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
e_{H}FGe_{H}. 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 e_{H}FGe_{H}. 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 FGmodule V in terms of the matrices in the standard generators
of G to construct explicitly the condensed e_{H}FGe_{H}module Ve_{H} in
terms of the matrices in the generating set for e_{H}FGe_{H}.
1.1 Faithful Condensation for M_{11} in Char. 2
Faithful condensation for the subgroup H with number 13 in the
table of marks. The group H is a Sylow 3subgroup.
The scalar product (1_{H}^{G},1_{H}^{G}) is 112.
The permutation character decomposes as
1a+10aabbcc+11aaa+44a^{4}+45a^{5}+55a^{7}. 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words z1,...,z6 in the standard generators generate
the algebra e_{H}FGe_{H}.
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 5subgroup.
The scalar product (1_{H}^{G},1_{H}^{G}) is 320.
The permutation character decomposes as
1a+10aabbcc+11aaa+16a^{4}b^{4}+44a^{8}+45a^{9}+55a^{11}. 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words z1,...,z4 in the standard generators generate the
algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 1216.
The permutation character decomposes as

1a+11aaabbb+45a^{5}+54a^{6}+55a^{7}b^{7}c^{7}+66a^{10}+99a^{11}+120a^{16} 
 
 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the elements z1,...,z5 in the standard generators generate e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 424.
The permutation character decomposes as

1a+11aabb+16aabb+45aaa+54a^{8}+55a^{5}bbcc+66a^{6}+99a^{7}+120a^{4} 
 
 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words z1,...,z5
in the standard generators
generate e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 412.
The permutation character decomposes as
1a+56a^{4}b^{4}+76a^{4}b^{4}+77aaab^{5}c^{5}+120a^{6}b^{6}c^{6}+133a^{7}b^{5}c^{5}+209a^{9}. 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words z1,z2 and z3 in the standard generators
generate e_{H}FGe_{H}.
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 3subgroup.
The scalar product (1_{H}^{G},1_{H}^{G}) is 5504.
The permutation character decomposes as

1a+21a^{5}+45a^{5}b^{5}+55a^{7}+99a^{11}+154a^{18}+210a^{26}+231a^{23} 
 +280a^{32}b^{32}+385a^{41}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
Since the condensed module (1_{H}^{G})e_{H} 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 1_{N}^{G}.
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 e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 145.
The permutation character decomposes as
1a+21aa+55aaa+99aa+154a^{6}+210aaa+231a^{5}+280aabb+385a^{7}. 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words
z1,...,z6 in the standard generators generate the
algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 24228.
The permutation character decomposes as

1a+14a^{4}b^{4}+21a^{7}b^{7}+36a^{4}+63a^{15}+70a^{16}b^{16}+90a^{22}+126a^{26}+160a^{28} 
 +175a^{35}+189a^{39}b^{39}+224a^{44}b^{44}+225a^{45}+288a^{60}+300a^{60}+336a^{64}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
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 (1_{H}^{G},1_{H}^{G}) is 660.
The permutation character 1_{H}^{G} decomposes as

1a+14aabb+21ab+36a^{4}+63a^{5}+70aabb+90a^{4}+126a^{8}+160a^{6} 
 +175a^{5}+189aaabbb+224a^{6}b^{6}+225a^{5}+288a^{10}+300a^{10}+336a^{12}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
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 (1_{H}^{G},1_{H}^{G}) is 95.
The permutation character 1_{H}^{G} decomposes as

1a+14ab+21ab+36aaa+63aaa+70ab+90aa+126aa+160aaa 
 +175aa+189ab+224aaabbb+288aaa+300aa+336a^{4}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words
z1,z5, and
z6 in the standard generators are a generating system for e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 23212.
The permutation character decomposes as

1a+22a^{4}+45aabb+230a^{14}+231a^{15}b^{9}c^{9}+253a^{13}+770a^{40}b^{40} 
 +896a^{40}b^{40}+990a^{47}b^{47}+1035a^{49}+2024a^{96}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
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 (1_{H}^{G},1_{H}^{G}) is 534.
The permutation character decomposes as

1a+22aaa+230a^{7}+231aaabbbccc+253a^{4}+770ab+896a^{6}b^{6} 
 +990aaabbb+1035a^{12}+2024a^{14}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words z1,...,z4 in the standard generators generate the
algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 100732.
The permutation character decomposes as

1a+22a^{4}+77a^{7}+154a^{8}b^{8}c^{8}+175a^{11}+231a^{15}+693a^{33}+770a^{40}b^{40}c^{40} 

+825a^{43}+896a^{40}b^{40}+1056a^{46}+1386a^{66}+1408a^{70}+1750a^{80} 
 +1925a^{91}b^{91}+2520a^{120}+2750a^{134}+3200a^{150}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
There is no generating system known for the condensed algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 746.
The permutation character decomposes as

1a+22a+77aaa+154aaa+175aaa+231a+693a^{7}+770a^{4}bc+825a^{5} 

+896a^{4}b^{4}+1056a^{6}+1386a^{5}+1408a^{6}+1750a^{8}+1925a^{9}b^{5}+2520a^{10} 
 
 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The condensations of the words z1,...,z5 and z8 in the standard generators generate
the algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 1430.
The permutation character decomposes as

1a+22aa+77a^{4}+154a^{4}bbcc+175a^{5}+231aa+693a^{8}+770a^{10}bbbccc 

+825a^{10}+896a^{5}b^{5}+1056a^{10}+1386a^{7}+1408a^{9}+1750a^{8}+1925a^{10}b^{10} 
 +2520a^{16}+2750a^{7}+3200a^{15}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
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 (1_{H}^{G},1_{H}^{G}) is 68932.
The permutation character decomposes as

1a+85a^{5}b^{5}+323a^{11}b^{11}+324a^{12}+646a^{24}b^{24}+816a^{32}+1140a^{44} 

+1215a^{45}b^{45}+1615a^{59}+1920a^{72}b^{72}c^{72}+1938a^{70}b^{70}+2432a^{90}+2754a^{102} 
 
 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
There is no generating system known for the condensed algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 7926.
The permutation character decomposes as

1a+85aabb+323a^{5}b^{5}+324a^{4}+646a^{7}b^{7}+816a^{8}+1140a^{18}+1215a^{18}b^{18} 
 +1615a^{23}+1920a^{24}b^{24}c^{24}+1938a^{25}b^{25}+2432a^{32}+2754a^{31}+3078a^{35}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
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 (1_{H}^{G},1_{H}^{G}) is 336224.
The permutation character decomposes as

1a+23aaa+45aaabbb+231a^{7}b^{7}+252a^{14}+253a^{15}+483a^{21}+770a^{28}b^{28} 

+990a^{38}b^{38}+1035a^{41}b^{37}c^{37}+1265a^{53}+1771a^{77}+2024a^{78}+2277a^{87} 
 +3312a^{120}+3520a^{132}+5313a^{189}+5544a^{210}+5796a^{210}+10395a^{385}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
There is no generating system known for the condensed algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 2884.
The corresponding permutation decomposes as:

1a+23aa+231ab+252a^{8}+253aa+483a^{7}+770ab+990ab+1035a^{10}bc 

+1265a^{9}+1771aaa+2024a^{10}+2277a^{8}+3312a^{18}+3520a^{20}+5313a^{24} 
 +5544a^{10}+5796a^{18}+10395a^{26}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
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 (1_{H}^{G},1_{H}^{G}) is 15318.
The permutation character decomposes as

1a+22aa+231aaa+252a^{4}+770a^{4}b^{4}+896aabb+1750a^{10}+3520a^{18}b^{18} 

+4500a^{16}+4752a^{20}+5103a^{21}+5544a^{26}+8019a^{33}b^{33}+8250a^{36}b^{36} 
 +9625a^{39}+9856a^{37}b^{37}+10395a^{43}b^{43}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
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 (1_{H}^{G},1_{H}^{G}) is 89941.
The permutation character decomposes as

1a+22aa+231aaa+252a^{6}+770a^{6}b^{6}+896a^{9}b^{9}+1750a^{22}+3520a^{38}b^{32} 

+4500a^{48}+4752a^{46}+5103a^{56}+5544a^{53}+8019a^{79}b^{79}+8250a^{82}b^{82} 
 +9625a^{100}+9856a^{99}b^{99}+10395a^{102}b^{102}. 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
There is no generating system known for the condensed algebra e_{H}FGe_{H}.
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 (1_{H}^{G},1_{H}^{G}) is 15318.
The permutation character decomposes as

1a+22aa+231aaa+252a^{4}+770a^{4}b^{4}+896aabb+1750a^{10}+3520a^{18}b^{18} 

+4500a^{16}+4752a^{20}+5103a^{21}+5544a^{26}+8019a^{33}b^{33}+8250a^{36}b^{36} 
 +9625a^{39}+9856a^{37}b^{37}+10395a^{43}b^{43} 

 

The dimensions of the modular irreducibles and their condensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
The elements corresponding to z2, z10, and z11 generate the condensed algebra. Klaus Lux, 21. March 1999.
1.23 Faithful condensation for U_{3}(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:
The dimensions of the modular irreducibles and their kondensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
1.24 Faithful condensation for U_{3}(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:
The dimensions of the modular irreducibles and their kondensed dimensions are:
The Cartan matrix of the block is:
The dimensions of the projective indecomposable modules and
their condensed dimensions are:
File translated from T_{E}X by T_{T}H, version 2.00.
On 3 Feb 1999, 21:53.