M(4,2,2) - Revision history

Charles Eaton at 14:47, 4 January 2019

2019-01-04T14:47:52Z

2018-11-22T15:31:59Z

2018-10-31T19:49:42Z

2018-10-31T17:56:18Z

2018-10-31T17:32:14Z

2018-10-31T17:05:54Z

2018-09-21T08:19:03Z

‎Projective indecomposable modules

2018-09-21T08:09:56Z

Covered and covering blocks

2018-09-21T08:06:12Z

Rearranged labelling for simples

2018-08-31T12:25:29Z

@@ Line 25: / Line 25: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>\mathcal{T}(B)=C_2</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
+|Pic-O = <math>\mathcal{T}(B)=C_2</math><ref>Every Morita equivalence is a source algebra equivalence by [[References#C|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
 |PIgroup = <math>S_4 \times C_2</math><ref>This follows from the result for [[M(4,2,3)]] since blocks in these classes are derived equivalent and so perfectly isometric</ref>
 |source? = Yes

@@ Line 8: / Line 8: @@
 |l(B) = 3
 |k-morita-frob = 1
-|Pic-k= <math>k^* \wr C_2</math>
+|Pic-k= <math>k^* \wr C_2</math><ref>This is an elementary calculation</ref>
 |cartan = <math>\left( \begin{array}{ccc}
 & 2 & 1 \\
@@ Line 26: / Line 26: @@
 |O-morita-frob = 1
 |Pic-O = <math>\mathcal{T}(B)=C_2</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
-|PIgroup = <math>S_4 \times C_2</math>
+|PIgroup = <math>S_4 \times C_2</math><ref>This follows from the result for [[M(4,2,3)]] since blocks in these classes are derived equivalent and so perfectly isometric</ref>
 |source? = Yes
 |sourcereps = <math>B_0(\mathcal{O}A_5)</math>

@@ Line 34: / Line 34: @@
 |coveringblocks = M(4,2,2) (complete)
 |coveredblocks = M(4,2,2) (complete)
-|pcoveringblocks = [[M(8,5,2)]] (complete)<ref>This follows from Theorem 2.15 of [[References|[EL18a]]] since a covering block cannot have defect group <math>C_4 \times C_2</math>.</ref>
+|pcoveringblocks = [[M(8,3,2)]], [[M(8,5,2)]] (complete)<ref>A covering block cannot have defect group <math>C_4 \times C_2</math>.</ref>
 }}

@@ Line 32: / Line 32: @@
 |k-derived = [[M(4,2,3)]]
 |O-derived-known? = Yes
-|coveringblocks = M(4,2,2)
+|coveringblocks = M(4,2,2) (complete)
-|coveredblocks = M(4,2,2)
+|coveredblocks = M(4,2,2) (complete)
 |pcoveringblocks = [[M(8,5,2)]] (complete)<ref>This follows from Theorem 2.15 of [[References|[EL18a]]] since a covering block cannot have defect group <math>C_4 \times C_2</math>.</ref>
 }}

@@ Line 25: / Line 25: @@
 \end{array}\right)</math>
 |O-morita-frob = 1
-|Pic-O = <math>\mathcal{T}(B)=C_2</math>
+|Pic-O = <math>\mathcal{T}(B)=C_2</math><ref>Every Morita equivalence is a source algebra equivalence by [[References|[CEKL13]]], so <math>{\rm Pic}(B)=\mathcal{T}(B)</math></ref>
 |PIgroup = <math>S_4 \times C_2</math>
 |source? = Yes

@@ Line 60: / Line 60: @@
     \begin{array}{ccc}
       & S_2 & \\
-       S_1 & & S_3 \\
+       \begin{array}{c} S_1 \\ S_2 \\ S_3 \\ \end{array} & \oplus & \begin{array}{c} S_3 \\ S_2 \\ S_1 \\ \end{array} \\
       & S_2 & \\
     \end{array},

@@ Line 31: / Line 31: @@
 |k-derived = [[M(4,2,3)]]
 |O-derived-known? = Yes
 }}
@@ Line 42: / Line 44: @@
 == Other notatable representatives ==
 == Projective indecomposable modules ==

@@ Line 10: / Line 10: @@
 |Pic-k= <math>k^* \wr C_2</math>
 |cartan = <math>\left( \begin{array}{ccc}
 & 2 & 1 \\
-& 1 & 2 \\
+& 4 & 2 \\
 \end{array} \right)</math>
 |defect-morita-inv? = Yes
@@ Line 19: / Line 19: @@
 |O-morita = <math>B_0(\mathcal{O}A_5)</math>
 |decomp = <math>\left( \begin{array}{ccc}
-& 0 & 0 \\
+& 1 & 0 \\
 & 1 & 0 \\
-& 0 & 1 \\
+& 1 & 1 \\
 & 1 & 1 \\
 \end{array}\right)</math>
@@ Line 54: / Line 54: @@
 <math>\begin{array}{ccc}
-   \begin{array}{ccc}
+  \begin{array}{c}
-     & S_1 & \\
+       S_1 \\
        S_2 \\
        S_3 \\
        S_1 \\
-       S_2 \\
+   \end{array},
     \end{array},
+&
     \begin{array}{c}
        S_3 \\
        S_1 \\
        S_2 \\
        S_3 \\
     \end{array}

@@ Line 1: / Line 1: @@
 {{blockbox
 |title = M(4,2,2) - <math>B_0(kA_5)</math>
-|image =
+|image = M(4,2,2)quiver.png
 |representative =  <math>B_0(kA_5)</math>
-|defect = <math>C_2 \times C_2</math>
+|defect = [[C2xC2|<math>C_2 \times C_2</math>]]
 |inertialquotients = <math>C_3</math>
 |k(B) = 4
@@ Line 83: / Line 83: @@
 All irreducible characters have height zero.