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| | == Blocks with basic algebras of dimension at most 12 == | | == Blocks with basic algebras of dimension at most 12 == |
| | + | [[Blocks with basic algebras of low dimension|Main article: Blocks with basic algebras of low dimension]] |
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| − | In [[References|[Li18]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9. | + | In [[References|[Li18]]] Markus Linckelmann calculated the <math>k</math>-algebras of dimension at most twelve which occur as basic algebras of blocks of finite groups, with the exception of one case of dimension 9 where no block with that basic algebra is identified. See [[Blocks with basic algebras of low dimension]] for a description of these results. |
| − | | |
| − | {| class="wikitable"
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| − | |-
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| − | ! scope="col"| Dimension
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| − | ! scope="col"| Class
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| − | ! scope="col"| Defect group
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| − | ! scope="col"| Representative
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| − | ! scope="col"| <math>\dim_k(Z(A))</math>
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| − | ! scope="col"| <math>l(A)</math>
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| − | ! scope="col"| Notes
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| − | |-
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| − | | 1 || [[M(1,1,1)]] || <math>1</math> || <math>k1</math> || 1 || 1 || Blocks of defect zero
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| − | |-
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| − | | 2 || [[M(2,1,1)]] || <math>C_2</math> || <math>kC_2</math> || 2 || 1 ||
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| − | |-
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| − | | 3 || [[M(3,1,1)]] || <math>C_3</math> || <math>kC_3</math> || 3 || 1 ||
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| − | |-
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| − | | 4 || [[M(4,1,1)]] || <math>C_4</math> || <math>kC_4</math> || 4 || 1 ||
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| − | |-
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| − | | 4 || [[M(4,2,1)]] || <math>C_2 \times C_2</math> || <math>k(C_2 \times C_2)</math> || 4 || 1 ||
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| − | |-
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| − | | 5 || [[M(5,1,1)]] || <math>C_5</math> || <math>kC_5</math> || 5 || 1 ||
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| − | |-
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| − | | 6 || [[M(3,1,2)]] || <math>C_3</math> || <math>kS_3</math> || 3 || 2 ||
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| − | |-
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| − | | 7 || [[M(5,1,3)]] || <math>C_5</math> || <math>B_0(kA_5)</math> || 4 || 2 ||
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| − | |-
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| − | | 7 || [[M(7,1,1)]] || <math>C_7</math> || <math>kC_7</math> || 7 || 1 ||
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| − | |-
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| − | | 8 || [[M(8,1,1)]] || <math>C_8</math> || <math>kC_8</math> || 8 || 1 ||
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| − | |-
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| − | | 8 || [[M(8,2,1)]] || <math>C_4 \times C_2</math> || <math>k(C_4 \times C_2)</math> || 8 || 1 ||
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| − | |-
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| − | | 8 || [[M(8,3,1)]] || <math>D_8</math> || <math>kD_8</math> || 5 || 1 ||
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| − | |-
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| − | | 8 || [[M(8,4,1)]] || <math>Q_8</math> || <math>kQ_8</math> || 5 || 1 ||
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| − | |-
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| − | | 8 || [[M(8,5,1)]] || <math>C_2 \times C_2 \times C_2</math> || <math>k(C_2 \times C_2 \times C_2)</math> || 8 || 1 ||
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| − | |-
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| − | | 8 || [[M(7,1,3)]] || <math>C_7</math> || <math>B_0(kPSL_2(13))</math> || 5 || 2 ||
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| − | |-
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| − | | 9 || [[M(9,1,1)]] || <math>C_9</math> || <math>kC_9</math> || 9 || 1 ||
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| − | |-
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| − | | 9 || [[M(9,1,3)]] || <math>C_9</math> || <math>B_0(kSL_2(8))</math> || 6 || 2 ||
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| − | |-
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| − | | 9 || [[M(9,2,1)]] || <math>C_3 \times C_3</math> || <math>k(C_3 \times C_3)</math> || 9 || 1 ||
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| − | |-
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| − | | 9 || [[M(9,2,2)]] || <math>C_3 \times C_3</math> || Faithful block of <math>k((C_3 \times C_3):D_8)</math>, in which <math>Z(D_8)</math> acts trivially || 6 || 1 ||
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| − | |-
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| − | | 9 || || <math>C_3 \times C_3</math> || ? || 6 || 2 || Unknown
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| − | |-
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| − | | 10 || [[M(5,1,2)]] || <math>C_5</math> || <math>kD_{10}</math> || 4 || 2 ||
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| − | |-
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| − | | 10 || [[M(11,1,3)]] || <math>C_{11}</math> || <math>B_0(kSL_2(32))</math> || 7 || 2 ||
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| − | |-
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| − | | 11 || [[M(8,3,3)]] || <math>D_8</math> || <math>kS_4</math> || 5 || 2 ||
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| − | |-
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| − | | 11 || [[M(7,1,6)]] || <math>C_7</math> || <math>B_0(kA_7)</math> || 5 || 3 ||
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| − | |-
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| − | | 11 || [[M(11,1,1)]] || <math>C_{11}</math> || <math>kC_{11}</math> || 11 || 1 ||
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| − | |-
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| − | | 11 || [[M(13,1,3)]] || <math>C_{13}</math> || <math>B_0(kPSL_2(25))</math> || 8 || 2 ||
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| − | |-
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| − | | 12 || [[M(4,2,3)]] || <math>C_2 \times C_2</math> || <math>kA_4</math> || 4 || 3 ||
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| − | |}
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