿ.)
  • 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 C_p×C_p.)
  • Holloway [pre-Hol].

Groups of Lie type in non-defining characteristic

• The principal block in characteristic p of the group G=G^F of rational points of a connected reductive group G defined over the field F_q 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 C_r×C_r, where r=p^a is the largest power of p dividing q+1.)
  • Kunugi and Waki [KunWak].
• 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.)
  • Holloway [pre-Hol].
• 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 C_r×C_r, for r=p^a some power of p.)
  • Hida and Miyachi [pre-HidMiy], Turner [pre-Tur].