Chapter 1
Faithful Condensation for Sporadic Groups

In this chapter, we determine a subgroup H suitable for condensation in the cases not covered in the example sections of Chapter . For the cases where the socle series of the projective indecomposable modules in the principal block B can be found in the literature we give an explicit generating set for the condensation algebra eHFGeH. This allows us to recover the results on the layers of the socle series but it also allows us to use Morita equivalence when we want to study a given module in B. The cases where the layers of the socle series are not yet known could in principle be attacked with the method presented in Chapter  but lack of time and lack of computer power and memory have prevented us from doing so. In each of the following sections we proceed as described in chapter . Finally, we give an explicit generating set for the condensation algebra eHFGeH. These generators are condensations of elements given as words in the standard generators of G and the definition of these words can be found in the Appendix, page . As mentioned above, this enables us given an FG-module V in terms of the matrices in the standard generators of G to construct explicitly the condensed eHFGeH-module VeH in terms of the matrices in the generating set for eHFGeH.

1.1  Faithful Condensation for M11 in Char. 2


Faithful condensation for the subgroup H with number 13 in the table of marks. The group H is a Sylow 3-subgroup.


The scalar product (1HG,1HG) is 112. The permutation character decomposes as

1a+10aabbcc+11aaa+44a4+45a5+55a7.

The dimensions of the modular irreducibles and their condensed dimensions are:

1
10
44
1
2
4
The Cartan matrix of the block is:

4
2
2
2
5
1
2
1
3
The dimensions of the projective indecomposable modules and their condensed dimensions are:

112
96
144
16
16
16

The condensations of the words z1,...,z6 in the standard generators generate the algebra eHFGeH.

1.2  Faithful Condensation for M_11 in Char. 3


Faithful condensation for subgroup H with number 6 in the table of marks. H is a Sylow 5-subgroup.


The scalar product (1HG,1HG) is 320. The permutation character decomposes as

1a+10aabbcc+11aaa+16a4b4+44a8+45a9+55a11.

The dimensions of the modular irreducibles and their condensed dimensions are:

1
5
5
10
10
10
24
1
1
1
2
2
2
4
The Cartan matrix of the block is:

5
3
3
0
2
2
1
3
4
3
1
1
2
2
3
3
4
1
2
1
2
0
1
1
2
0
0
1
2
1
2
0
3
1
1
2
2
1
0
1
3
1
1
2
2
1
1
1
2
The dimensions of the projective indecomposable modules and their condensed dimensions are:

99
126
126
54
81
81
99
23
26
26
10
17
17
19

The condensations of the words z1,...,z4 in the standard generators generate the algebra eHFGeH.

1.3  Faithful Condensation for M_12 in Char. 2


Faithful condensation for the subgroup H with number 35 in the table of marks. The order of H is 9.


The scalar product (1HG,1HG) is 1216. The permutation character decomposes as

1a+11aaabbb+45a5+54a6+55a7b7c7+66a10+99a11+120a16
+144a16+176a16.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
10
44
1
2
4
The Cartan matrix of the block is:

16
16
12
16
20
14
12
14
14
The dimensions of the projective indecomposable modules and their condensed dimensions are:

704
832
768
96
112
96

The condensations of the elements z1,...,z5 in the standard generators generate eHFGeH.

1.4  Faithful Condensation for M_12 in Char. 3


Faithful condensation for the subgroup H with number 50 in the table of marks. The order of the subgroup is 16.


The scalar product (1HG,1HG) is 424. The permutation character decomposes as

1a+11aabb+16aabb+45aaa+54a8+55a5bbcc+66a6+99a7+120a4
+144a10+176a10.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
10
10
15
15
34
45
45
1
1
1
1
1
2
1
1
The Cartan matrix of the block is:

14
4
4
5
5
5
2
2
4
4
2
1
1
2
2
1
4
2
4
1
1
2
1
2
5
1
1
4
3
2
2
2
5
1
1
3
4
2
2
2
5
2
2
2
2
3
1
1
2
2
1
2
2
1
3
2
2
1
2
2
2
1
2
3
The dimensions of the projective indecomposable modules and their condensed dimensions are:

594
297
297
378
378
297
351
351
46
19
19
22
22
21
16
16
The condensations of the words z1,...,z5 in the standard generators generate eHFGeH.

1.5  Faithful Condensation for J_1 in Char. 2


Faithful condensation for the subgroup H with number 21 in the table of marks. The order of H is 21.


The scalar product (1HG,1HG) is 412. The permutation character decomposes as

1a+56a4b4+76a4b4+77aaab5c5+120a6b6c6+133a7b5c5+209a9.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
20
56
56
76
1
2
2
2
2
The Cartan matrix of the block is:

8
4
4
4
4
4
4
3
3
1
4
3
4
2
2
4
3
2
4
2
4
1
2
2
4
The dimensions of the projective indecomposable modules and their condensed dimensions are:

840
496
552
552
552
40
26
26
26
22
The condensations of the words z1,z2 and z3 in the standard generators generate eHFGeH.

1.6  Faithful Condensation for M_22 in Char. 2


Faithful condensation for the subgroup H with number 26 in the table of marks. The subgroup H is a Sylow 3-subgroup.


The scalar product (1HG,1HG) is 5504. The permutation character decomposes as

1a+21a5+45a5b5+55a7+99a11+154a18+210a26+231a23
+280a32b32+385a41.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
10
10
34
70
70
98
1
2
2
2
6
6
10
The Cartan matrix of the block is:

92
42
42
50
12
12
20
42
21
20
23
5
5
9
42
20
21
23
5
5
9
50
23
23
29
7
7
10
12
5
5
7
3
2
2
12
5
5
7
2
3
2
20
9
9
10
2
2
6
The dimensions of the projective indecomposable modules and their condensed dimensions are:

6272
2816
2816
3456
896
896
1408
704
320
320
384
96
96
160

Since the condensed module (1HG)eH is of dimension 5504, we prefer to use Theorem in its stronger version, i.e., we take the condensed module on the cosets of the normalizer N of H in G, which is of order 72. The condensed permutation module has dimension 688 as can be derived from the permutation character 1NG. The coset N spans the whole space under the condensations of the words z1, z2, z3, z4 and z7 in the standard generators. These elements together with the condensations of the generators of the normalizer given in the table of marks therefore generate eHFGeH.

1.7  Faithful Condensation for M_22 in Char. 3


Faithful condensation for the subgroup H with number 106 in the table of marks. The order of the subgroup H is 64.


The scalar product (1HG,1HG) is 145. The permutation character decomposes as

1a+21aa+55aaa+99aa+154a6+210aaa+231a5+280aabb+385a7.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
49
49
55
231
1
1
1
3
1
The Cartan matrix of the block is:

3
2
2
2
1
2
3
2
2
2
2
2
3
2
2
2
2
2
3
1
1
2
2
1
3
The dimensions of the projective indecomposable modules and their condensed dimensions are:

540
819
819
594
945
14
15
15
16
11

The condensations of the words z1,...,z6 in the standard generators generate the algebra eHFGeH.

1.8  Faithful condensation for J_2 in char. 2


Faithful condensation for the subgroup H with number 11 in the table of marks. The order of H is 5.


The scalar product (1HG,1HG) is 24228. The permutation character decomposes as

1a+14a4b4+21a7b7+36a4+63a15+70a16b16+90a22+126a26+160a28
+175a35+189a39b39+224a44b44+225a45+288a60+300a60+336a64.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
6
6
14
14
36
84
1
2
2
4
4
4
12
The Cartan matrix of the block is:

112
76
76
40
40
40
32
76
54
52
28
28
28
22
76
52
54
28
28
28
22
40
28
28
20
16
14
14
40
28
28
16
20
14
14
40
28
28
14
14
17
11
32
22
22
14
14
11
13
The dimensions of the projective indecomposable modules and their condensed dimensions are:

6272
4352
4352
2560
2560
2304
2176
1280
888
888
520
520
464
432

For further details, see chapter 3.

1.9  The P.I.M.s of J_2 in Char. 3


We get faithful condensation for the subgroup H with number 76 in the table of marks. The order of H is 32.


The scalar product (1HG,1HG) is 660. The permutation character 1HG decomposes as

1a+14aabb+21ab+36a4+63a5+70aabb+90a4+126a8+160a6
+175a5+189aaabbb+224a6b6+225a5+288a10+300a10+336a12.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
13
13
21
21
57
57
133
1
1
1
1
1
1
1
3
The Cartan matrix of the block is:

9
6
6
2
2
1
1
6
6
9
6
1
2
4
2
6
6
6
9
2
1
2
4
6
2
1
2
4
2
0
1
4
2
2
1
2
4
1
0
4
1
4
2
0
1
3
1
2
1
2
4
1
0
1
3
2
6
6
6
4
4
2
2
9
The dimensions of the projective indecomposable modules and their condensed dimensions are:

1161
1404
1404
756
756
594
594
1755
45
48
48
24
24
18
18
57

We take the condensations of the words z9 and z10 in the standard generators. Then these elements together with the condensation of z7(z9,z10) and the condensations of the standard generators generate the condensed algebra.

1.10  The P.I.M.s of J_2 in Char. 5

We get faithful condensation for the subgroup H with number 111 in the table of marks. The order of the subgroup H is 96.


The scalar product (1HG,1HG) is 95. The permutation character 1HG decomposes as

1a+14ab+21ab+36aaa+63aaa+70ab+90aa+126aa+160aaa
+175aa+189ab+224aaabbb+288aaa+300aa+336a4.

The dimensions of the modular irreducibles and their condensed dimensions are:

1
14
21
41
85
189
1
1
1
1
1
1
The Cartan matrix of the block is:

3
1
2
1
0
0
1
6
3
0
1
3
2
3
7
2
1
3
1
0
2
3
2
1
0
1
1
2
3
2
0
3
3
1
2
6
The dimensions of the projective indecomposable modules and their condensed dimensions are:

100
800
925
525
750
1450
7
14
18
9
9
15
The condensations of the words z1,z5, and z6 in the standard generators are a generating system for eHFGeH.

1.11  Faithful Condensation for M_23 in Char. 2

Faithful condensation for the subgroup H with number 49 in the table of marks. The order of the subgroup H is 21.


The scalar product (1HG,1HG) is 23212. The permutation character decomposes as

1a+22a4+45aabb+230a14+231a15b9c9+253a13+770a40b40
+896a40b40+990a47b47+1035a49+2024a96.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
11
11
44
44
120
220
220
252
1
2
2
1
1
8
7
7
12
The Cartan matrix of the block is:

46
39
39
28
28
16
17
17
24
39
40
39
26
26
16
17
16
22
39
39
40
26
26
16
16
17
22
28
26
26
21
18
12
10
11
14
28
26
26
18
21
12
11
10
14
16
16
16
12
12
8
6
6
8
17
17
16
10
11
6
9
7
10
17
16
17
11
10
6
7
9
10
24
22
22
14
14
8
10
10
15
The dimensions of the projective indecomposable modules and their condensed dimensions are:

18816
17920
17920
11904
11904
7040
8064
8064
10880
912
872
872
582
582
348
384
384
524

There is a better choice for H than the one offered above. This subgroup has been used in [] to determine the socle series of the projective indecomposable modules for G in the principal block.

1.12  Faithful Condensation for M_23 in Char. 3


Faithful condensation for the subgroup H with number 136 in the table of marks. The order of the subgroup H is 160.


The scalar product (1HG,1HG) is 534. The permutation character decomposes as

1a+22aaa+230a7+231aaabbbccc+253a4+770ab+896a6b6
+990aaabbb+1035a12+2024a14.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
22
104
104
253
770
770
1
3
2
2
4
1
1
The Cartan matrix of the block is:

2
1
1
1
1
1
1
1
5
3
3
1
2
2
1
3
3
2
1
2
1
1
3
2
3
1
1
2
1
1
1
1
2
1
1
1
2
2
1
1
3
1
1
2
1
2
1
1
3
The dimensions of the projective indecomposable modules and their condensed dimensions are:

2025
4068
3150
3150
2277
3690
3690
15
36
27
27
18
21
21

The condensations of the words z1,...,z4 in the standard generators generate the algebra eHFGeH.

1.13  Faithful Condensation for HS in Char. 2


Faithful condensation for the subgroup H with number 146 in the table of marks. The order of the subgroup H is 21.


The scalar product (1HG,1HG) is 100732. The permutation character decomposes as

1a+22a4+77a7+154a8b8c8+175a11+231a15+693a33+770a40b40c40
+825a43+896a40b40+1056a46+1386a66+1408a70+1750a80
+1925a91b91+2520a120+2750a134+3200a150.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
20
56
132
518
1000
1
2
4
4
22
42
The Cartan matrix of the block is:

416
264
92
120
72
42
264
172
62
76
48
25
92
62
27
26
18
10
120
76
26
36
20
12
72
48
18
20
16
7
42
25
10
12
7
7
The dimensions of the projective indecomposable modules and their condensed dimensions are:

105984
67072
25600
30208
19968
13312
5140
3266
1244
1464
966
628
There is no generating system known for the condensed algebra eHFGeH.

1.14  Faithful Condensation for HS in Char. 3


Faithful condensation for the subgroup H with number 482 in the table of marks. The order of the subgroup is 256.


The scalar product (1HG,1HG) is 746. The permutation character decomposes as

1a+22a+77aaa+154aaa+175aaa+231a+693a7+770a4bc+825a5
+896a4b4+1056a6+1386a5+1408a6+1750a8+1925a9b5+2520a10
+2750a9+3200a12.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
22
154
321
748
1176
1253
1
1
3
2
3
5
2
The Cartan matrix of the block is:

4
1
1
0
2
2
2
1
4
1
1
2
1
2
1
1
3
1
0
0
2
0
1
1
2
0
1
2
2
2
0
0
3
2
1
2
1
0
1
2
3
2
2
2
2
2
1
2
4
The dimensions of the projective indecomposable modules and their condensed dimensions are:

6534
5742
3312
4500
5895
7875
9108
28
25
17
17
25
30
35
The condensations of the words z1,...,z5 and z8 in the standard generators generate the algebra eHFGeH.

1.15  Faithful condensation for HS in char. 5

Faithful condensation for the subgroup H with number 456 in the table of marks. The order of the subgroup H is 192.


The scalar product (1HG,1HG) is 1430. The permutation character decomposes as

1a+22aa+77a4+154a4bbcc+175a5+231aa+693a8+770a10bbbccc
+825a10+896a5b5+1056a10+1386a7+1408a9+1750a8+1925a10b10
+2520a16+2750a7+3200a15.

The dimensions of the modular irreducibles and their condensed dimensions are:

1
21
55
98
133
133
210
280
280
518
1
1
2
1
1
1
1
1
1
3
The Cartan matrix of the block is:

7
6
4
5
1
1
2
1
1
2
6
16
5
12
6
6
8
3
3
7
4
5
5
6
2
2
1
0
0
2
5
12
6
18
6
6
8
5
5
11
1
6
2
6
4
3
3
1
1
3
1
6
2
6
3
4
3
1
1
3
2
8
1
8
3
3
10
6
6
6
1
3
0
5
1
1
6
6
5
5
1
3
0
5
1
1
6
5
6
5
2
7
2
11
3
3
6
5
5
10
The dimensions of the projective indecomposable modules and their condensed dimensions are:

3125
10375
2750
14125
4500
4500
10375
7750
7750
11375
38
91
36
110
38
38
66
43
43
76

The elements corresponding to z1,z2,z3,z4,z5,z6 generate the condensation algebra.

1.16  Faithful Condensation for J_3 in Char. 2


Faithful condensation for the subgroup H with number 54 in the table of marks. The order of the subgroup H is 27.


The scalar product (1HG,1HG) is 68932. The permutation character decomposes as

1a+85a5b5+323a11b11+324a12+646a24b24+816a32+1140a44
+1215a45b45+1615a59+1920a72b72c72+1938a70b70+2432a90+2754a102
+3078a114.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
78
78
80
84
84
244
322
322
966
1
2
2
2
4
4
10
10
10
34
The Cartan matrix of the block is:

84
16
16
24
32
32
26
20
20
14
16
7
6
8
9
9
8
3
4
6
16
6
7
8
9
9
8
4
3
6
24
8
8
19
18
18
12
4
4
12
32
9
9
18
21
20
12
6
6
12
32
9
9
18
20
21
12
6
6
12
26
8
8
12
12
12
13
6
6
7
20
3
4
4
6
6
6
7
4
2
20
4
3
4
6
6
6
4
7
2
14
6
6
12
12
12
7
2
2
9
The dimensions of the projective indecomposable modules and their condensed dimensions are:

42624
13184
13184
22912
24704
24704
18048
8832
8832
15616
1588
484
484
846
916
916
666
328
328
574

There is no generating system known for the condensed algebra eHFGeH.

1.17  Faithful condensation for J_3 in char. 3


Faithful condensation for the subgroup H with number 90 in the table of marks. The order of the subgroup H is 80.


The scalar product (1HG,1HG) is 7926. The permutation character decomposes as

1a+85aabb+323a5b5+324a4+646a7b7+816a8+1140a18+1215a18b18
+1615a23+1920a24b24c24+1938a25b25+2432a32+2754a31+3078a35.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
18
18
84
84
153
153
934
1
1
1
1
1
1
1
12
The Cartan matrix of the block is:

62
26
26
47
47
40
40
19
26
16
14
23
23
23
22
10
26
14
16
23
23
22
23
10
47
23
23
40
39
37
37
16
47
23
23
39
40
37
37
16
40
23
22
37
37
40
38
14
40
22
23
37
37
38
40
14
19
10
10
16
16
14
14
8
The dimensions of the projective indecomposable modules and their condensed dimensions are:

38880
20655
20655
33777
33777
32076
32076
14823
516
267
267
438
438
405
405
195

The elements corresponding to z3,z5, z3*z5, and the second generator of the normalizer of the condensation subgroup as given in GAP library of table of marks generate the condensation algebra.

1.18  Faithful Condensation for M_24 in Char. 2

Faithful condensation for the subgroup H whose number in the table of marks is 303. The order of the subgroup H is 27.


The scalar product (1HG,1HG) is 336224. The permutation character decomposes as

1a+23aaa+45aaabbb+231a7b7+252a14+253a15+483a21+770a28b28
+990a38b38+1035a41b37c37+1265a53+1771a77+2024a78+2277a87
+3312a120+3520a132+5313a189+5544a210+5796a210+10395a385.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
11
11
44
44
120
220
220
252
320
320
1242
1792
1
1
1
2
2
6
6
6
8
8
8
46
64
The Cartan matrix of the block is:

342
361
361
245
245
208
120
120
160
50
50
46
18
361
399
396
264
263
228
130
130
172
54
56
49
21
361
396
399
263
264
228
130
130
172
56
54
49
21
245
264
263
182
179
152
87
87
114
34
35
33
13
245
263
264
179
182
152
87
87
114
35
34
33
13
208
228
228
152
152
132
74
74
100
32
32
28
12
120
130
130
87
87
74
47
45
56
19
18
18
8
120
130
130
87
87
74
45
47
56
18
19
18
8
160
172
172
114
114
100
56
56
80
25
25
20
8
50
54
56
34
35
32
19
18
25
11
9
8
4
50
56
54
35
34
32
18
19
25
9
11
8
4
46
49
49
33
33
28
18
18
20
8
8
10
4
18
21
21
13
13
12
8
8
8
4
4
4
3

The dimensions of the projective indecomposable modules and their condensed dimensions are:

269312
293888
293888
193536
193536
168960
102400
102400
10080
10992
10992
7264
7264
6320
3808
3808
125952
46080
46080
45056
21504
4704
1696
1696
1664
784
There is no generating system known for the condensed algebra eHFGeH.

1.19  Faithful Condensation for M_24 in Char. 3


Faithful condensation for the subgroup H with number 1234 in the table of marks. The order of the subgroup H is 320.


The scalar product (1HG,1HG) is 2884. The corresponding permutation decomposes as:

1a+23aa+231ab+252a8+253aa+483a7+770ab+990ab+1035a10bc
+1265a9+1771aaa+2024a10+2277a8+3312a18+3520a20+5313a24
+5544a10+5796a18+10395a26.

The dimensions of the modular irreducibles and their condensed dimensions are:

1
22
231
483
770
770
1243
1
1
1
7
1
1
8
The Cartan matrix of the block is:

5
4
3
3
4
4
2
4
8
6
3
5
5
4
3
6
9
3
6
6
3
3
3
3
4
4
4
2
4
5
6
4
8
7
3
4
5
6
4
7
8
3
2
4
3
2
3
3
3
The dimensions of the projective indecomposable modules and their condensed dimensions are:

10881
15687
16632
11340
18711
18711
10098
57
81
75
61
82
82
53

The elements corresponding to z1,z2,z3,z4,z5,z6,z7,z8,z9,z10 and z7(z7,z8) generate the condensation algebra.

1.20  Faithful Condensation for McL in Char. 2

Faithful condensation for the subgroup H with number 245 in the table of marks. The order of the subgroup H is 243.


The scalar product (1HG,1HG) is 15318. The permutation character decomposes as

1a+22aa+231aaa+252a4+770a4b4+896aabb+1750a10+3520a18b18
+4500a16+4752a20+5103a21+5544a26+8019a33b33+8250a36b36
+9625a39+9856a37b37+10395a43b43.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
22
230
748
748
2124
2124
3584
1
2
2
2
2
6
6
14
The Cartan matrix of the block is:

36
32
18
20
20
6
6
14
32
88
24
36
36
20
20
24
18
24
17
10
10
7
7
7
20
36
10
20
16
8
6
12
20
36
10
16
20
6
8
12
6
20
7
8
6
7
5
5
6
20
7
6
8
5
7
5
14
24
7
12
12
5
5
9
The dimensions of the projective indecomposable modules and their condensed dimensions are:

110464
232320
74240
102784
102784
55936
55936
73600
484
976
322
436
436
230
230
310

The elements corresponding to z6, z8, and z11 generate the condensed algebra. Klaus Lux, 1. Feb. 1999.

1.21  Faithful Condensation for McL in Char. 3

Faithful condensation for the subgroup H with number 179 in the table of marks. The order of the subgroup H is 100.


The scalar product (1HG,1HG) is 89941. The permutation character decomposes as

1a+22aa+231aaa+252a6+770a6b6+896a9b9+1750a22+3520a38b32
+4500a48+4752a46+5103a56+5544a53+8019a79b79+8250a82b82
+9625a100+9856a99b99+10395a102b102.
The dimensions of the modular irreducibles and their condensed dimensions are:

1
21
104
104
210
560
605
605
1498
2794
1
1
1
1
2
4
6
6
16
27
The Cartan matrix of the block is:

262
174
134
134
174
226
40
40
51
39
174
133
91
91
117
144
21
21
39
26
134
91
70
69
90
116
20
20
27
20
134
91
69
70
90
116
20
20
27
20
174
117
90
90
129
162
27
27
33
27
226
144
116
116
162
211
37
37
42
36
40
21
20
20
27
37
9
8
6
6
40
21
20
20
27
37
8
9
6
6
51
39
27
27
33
42
6
6
13
8
39
26
20
20
27
36
6
6
8
8
The dimensions of the projective indecomposable modules and their condensed dimensions are:

428652
283581
220887
220887
296703
387828
67068
67068
4305
2877
2220
2220
2958
3858
663
663
86022
72171
874
719

There is no generating system known for the condensed algebra eHFGeH.

1.22  Faithful Condensation for McL in Char. 5


Faithful condensation for the subgroup H with number 245 in the table of marks. The order of the subgroup H is 243.


The scalar product (1HG,1HG) is 15318. The permutation character decomposes as

1a+22aa+231aaa+252a4+770a4b4+896aabb+1750a10+3520a18b18
+4500a16+4752a20+5103a21+5544a26+8019a33b33+8250a36b36
+9625a39+9856a37b37+10395a43b43
The dimensions of the modular irreducibles and their condensed dimensions are:

1
21
210
230
560
896
896
1200
1200
3038
3245
3245
1
1
2
2
2
2
2
7
7
12
11
11
The Cartan matrix of the block is:

9
11
3
2
8
8
8
5
5
3
1
1
11
24
8
5
14
15
15
8
8
10
2
2
3
8
7
1
8
5
5
3
3
3
0
0
2
5
1
3
0
1
1
0
0
2
0
0
8
14
8
0
23
15
15
12
12
6
2
2
8
15
5
1
15
16
15
9
9
7
2
2
8
15
5
1
15
15
16
9
9
7
2
2
5
8
3
0
12
9
9
8
7
3
2
1
5
8
3
0
12
9
9
7
8
3
1
2
3
10
3
2
6
7
7
3
3
6
1
1
1
2
0
0
2
2
2
2
1
1
2
0
1
2
0
0
2
2
2
1
2
1
0
2
The dimensions of the projective indecomposable modules and their condensed dimensions are:

47750
100625
31625
8875
101750
93625
93625
60500
60500
206
425
141
43
428
381
381
253
253
49125
17875
17875
199
70
70

The elements corresponding to z2, z10, and z11 generate the condensed algebra. Klaus Lux, 21. March 1999.

1.23  Faithful condensation for U3(3) in char. 2


Faitfful condensation for the subgroup with number 3 in the table of marks. The order of the subgroup is 3.


The corresponding permutation character is:

2016
0
0
6
0
0
0
0
0
0
0
0
0
0
The dimensions of the modular irreducibles and their kondensed dimensions are:

1
6
14
1
2
4
The Cartan matrix of the block is:

16
16
8
16
19
9
8
9
7
The dimensions of the projective indecomposable modules and their condensed dimensions are:

224
256
160
80
90
54

1.24  Faithful condensation for U3(3) in char. 3


Faitfful condensation for the subgroup witj number 7 in the table of marks. The order of the subgroup is 4.


The corresponding permutation character is:

1512
24
0
0
0
0
8
0
0
0
0
0
0
0
The dimensions of the modular irreducibles and their kondensed dimensions are:

1
3
3
6
6
7
15
15
1
1
1
2
2
1
3
3
The Cartan matrix of the block is:

10
2
2
4
4
5
1
1
2
5
4
0
1
4
1
2
2
4
5
1
0
4
2
1
4
0
1
4
1
2
2
0
4
1
0
1
4
2
0
2
5
4
4
2
2
7
2
2
1
1
2
2
0
2
3
0
1
2
1
0
2
2
0
3
The dimensions of the projective indecomposable modules and their condensed dimensions are:

135
108
108
81
81
162
81
81
41
26
26
23
23
40
19
19

File translated from TEX by TTH, version 2.00.
On 3 Feb 1999, 21:53.