Difference between revisions of "Classification by p-group"
(→Blocks for p=2: [KoLa20] reference) |
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Line 31: | Line 31: | ||
|8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | |8 || [[C4xC2|2]] || [[C4xC2|<math>C_4 \times C_2</math>]] ||1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || || | ||
|- | |- | ||
− | |8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || | + | |8 || [[D8|3]] || [[D8|<math>D_8</math>]] ||6(?) || <math>k</math> || <math>k</math> || [[References|[Er87] ]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] |
|- | |- | ||
|8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || | |8 || [[Q8|4]] || [[Q8|<math>Q_8</math>]] ||3(3) || <math>\mathcal{O}</math> || <math>k</math> || [[References|[Er88a], [Er88b], [HKL07], [Ei16]]] || | ||
Line 63: | Line 63: | ||
|16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || | |16 || [[M16|6]] || [[M16|<math>M_{16}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References|[CG12], [Sa12b] ]] || | ||
|- | |- | ||
− | |16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || | + | |16 || [[D16|7]] || [[D16|<math>D_{16}</math>]] || <math>k</math>|| 5(?) || <math>k</math> || <math>k</math> || [[References|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] |
|- | |- | ||
|16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 8(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes | |16 || [[SD16|8]] || [[SD16|<math>SD_{16}</math>]] || <math>k</math> || 8(?) || || || [[References|[Er88c], [Er90b]]] || Two other possible classes | ||
Line 129: | Line 129: | ||
|32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || | |32 || [[M32|17]] || [[M32|<math>M_{32}</math>]] || <math>\mathcal{O}</math> || 1(1) || <math>\mathcal{O}</math> || <math>\mathcal{O}</math> || [[References#C|[CG12], [Sa12b]]] || | ||
|- | |- | ||
− | |32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || | + | |32 || [[D32|18]] || [[D32|<math>D_{32}</math>]] || <math>k</math> || 5(?) || <math>k</math> || <math>k</math> || [[References#E|[Er87]]] || Principal blocks classified up to source algebra equivalence in [[References#K|[KoLa20]]] |
|- | |- | ||
|32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || || | |32 || [[SD32|19]] || [[SD32|<math>SD_{32}</math>]] || <math>k</math> || || || || || |
Revision as of 18:08, 13 January 2020
Classification of Morita equivalences for blocks with a given defect group
On this page we list classifications of Morita equivalence classes for each isomorphism class of p-groups in turn. Generic classifications for classes of p-groups can be found here.
See this page for a description of the labelling conventions.
Contents
Blocks for [math] p=2 [/math]
[math]1 \leq |D| \leq 8[/math] | |||||||
[math]|D|[/math] | SmallGroup | Isotype | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
---|---|---|---|---|---|---|---|
1 | 1 | [math]1[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
2 | 1 | [math]C_2[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
4 | 1 | [math]C_4[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
4 | 2 | [math]C_2 \times C_2[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Er82], [Li94] | |
8 | 1 | [math]C_8[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
8 | 2 | [math]C_4 \times C_2[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
8 | 3 | [math]D_8[/math] | 6(?) | [math]k[/math] | [math]k[/math] | [Er87] | Principal blocks classified up to source algebra equivalence in [KoLa20] |
8 | 4 | [math]Q_8[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]k[/math] | [Er88a], [Er88b], [HKL07], [Ei16] | |
8 | 5 | [math]C_2 \times C_2 \times C_2[/math] | 8(8) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Ea16] | Uses CFSG |
[math]|D|=16[/math] | ||||||||
[math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t)? | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
---|---|---|---|---|---|---|---|---|
16 | 1 | [math]C_{16}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
16 | 2 | [math]C_4 \times C_4[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKKS14] | |
16 | 3 | MNA(2,1) | No | 3(?) | No | [Sa11] | Block invariants known | |
16 | 4 | [math]C_4:C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
16 | 5 | [math]C_8 \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
16 | 6 | [math]M_{16}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
16 | 7 | [math]D_{16}[/math] | [math]k[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | Principal blocks classified up to source algebra equivalence in [KoLa20] |
16 | 8 | [math]SD_{16}[/math] | [math]k[/math] | 8(?) | [Er88c], [Er90b] | Two other possible classes | ||
16 | 9 | [math]Q_{16}[/math] | No | 6(?) | [math]k[/math] | [Er88a], [Er88b], [Ho97] | Two possibly infinite families when [math]l(B)=2[/math]. Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16] | |
16 | 10 | [math]C_4 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL18a] | |
16 | 11 | [math]D_8 \times C_2[/math] | No | [Sa12] | Block invariants known | |||
16 | 12 | [math]Q_8 \times C_2[/math] | No | [Sa13] | Block invariants known | |||
16 | 13 | [math]D_8*C_4[/math] | No | 3(?) | No | [Sa13b] | Block invariants known | |
16 | 14 | [math](C_2)^4[/math] | [math]\mathcal{O}[/math] | 16(16) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Ea18] |
The table for defect groups of order 32 takes as its starting point Table 13.1 of Sambale's book [Sa14].
[math]|D|=32[/math] | ||||||||
[math]|D|[/math] | SmallGroup | Isotype | Donovan (w.r.t)? | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
---|---|---|---|---|---|---|---|---|
32 | 1 | [math]C_{32}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
32 | 2 | [math]MNA(2,2)[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKS12] | |
32 | 3 | [math]C_8 \times C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
32 | 4 | [math]C_8:C_4[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
32 | 5 | [math]MNA(3,1)[/math] | No | [Sa11] | Invariants known | |||
32 | 6 | [math]MNA(2,1):C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 7 | SmallGroup(32,7) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | [math]M_{16}:C_2[/math] |
32 | 8 | [math]2.MNA(2,1)[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 9 | [math]D_8:C_4[/math] | No | [Sa14,10.23] | Invariants known | |||
32 | 10 | [math]Q_8:C_4[/math] | No | [Sa14,10.25] | Invariants known | |||
32 | 11 | [math]C_4 \wr C_2[/math] | No | [Ku80] | Invariants known | |||
32 | 12 | [math]C_4:C_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
32 | 13 | [math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^3b \rangle[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
32 | 14 | [math]C_8:C_4=\langle a,b|a^8=b^4=1, ba=a^7b \rangle[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
32 | 15 | SmallGroup(32,15) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
32 | 16 | [math]C_{16} \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
32 | 17 | [math]M_{32}[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [CG12], [Sa12b] | |
32 | 18 | [math]D_{32}[/math] | [math]k[/math] | 5(?) | [math]k[/math] | [math]k[/math] | [Er87] | Principal blocks classified up to source algebra equivalence in [KoLa20] |
32 | 19 | [math]SD_{32}[/math] | [math]k[/math] | |||||
32 | 20 | [math]Q_{32}[/math] | No | [Er88a], [Er88b], [Ho97] | Two possibly infinite families when [math]l(B)=2[/math]. Classified over [math]\mathcal{O}[/math] when [math]l(B)=3[/math] in [Ei16] | |||
32 | 21 | [math]C_4 \times C_4 \times C_2[/math] | [math]\mathcal{O}[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EKKS14] | |
32 | 22 | [math]MNA(2,1) \times C_2[/math] | No | [Sa14,10.25] | Invariants known | |||
32 | 23 | [math](C_4:C_4) \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 24 | SmallGroup(32,24) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 25 | [math]D_8 \times C_4[/math] | No | [Sa14,9.7] |
Invariants known | |||
32 | 26 | [math]Q_8 \times C_4[/math] | No | [Sa14,9.28] | Invariants known | |||
32 | 27 | SmallGroup(32,27) | No | |||||
32 | 28 | SmallGroup(32,28) | No | [Sa14,13.11] | Invariants known | |||
32 | 29 | SmallGroup(32,29) | No | [Sa14,13.11] | Invariants known | |||
32 | 30 | SmallGroup(32,30) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 31 | SmallGroup(32,31) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 32 | SmallGroup(32,32) | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 33 | SmallGroup(32,33) | No | [Sa14,13.12] | Invariants partly known | |||
32 | 34 | SmallGroup(32,34) | No | [Sa14,13.12] | Invariants partly known | |||
32 | 35 | [math]C_4:Q_8[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 36 | [math]C_8 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [EL18a] | |
32 | 37 | [math]M_{16} \times C_2[/math] | [math]\mathcal{O}[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | [Sa14] | |
32 | 38 | [math]D_8*C_8[/math] | No | [Sa14,9.18] | Invariants known | |||
32 | 39 | [math]D_{16} \times C_2[/math] | No | [Sa14,9.7] | Invariants known | |||
32 | 40 | [math]SD_{16} \times C_2[/math] | No | [Sa14,9.37] | Invariants known | |||
32 | 41 | [math]Q_{16} \times C_2[/math] | No | [Sa14,9.28] | Invariants known | |||
32 | 42 | [math]D_{16}*C_4[/math] | No | [Sa14,9.18] | Invariants known | |||
32 | 43 | SmallGroup(32,43) | No | |||||
32 | 44 | SmallGroup(32,44) | No | |||||
32 | 45 | [math]C_4 \times C_2 \times C_2 \times C_2[/math] | [math]\mathcal{O}[/math] | [Sa14, 13.9] | Invariants known | |||
32 | 46 | [math]D_8 \times C_2 \times C_2[/math] | No | |||||
32 | 47 | [math]Q_8 \times C_2 \times C_2[/math] | No | |||||
32 | 48 | [math](D_8*C_4) \times C_2[/math] | No | |||||
32 | 49 | [math]D_8*D_8[/math] | No | [Sa13c] | Invariants partly known | |||
32 | 50 | [math]D_8*Q_8[/math] | No | [Sa13c] | Invariants partly known | |||
32 | 51 | [math](C_2)^5[/math] | [math]\mathcal{O}[/math] | 34 (34) | [math]\mathcal{O}[/math] | [Ar19] | Derived eq. classes determined for 30 of the 34 Morita eq. classes. |
Blocks for [math]p=3[/math]
[math]1 \leq |D| \leq 27[/math] | |||||||
[math]|D|[/math] | SmallGroup | Isotype | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
---|---|---|---|---|---|---|---|
1 | 1 | [math]1[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
3 | 1 | [math]C_3[/math] | 2(2) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
9 | 1 | [math]C_9[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
9 | 2 | [math]C_3 \times C_3[/math] | |||||
27 | 1 | [math]C_{27}[/math] | 3(3) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
27 | 2 | [math]C_9 \times C_3[/math] | |||||
27 | 3 | [math]3_+^{1+2}[/math] | |||||
27 | 4 | [math]3_-^{1+2}[/math] | |||||
27 | 5 | [math]C_3 \times C_3 \times C_3[/math] |
Blocks for [math]p=5[/math]
[math]5 \leq |D| \leq 25[/math] | |||||||
[math]|D|[/math] | SmallGroup | Isotype | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
---|---|---|---|---|---|---|---|
1 | 1 | [math]1[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
5 | 1 | [math]C_5[/math] | 6(6) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
25 | 1 | [math]C_{25}[/math] | 6(6) | No | [math]\mathcal{O}[/math] | Max 12 classes | |
25 | 2 | [math]C_5 \times C_5[/math] |
Blocks for [math]p\geq 7[/math]
[math]|D|[/math] | |||||||
[math]|D|[/math] | SmallGroup | Isotype | Known [math]k[/math]-([math]\mathcal{O}[/math]-)classes | Complete (w.r.t.)? | Derived equiv classes (w.r.t)? | References | Notes |
---|---|---|---|---|---|---|---|
1 | 1 | [math]1[/math] | 1(1) | [math]\mathcal{O}[/math] | [math]\mathcal{O}[/math] | ||
7 | 1 | [math]C_7[/math] | 14(14) | No | [math]\mathcal{O}[/math] | Max 21 classes | |
11 | 1 | [math]C_{11}[/math] | No | [math]\mathcal{O}[/math] | |||
13 | 1 | [math]C_{13}[/math] | No | [math]\mathcal{O}[/math] | |||
17 | 1 | [math]C_{17}[/math] | No | [math]\mathcal{O}[/math] | |||
19 | 1 | [math]C_{19}[/math] | No | [math]\mathcal{O}[/math] | |||
23 | 1 | [math]C_{23}[/math] | No | [math]\mathcal{O}[/math] |