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Permutation representations of finite groupsand association schemes (Part 3)
Akihiro Munemasa
August 15, 2018
G2R2, Novosibirsk
1
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Contents
1 Notation and motivation
2 Association schemes
3 ⇤-Algebras in a matrix algebra
4 Commutative case: eigenmatrix and Krein parameters
5 The center of the group algebra and the character table
6 A formula for central idempotents
7 The splitting field
8 Applications
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Review
In the previous lecture, we discussed the adjacency algebra C[R] of
an association scheme R. In this lecture we always take F = C.
If R is the set of 2-orbits of an action (G ;⌦), then
C[R] = VC(G ,⌦)
which is the centralizer in M⌦(C) of
{Pg | g 2 G}.
That is, the two algebras
C[R] and C[{Pg | g 2 G}]
centralize each other.
Note that C[{Pg | g 2 G}] is not a group algebra since Pg (g 2 G )
are not necessarily linearly independent.
Then what is dimC[{Pg | g 2 G}]?2
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Review
In the previous lecture, we discussed the adjacency algebra C[R] of
an association scheme R. In this lecture we always take F = C.
If R is the set of 2-orbits of an action (G ;⌦), then
C[R] = VC(G ,⌦)
which is the centralizer in M⌦(C) of
{Pg | g 2 G}.
That is, the two algebras
C[R] and C[{Pg | g 2 G}]
centralize each other.
Note that C[{Pg | g 2 G}] is not a group algebra since Pg (g 2 G )
are not necessarily linearly independent.
Then what is dimC[{Pg | g 2 G}]?3
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⇤-algebrasFor A 2 Mn(C), write A⇤
= A>.
Definition
A subalgebra A of Mn(C) is a ⇤-algebra if
A 2 A =) A⇤ 2 A.
If (G ;⌦) is an action, then {Pg | g 2 G} is closed under
transposition: P>g = Pg�1 . Indeed,
((Pg )>)↵� = (Pg )�↵ = ��g↵ = �↵g�1� = (Pg�1)↵�.
Thus C[{Pg | g 2 G}] is a ⇤-algebra.
If R is an association scheme, then {A(R) | R 2 R} is closed under
transposition. Thus C[R] is a ⇤-algebra.4
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⇤-algebrasFor A 2 Mn(C), write A⇤
= A>.
Definition
A subalgebra A of Mn(C) is a ⇤-algebra if
A 2 A =) A⇤ 2 A.
If (G ;⌦) is an action, then {Pg | g 2 G} is closed under
transposition: P>g = Pg�1 . Indeed,
((Pg )>)↵� = (Pg )�↵ = ��g↵ = �↵g�1� = (Pg�1)↵�.
Thus C[{Pg | g 2 G}] is a ⇤-algebra.
If R is an association scheme, then {A(R) | R 2 R} is closed under
transposition. Thus C[R] is a ⇤-algebra.5
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⇤-algebrasFor A 2 Mn(C), write A⇤
= A>.
Definition
A subalgebra A of Mn(C) is a ⇤-algebra if
A 2 A =) A⇤ 2 A.
If (G ;⌦) is an action, then {Pg | g 2 G} is closed under
transposition: P>g = Pg�1 . Indeed,
((Pg )>)↵� = (Pg )�↵ = ��g↵ = �↵g�1� = (Pg�1)↵�.
Thus C[{Pg | g 2 G}] is a ⇤-algebra.
If R is an association scheme, then {A(R) | R 2 R} is closed under
transposition. Thus C[R] is a ⇤-algebra.6
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The standard module
Let A ✓ Mn(C) be a ⇤-algebra. Since Cnis a left Mn(C)-module, it
is also an A-module, which is called the standard module.
Recall that an A-module W is irreducible if it has no nontrivial
A-submodule.
Proposition
Let Cn ◆ V be an A-submodule of a ⇤-algebra A ✓ Mn(C). ThenV?
is also an A-submodule.
Thus, the standard module Cnis an orthogonal direct sum of
irreducible A-modules. This means that, 9 unitary matrix U ,
U⇤AU = algebra of block diagonal matrices
and each block (say, size m ⇥m) acts on Cmirreducibly, hence
isomorphic to Mm(C) by Wedderburn’s theorem.
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The standard module
Let A ✓ Mn(C) be a ⇤-algebra. Since Cnis a left Mn(C)-module, it
is also an A-module, which is called the standard module.
Recall that an A-module W is irreducible if it has no nontrivial
A-submodule.
Proposition
Let Cn ◆ V be an A-submodule of a ⇤-algebra A ✓ Mn(C). ThenV?
is also an A-submodule.
Thus, the standard module Cnis an orthogonal direct sum of
irreducible A-modules. This means that, 9 unitary matrix U ,
U⇤AU = algebra of block diagonal matrices
and each block (say, size m ⇥m) acts on Cmirreducibly, hence
isomorphic to Mm(C) by Wedderburn’s theorem.
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The standard module
Let A ✓ Mn(C) be a ⇤-algebra. Since Cnis a left Mn(C)-module, it
is also an A-module, which is called the standard module.
Recall that an A-module W is irreducible if it has no nontrivial
A-submodule.
Proposition
Let Cn ◆ V be an A-submodule of a ⇤-algebra A ✓ Mn(C). ThenV?
is also an A-submodule.
Thus, the standard module Cnis an orthogonal direct sum of
irreducible A-modules. This means that, 9 unitary matrix U ,
U⇤AU = algebra of block diagonal matrices
and each block (say, size m ⇥m) acts on Cmirreducibly, hence
isomorphic to Mm(C) by Wedderburn’s theorem.
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Wedderburn’s theorem (cf. Rotman’s book)
Upon block-diagonalizing A, we find a block B ✓ Mm(C) such that
Cmis an irreducible B-module, and it is faithful
=) B is simple ( 6 9 nontrivial ideal)
=) B =
kM
i=1
Li pairwise isomorphic minimal left ideals
B ⇠= EndB(B)op
⇠= EndB(
kM
i=1
Li)op
⇠= Mk(C)op (by Schur’s lemma)
⇠= Mk(C)=) the only irreducible B-module has dim = k
=) m = k , so B = Mm(C).10
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Wedderburn’s theorem (cf. Rotman’s book)
Upon block-diagonalizing A, we find a block B ✓ Mm(C) such that
Cmis an irreducible B-module, and it is faithful
=) B is simple ( 6 9 nontrivial ideal)
=) B =
kM
i=1
Li pairwise isomorphic minimal left ideals
B ⇠= EndB(B)op
⇠= EndB(
kM
i=1
Li)op
⇠= Mk(C)op (by Schur’s lemma)
⇠= Mk(C)=) the only irreducible B-module has dim = k
=) m = k , so B = Mm(C).11
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Block diagonalization
U⇤AU =
2
66666664
. . .
. . .
Mm(C). . .
. . .
3
77777775
2
666664
...
�Cm
�...
3
777775= Cn.
Be careful! Isomorphic modules may appear more than once. What if
U⇤AU =
⇢A 0
0 A
�| A 2 Mm(C)
�? 6=
Mm(C) 0
0 Mm(C)
�.
It is block-diagonal with two blocks, but A ⇠= Mm(C).
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Block diagonalization
U⇤AU =
2
66666664
. . .
. . .
Mm(C). . .
. . .
3
77777775
2
666664
...
�Cm
�...
3
777775= Cn.
Be careful! Isomorphic modules may appear more than once. What if
U⇤AU =
⇢A 0
0 A
�| A 2 Mm(C)
�? 6=
Mm(C) 0
0 Mm(C)
�.
It is block-diagonal with two blocks, but A ⇠= Mm(C).
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Representation of an algebra
A (matrix) representation of a C-algebra A is a C-algebrahomomorphism � : A ! Mm(C). This is equivalent to giving an
A-module of dimension m.
Since we only consider ⇤-algebras, we require a representation to be
⇤-representation, that is
�(A⇤) = �(A)⇤.
Block diagonalization by a unitary matrix satisfies this requirement,
since
(U⇤AU)⇤= U⇤A⇤U .
Our representations are of the form “first conjugating by a unitary
matrix, then extracting a block”.
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Representation of an algebra
A (matrix) representation of a C-algebra A is a C-algebrahomomorphism � : A ! Mm(C). This is equivalent to giving an
A-module of dimension m.
Since we only consider ⇤-algebras, we require a representation to be
⇤-representation, that is
�(A⇤) = �(A)⇤.
Block diagonalization by a unitary matrix satisfies this requirement,
since
(U⇤AU)⇤= U⇤A⇤U .
Our representations are of the form “first conjugating by a unitary
matrix, then extracting a block”.
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Representation of an algebra
A (matrix) representation of a C-algebra A is a C-algebrahomomorphism � : A ! Mm(C). This is equivalent to giving an
A-module of dimension m.
Since we only consider ⇤-algebras, we require a representation to be
⇤-representation, that is
�(A⇤) = �(A)⇤.
Block diagonalization by a unitary matrix satisfies this requirement,
since
(U⇤AU)⇤= U⇤A⇤U .
Our representations are of the form “first conjugating by a unitary
matrix, then extracting a block”.
16
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Representation of an algebra
A (matrix) representation of a C-algebra A is a C-algebrahomomorphism � : A ! Mm(C). This is equivalent to giving an
A-module of dimension m.
Since we only consider ⇤-algebras, we require a representation to be
⇤-representation, that is
�(A⇤) = �(A)⇤.
Block diagonalization by a unitary matrix satisfies this requirement,
since
(U⇤AU)⇤= U⇤A⇤U .
Our representations are of the form “first conjugating by a unitary
matrix, then extracting a block”.
17
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Block diaognalization
The block diaognalization of a ⇤-algebra A ✓ Mn(C) is described as
U⇤AU =
2
66666664
. . .
�i(A). . .
�i(A). . .
3
77777775
(�i(A) is repeated mi times).
�i : A ! Mni (C) is a ⇤-representation, and �i is surjective by
Wedderburn’s theorem.8><
>:
�i(A). . .
�i(A)
�����A 2 A
9>=
>;⇠= Mni (C).
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Block diaognalization
The block diaognalization of a ⇤-algebra A ✓ Mn(C) is described as
U⇤AU =
2
66666664
. . .
�i(A). . .
�i(A). . .
3
77777775
(�i(A) is repeated mi times).
�i : A ! Mni (C) is a ⇤-representation, and �i is surjective by
Wedderburn’s theorem.8><
>:
�i(A). . .
�i(A)
�����A 2 A
9>=
>;⇠= Mni (C).
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Centralizer (= matrices which commute with givenones)
What is the centralizer of
U⇤AU =
2
66666664
. . .
�i(A). . .
�i(A). . .
3
77777775
(�i(A) is repeated mi times).
20
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Centralizer (I)
Centralizer of Mn(C) is CI (scalar matrices).
The centralizer of
U⇤AU =
⇢A 0
0 A
�| A 2 Md(C)
�?
is CI CICI CI
�=
⇢a11I a12Ia21I a22I
�|a11 a12a21 a22
�2 M2(C)
�.
21
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Centralizer (I)
Centralizer of Mn(C) is CI (scalar matrices).
The centralizer of
U⇤AU =
⇢A 0
0 A
�| A 2 Md(C)
�?
is CI CICI CI
�=
⇢a11I a12Ia21I a22I
�|a11 a12a21 a22
�2 M2(C)
�.
22
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Centralizer (I)
Centralizer of Mn(C) is CI (scalar matrices).
The centralizer of
U⇤AU =
⇢A 0
0 A
�| A 2 Md(C)
�?
is CI CICI CI
�=
⇢a11I a12Ia21I a22I
�|a11 a12a21 a22
�2 M2(C)
�.
23
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Centralizer (II)
The centralizer of
U⇤AU =
8><
>:
2
64A
. . .
A
3
75 | A 2 Md(C)
9>=
>;(m times)
in Mdm(C) is8><
>:
2
64a11Id · · · a1mId...
. . ....
am1Id · · · ammId
3
75 | [aij ] 2 Mm(C)
9>=
>;.
Md(C)⌦ Im and Id ⌦Mm(C) centralize each other.
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Centralizer (II)
The centralizer of
U⇤AU =
8><
>:
2
64A
. . .
A
3
75 | A 2 Md(C)
9>=
>;(m times)
in Mdm(C) is8><
>:
2
64a11Id · · · a1mId...
. . ....
am1Id · · · ammId
3
75 | [aij ] 2 Mm(C)
9>=
>;.
Md(C)⌦ Im and Id ⌦Mm(C) centralize each other.
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Md(C)⌦ Im and Id ⌦Mm(C) centralize each other
The centralizer of2
66666664
. . .
�i(A). . .
�i(A). . .
3
77777775
=
2
64
. . .
Mni (C)⌦ Imi
. . .
3
75 ,
where �i(A) is repeated mi times, A 2 Mni (C), is2
64
. . .
Ini ⌦Mmi (C). . .
3
75 .
Centralizer(Centralizer(A)) = A (The double centralizer theorem).26
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Md(C)⌦ Im and Id ⌦Mm(C) centralize each other
The centralizer of2
66666664
. . .
�i(A). . .
�i(A). . .
3
77777775
=
2
64
. . .
Mni (C)⌦ Imi
. . .
3
75 ,
where �i(A) is repeated mi times, A 2 Mni (C), is2
64
. . .
Ini ⌦Mmi (C). . .
3
75 .
Centralizer(Centralizer(A)) = A (The double centralizer theorem).27
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The Johnson scheme J(n, 2)
It is the action (Sn;⌦), where
⌦ = {↵ | ↵ ✓ {1, . . . , n}, |↵| = 2}.
Its adjacency algebra C[R] ✓ M⌦(C) is isomorphic to C3. The
(block) diagonal form is
2
64
. . .
Mni (C)⌦ Imi
. . .
3
75 =
2
4CIm0
CIm1
CIm2
3
5
C[{Pg | g 2 Sn}] ⇠=
2
4Mm0(C)
Mm1(C)Mm2(C)
3
5
multiplicity-free () ni = 1 (8i).28
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The Johnson scheme J(n, 2)
It is the action (Sn;⌦), where
⌦ = {↵ | ↵ ✓ {1, . . . , n}, |↵| = 2}.
Its adjacency algebra C[R] ✓ M⌦(C) is isomorphic to C3. The
(block) diagonal form is
2
64
. . .
Mni (C)⌦ Imi
. . .
3
75 =
2
4CIm0
CIm1
CIm2
3
5
C[{Pg | g 2 Sn}] ⇠=
2
4Mm0(C)
Mm1(C)Mm2(C)
3
5
multiplicity-free () ni = 1 (8i).29
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The Johnson scheme J(n, 2)
It is the action (Sn;⌦), where
⌦ = {↵ | ↵ ✓ {1, . . . , n}, |↵| = 2}.
Its adjacency algebra C[R] ✓ M⌦(C) is isomorphic to C3. The
(block) diagonal form is
2
64
. . .
Mni (C)⌦ Imi
. . .
3
75 =
2
4CIm0
CIm1
CIm2
3
5
C[{Pg | g 2 Sn}] ⇠=
2
4Mm0(C)
Mm1(C)Mm2(C)
3
5
multiplicity-free () ni = 1 (8i).30
![Page 32: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/32.jpg)
The group algebra C[G ] (I)The group algebra of a group G is the algebra with basis indexed by
the elements of G , such that multiplication is done just as in G .
The algebra of the right regular representation (GR ;G )
A = C[{Pg | g 2 G}]
is such an algebra. Since this is a ⇤-algebra, its block diagonal form is
2
66666664
. . .
�i(A). . .
�i(A). . .
3
77777775
=
2
664
. . .
Mni (C)⌦ Imi
. . .
3
775
We aim to show mi = ni .31
![Page 33: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/33.jpg)
The group algebra C[G ] (I)The group algebra of a group G is the algebra with basis indexed by
the elements of G , such that multiplication is done just as in G .
The algebra of the right regular representation (GR ;G )
A = C[{Pg | g 2 G}]
is such an algebra. Since this is a ⇤-algebra, its block diagonal form is
2
66666664
. . .
�i(A). . .
�i(A). . .
3
77777775
=
2
664
. . .
Mni (C)⌦ Imi
. . .
3
775
We aim to show mi = ni .32
![Page 34: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/34.jpg)
The group algebra C[G ] satisfies mi = ni
U⇤AU =
2
664
. . .
Mni (C)⌦ Imi
. . .
3
775
The size |G | of the matrix isP
nimi .
The dimension of the algebra is |G | =P
n2i .The centralizer of A is the algebra of left regular representation, its
dimension is
|G | =X
m2i .
Equality in Cauchy–Schwarz inequality forces mi = ni (8i). Thisproves
C[G ] ⇠=
2
664
. . .
Mni (C)⌦ Ini. . .
3
775 .
33
![Page 35: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/35.jpg)
The group algebra C[G ] satisfies mi = ni
U⇤AU =
2
664
. . .
Mni (C)⌦ Imi
. . .
3
775
The size |G | of the matrix isP
nimi .
The dimension of the algebra is |G | =P
n2i .The centralizer of A is the algebra of left regular representation, its
dimension is
|G | =X
m2i .
Equality in Cauchy–Schwarz inequality forces mi = ni (8i). Thisproves
C[G ] ⇠=
2
664
. . .
Mni (C)⌦ Ini. . .
3
775 .
34
![Page 36: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/36.jpg)
The group algebra C[G ] satisfies mi = ni
U⇤AU =
2
664
. . .
Mni (C)⌦ Imi
. . .
3
775
The size |G | of the matrix isP
nimi .
The dimension of the algebra is |G | =P
n2i .The centralizer of A is the algebra of left regular representation, its
dimension is
|G | =X
m2i .
Equality in Cauchy–Schwarz inequality forces mi = ni (8i). Thisproves
C[G ] ⇠=
2
664
. . .
Mni (C)⌦ Ini. . .
3
775 .
35
![Page 37: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/37.jpg)
The group algebra C[G ] satisfies mi = ni
U⇤AU =
2
664
. . .
Mni (C)⌦ Imi
. . .
3
775
The size |G | of the matrix isP
nimi .
The dimension of the algebra is |G | =P
n2i .The centralizer of A is the algebra of left regular representation, its
dimension is
|G | =X
m2i .
Equality in Cauchy–Schwarz inequality forces mi = ni (8i). Thisproves
C[G ] ⇠=
2
664
. . .
Mni (C)⌦ Ini. . .
3
775 .
36
![Page 38: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/38.jpg)
Summary
The centralizer algebra VC(G ,⌦), or more generally, the adjacency
algebra C[R] of an association scheme, is (abstractly) isomorphic to
M
i
Mni (C).
In terms of block diagonal matrices, Mni (C) appear mi times in the
diagonal.
In particular, the group algebra:
C[G ] ⇠= the algebra of the right regular representation
⇠= the algebra of the left regular representation
= centralizer of the right regular representation
is also isomorphic to the direct sum of the full matrix algebras.
J. Rotman, “Advanced Abstract Algebra”, attributes this to
T. Molien.37
![Page 39: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/39.jpg)
Summary
The centralizer algebra VC(G ,⌦), or more generally, the adjacency
algebra C[R] of an association scheme, is (abstractly) isomorphic to
M
i
Mni (C).
In terms of block diagonal matrices, Mni (C) appear mi times in the
diagonal.
In particular, the group algebra:
C[G ] ⇠= the algebra of the right regular representation
⇠= the algebra of the left regular representation
= centralizer of the right regular representation
is also isomorphic to the direct sum of the full matrix algebras.
J. Rotman, “Advanced Abstract Algebra”, attributes this to
T. Molien.38
![Page 40: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/40.jpg)
Summary
The centralizer algebra VC(G ,⌦), or more generally, the adjacency
algebra C[R] of an association scheme, is (abstractly) isomorphic to
M
i
Mni (C).
In terms of block diagonal matrices, Mni (C) appear mi times in the
diagonal.
In particular, the group algebra:
C[G ] ⇠= the algebra of the right regular representation
⇠= the algebra of the left regular representation
= centralizer of the right regular representation
is also isomorphic to the direct sum of the full matrix algebras.
J. Rotman, “Advanced Abstract Algebra”, attributes this to
T. Molien.39
![Page 41: Permutation representations of finite groups and ...math.nsc.ru/conference/g2/g2r2/files/pdf/munemasa-lecture3.pdf · Contents 1 Notation and motivation 2 Association schemes 3 ⇤-Algebras](https://reader033.vdocuments.net/reader033/viewer/2022042310/5ed83d880fa3e705ec0e19ae/html5/thumbnails/41.jpg)