representations of finite groups - mathematical...
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Representations of finite groups
Meinolf Geck
Aberdeen University
Prospects in Mathematics
Edinburgh, December 2010
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 1 / 10
The Classification Theorem
Theorem. Let G be a finite simple group.
Then G is isomorphic to:
Z=pZ where p is a prime number;
An where n > 5 (the group of even permutations of n letters);
a simple group of Lie type, e.g., PSLn(Fq), : : :, E8(Fq);
one of 26 sporadic simple groups:
I smallest: Mathieu group M11 (of order 7; 920);
I largest: Fischer–Griess Monster M (of order � 8� 1053).
The theorem was announced in 1981. “2nd generation proof”:
M. Aschbacher, S. D. Smith: The classification of quasithin groups,
Math. Surveys and Monographs, AMS, vol. 111/112 (2004), � 1220 pp.
D. Gorenstein, R. Lyons, R. Solomon: The classification of the
finite simple groups, Math. Surveys and Monographs, AMS, vol. 40.1–40.6
(1994–2005), � 2140 pp.; to be continued.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 2 / 10
The Classification Theorem
Theorem. Let G be a finite simple group. Then G is isomorphic to:
Z=pZ where p is a prime number;
An where n > 5 (the group of even permutations of n letters);
a simple group of Lie type, e.g., PSLn(Fq), : : :, E8(Fq);
one of 26 sporadic simple groups:
I smallest: Mathieu group M11 (of order 7; 920);
I largest: Fischer–Griess Monster M (of order � 8� 1053).
The theorem was announced in 1981. “2nd generation proof”:
M. Aschbacher, S. D. Smith: The classification of quasithin groups,
Math. Surveys and Monographs, AMS, vol. 111/112 (2004), � 1220 pp.
D. Gorenstein, R. Lyons, R. Solomon: The classification of the
finite simple groups, Math. Surveys and Monographs, AMS, vol. 40.1–40.6
(1994–2005), � 2140 pp.; to be continued.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 2 / 10
The Classification Theorem
Theorem. Let G be a finite simple group. Then G is isomorphic to:
Z=pZ where p is a prime number;
An where n > 5 (the group of even permutations of n letters);
a simple group of Lie type, e.g., PSLn(Fq), : : :, E8(Fq);
one of 26 sporadic simple groups:
I smallest: Mathieu group M11 (of order 7; 920);
I largest: Fischer–Griess Monster M (of order � 8� 1053).
The theorem was announced in 1981. “2nd generation proof”:
M. Aschbacher, S. D. Smith: The classification of quasithin groups,
Math. Surveys and Monographs, AMS, vol. 111/112 (2004), � 1220 pp.
D. Gorenstein, R. Lyons, R. Solomon: The classification of the
finite simple groups, Math. Surveys and Monographs, AMS, vol. 40.1–40.6
(1994–2005), � 2140 pp.; to be continued.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 2 / 10
The Classification Theorem
Theorem. Let G be a finite simple group. Then G is isomorphic to:
Z=pZ where p is a prime number;
An where n > 5 (the group of even permutations of n letters);
a simple group of Lie type, e.g., PSLn(Fq), : : :, E8(Fq);
one of 26 sporadic simple groups:
I smallest: Mathieu group M11 (of order 7; 920);
I largest: Fischer–Griess Monster M (of order � 8� 1053).
The theorem was announced in 1981. “2nd generation proof”:
M. Aschbacher, S. D. Smith: The classification of quasithin groups,
Math. Surveys and Monographs, AMS, vol. 111/112 (2004), � 1220 pp.
D. Gorenstein, R. Lyons, R. Solomon: The classification of the
finite simple groups, Math. Surveys and Monographs, AMS, vol. 40.1–40.6
(1994–2005), � 2140 pp.; to be continued.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 2 / 10
The Classification Theorem
Theorem. Let G be a finite simple group. Then G is isomorphic to:
Z=pZ where p is a prime number;
An where n > 5 (the group of even permutations of n letters);
a simple group of Lie type, e.g., PSLn(Fq), : : :, E8(Fq);
one of 26 sporadic simple groups:
I smallest: Mathieu group M11 (of order 7; 920);
I largest: Fischer–Griess Monster M (of order � 8� 1053).
The theorem was announced in 1981. “2nd generation proof”:
M. Aschbacher, S. D. Smith: The classification of quasithin groups,
Math. Surveys and Monographs, AMS, vol. 111/112 (2004), � 1220 pp.
D. Gorenstein, R. Lyons, R. Solomon: The classification of the
finite simple groups, Math. Surveys and Monographs, AMS, vol. 40.1–40.6
(1994–2005), � 2140 pp.; to be continued.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 2 / 10
The Classification Theorem
Theorem. Let G be a finite simple group. Then G is isomorphic to:
Z=pZ where p is a prime number;
An where n > 5 (the group of even permutations of n letters);
a simple group of Lie type, e.g., PSLn(Fq), : : :, E8(Fq);
one of 26 sporadic simple groups:
I smallest: Mathieu group M11 (of order 7; 920);
I largest: Fischer–Griess Monster M (of order � 8� 1053).
The theorem was announced in 1981.
“2nd generation proof”:
M. Aschbacher, S. D. Smith: The classification of quasithin groups,
Math. Surveys and Monographs, AMS, vol. 111/112 (2004), � 1220 pp.
D. Gorenstein, R. Lyons, R. Solomon: The classification of the
finite simple groups, Math. Surveys and Monographs, AMS, vol. 40.1–40.6
(1994–2005), � 2140 pp.; to be continued.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 2 / 10
The Classification Theorem
Theorem. Let G be a finite simple group. Then G is isomorphic to:
Z=pZ where p is a prime number;
An where n > 5 (the group of even permutations of n letters);
a simple group of Lie type, e.g., PSLn(Fq), : : :, E8(Fq);
one of 26 sporadic simple groups:
I smallest: Mathieu group M11 (of order 7; 920);
I largest: Fischer–Griess Monster M (of order � 8� 1053).
The theorem was announced in 1981. “2nd generation proof”:
M. Aschbacher, S. D. Smith: The classification of quasithin groups,
Math. Surveys and Monographs, AMS, vol. 111/112 (2004), � 1220 pp.
D. Gorenstein, R. Lyons, R. Solomon: The classification of the
finite simple groups, Math. Surveys and Monographs, AMS, vol. 40.1–40.6
(1994–2005), � 2140 pp.; to be continued.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 2 / 10
Why is the Classification useful?
Let G be a finite group and C be a category. A representation of G in
C is a group homomorphism � : G ! Aut(X ) for some object X 2 C.
C: category of sets permutation representations;
C: category of vector spaces linear representations.
Finite simple groups have “interesting” representations (e.g., on Lie
algebras, geometries) and hence carry a lot of structure. The general
finite group is very complex but does not possess much structure.
Aschbacher: “The Classification is a tool for passing from the highly
complex unstructured universe of the general finite group to the much less
complex but highly structured universe of the finite simple group.
If one can reduce a problem to the simple case then at the same time one
has avoided a great deal of complexity and made it possible to take
advantage of the structure of simple groups in solving the problem.”
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 3 / 10
Why is the Classification useful?
Let G be a finite group and C be a category. A representation of G in
C is a group homomorphism � : G ! Aut(X ) for some object X 2 C.
C: category of sets permutation representations;
C: category of vector spaces linear representations.
Finite simple groups have “interesting” representations (e.g., on Lie
algebras, geometries) and hence carry a lot of structure. The general
finite group is very complex but does not possess much structure.
Aschbacher: “The Classification is a tool for passing from the highly
complex unstructured universe of the general finite group to the much less
complex but highly structured universe of the finite simple group.
If one can reduce a problem to the simple case then at the same time one
has avoided a great deal of complexity and made it possible to take
advantage of the structure of simple groups in solving the problem.”
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 3 / 10
Why is the Classification useful?
Let G be a finite group and C be a category. A representation of G in
C is a group homomorphism � : G ! Aut(X ) for some object X 2 C.
C: category of sets permutation representations;
C: category of vector spaces linear representations.
Finite simple groups have “interesting” representations (e.g., on Lie
algebras, geometries) and hence carry a lot of structure. The general
finite group is very complex but does not possess much structure.
Aschbacher: “The Classification is a tool for passing from the highly
complex unstructured universe of the general finite group to the much less
complex but highly structured universe of the finite simple group.
If one can reduce a problem to the simple case then at the same time one
has avoided a great deal of complexity and made it possible to take
advantage of the structure of simple groups in solving the problem.”
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 3 / 10
Why is the Classification useful?
Let G be a finite group and C be a category. A representation of G in
C is a group homomorphism � : G ! Aut(X ) for some object X 2 C.
C: category of sets permutation representations;
C: category of vector spaces linear representations.
Finite simple groups have “interesting” representations (e.g., on Lie
algebras, geometries) and hence carry a lot of structure. The general
finite group is very complex but does not possess much structure.
Aschbacher: “The Classification is a tool for passing from the highly
complex unstructured universe of the general finite group to the much less
complex but highly structured universe of the finite simple group.
If one can reduce a problem to the simple case then at the same time one
has avoided a great deal of complexity and made it possible to take
advantage of the structure of simple groups in solving the problem.”
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 3 / 10
Why is the Classification useful?
Let G be a finite group and C be a category. A representation of G in
C is a group homomorphism � : G ! Aut(X ) for some object X 2 C.
C: category of sets permutation representations;
C: category of vector spaces linear representations.
Finite simple groups have “interesting” representations (e.g., on Lie
algebras, geometries) and hence carry a lot of structure. The general
finite group is very complex but does not possess much structure.
Aschbacher: “The Classification is a tool for passing from the highly
complex unstructured universe of the general finite group to the much less
complex but highly structured universe of the finite simple group.
If one can reduce a problem to the simple case then at the same time one
has avoided a great deal of complexity and made it possible to take
advantage of the structure of simple groups in solving the problem.”
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 3 / 10
Linear representations
Let G be a finite group, k a field, and V a vector space over k .
Representation of G on V : a group homomorphism � : G ! GL(V ).
V equipped with linear action of G ; say that V is a G -module.
V 6= f0g irreducible: the only G -invariant subspaces are f0g and V .
Irrk(G ) = f irreducible G -modules over k g= �.
“Semisimple case”: char(k) = 0.
Then every representation is a direct sum of irreducible ones.
“Modular case”: char(k) = ` > 0.
If ` divides jG j, then complete reducibility fails
R. Brauer’s theory of blocks, decomposition numbers, : : :
Aim: Determine Irrk(G ) for G any finite simple group, any field k .
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 4 / 10
Linear representations
Let G be a finite group, k a field, and V a vector space over k .
Representation of G on V : a group homomorphism � : G ! GL(V ).
V equipped with linear action of G ; say that V is a G -module.
V 6= f0g irreducible: the only G -invariant subspaces are f0g and V .
Irrk(G ) = f irreducible G -modules over k g= �.
“Semisimple case”: char(k) = 0.
Then every representation is a direct sum of irreducible ones.
“Modular case”: char(k) = ` > 0.
If ` divides jG j, then complete reducibility fails
R. Brauer’s theory of blocks, decomposition numbers, : : :
Aim: Determine Irrk(G ) for G any finite simple group, any field k .
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 4 / 10
Linear representations
Let G be a finite group, k a field, and V a vector space over k .
Representation of G on V : a group homomorphism � : G ! GL(V ).
V equipped with linear action of G ; say that V is a G -module.
V 6= f0g irreducible: the only G -invariant subspaces are f0g and V .
Irrk(G ) = f irreducible G -modules over k g= �.
“Semisimple case”: char(k) = 0.
Then every representation is a direct sum of irreducible ones.
“Modular case”: char(k) = ` > 0.
If ` divides jG j, then complete reducibility fails
R. Brauer’s theory of blocks, decomposition numbers, : : :
Aim: Determine Irrk(G ) for G any finite simple group, any field k .
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 4 / 10
Linear representations
Let G be a finite group, k a field, and V a vector space over k .
Representation of G on V : a group homomorphism � : G ! GL(V ).
V equipped with linear action of G ; say that V is a G -module.
V 6= f0g irreducible: the only G -invariant subspaces are f0g and V .
Irrk(G ) = f irreducible G -modules over k g= �.
“Semisimple case”: char(k) = 0.
Then every representation is a direct sum of irreducible ones.
“Modular case”: char(k) = ` > 0.
If ` divides jG j, then complete reducibility fails
R. Brauer’s theory of blocks, decomposition numbers, : : :
Aim: Determine Irrk(G ) for G any finite simple group, any field k .
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 4 / 10
Linear representations
Let G be a finite group, k a field, and V a vector space over k .
Representation of G on V : a group homomorphism � : G ! GL(V ).
V equipped with linear action of G ; say that V is a G -module.
V 6= f0g irreducible: the only G -invariant subspaces are f0g and V .
Irrk(G ) = f irreducible G -modules over k g= �.
“Semisimple case”: char(k) = 0.
Then every representation is a direct sum of irreducible ones.
“Modular case”: char(k) = ` > 0.
If ` divides jG j, then complete reducibility fails
R. Brauer’s theory of blocks, decomposition numbers, : : :
Aim: Determine Irrk(G ) for G any finite simple group, any field k .
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 4 / 10
Linear representations
Let G be a finite group, k a field, and V a vector space over k .
Representation of G on V : a group homomorphism � : G ! GL(V ).
V equipped with linear action of G ; say that V is a G -module.
V 6= f0g irreducible: the only G -invariant subspaces are f0g and V .
Irrk(G ) = f irreducible G -modules over k g= �.
“Semisimple case”: char(k) = 0.
Then every representation is a direct sum of irreducible ones.
“Modular case”: char(k) = ` > 0.
If ` divides jG j, then complete reducibility fails
R. Brauer’s theory of blocks, decomposition numbers, : : :
Aim: Determine Irrk(G ) for G any finite simple group, any field k .
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 4 / 10
Local-global problems
Let G be a finite group, p be a prime number.
Guiding Principle (Brauer, : : :): Global representation-theoretic
invariants of G should be determined “p-locally”.
Write jG j = pam where a > 0 and p - m.
Let P � G be a subgroup such that jP j = pa (“Sylow subgroup”).
Then the normaliser NG (P) is called a “p-local subgroup of G”.
Simplest example of an unsolved local-global problem:
McKay Conjecture (1971)
jfV 2 IrrC(G ) j p - dim V gj| {z }global
= jfV 0 2 IrrC(NG (P)) j p - dim V 0gj| {z }local
.
I. M. Isaacs, G. Malle, G. Navarro: A reduction theorem for the McKay
conjecture, Invent. Math. 170 (2007), 33–101.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 5 / 10
Local-global problems
Let G be a finite group, p be a prime number.
Guiding Principle (Brauer, : : :): Global representation-theoretic
invariants of G should be determined “p-locally”.
Write jG j = pam where a > 0 and p - m.
Let P � G be a subgroup such that jP j = pa (“Sylow subgroup”).
Then the normaliser NG (P) is called a “p-local subgroup of G”.
Simplest example of an unsolved local-global problem:
McKay Conjecture (1971)
jfV 2 IrrC(G ) j p - dim V gj| {z }global
= jfV 0 2 IrrC(NG (P)) j p - dim V 0gj| {z }local
.
I. M. Isaacs, G. Malle, G. Navarro: A reduction theorem for the McKay
conjecture, Invent. Math. 170 (2007), 33–101.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 5 / 10
Local-global problems
Let G be a finite group, p be a prime number.
Guiding Principle (Brauer, : : :): Global representation-theoretic
invariants of G should be determined “p-locally”.
Write jG j = pam where a > 0 and p - m.
Let P � G be a subgroup such that jP j = pa (“Sylow subgroup”).
Then the normaliser NG (P) is called a “p-local subgroup of G”.
Simplest example of an unsolved local-global problem:
McKay Conjecture (1971)
jfV 2 IrrC(G ) j p - dim V gj| {z }global
= jfV 0 2 IrrC(NG (P)) j p - dim V 0gj| {z }local
.
I. M. Isaacs, G. Malle, G. Navarro: A reduction theorem for the McKay
conjecture, Invent. Math. 170 (2007), 33–101.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 5 / 10
Local-global problems
Let G be a finite group, p be a prime number.
Guiding Principle (Brauer, : : :): Global representation-theoretic
invariants of G should be determined “p-locally”.
Write jG j = pam where a > 0 and p - m.
Let P � G be a subgroup such that jP j = pa (“Sylow subgroup”).
Then the normaliser NG (P) is called a “p-local subgroup of G”.
Simplest example of an unsolved local-global problem:
McKay Conjecture (1971)
jfV 2 IrrC(G ) j p - dim V gj| {z }global
= jfV 0 2 IrrC(NG (P)) j p - dim V 0gj| {z }local
.
I. M. Isaacs, G. Malle, G. Navarro: A reduction theorem for the McKay
conjecture, Invent. Math. 170 (2007), 33–101.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 5 / 10
Local-global problems
Let G be a finite group, p be a prime number.
Guiding Principle (Brauer, : : :): Global representation-theoretic
invariants of G should be determined “p-locally”.
Write jG j = pam where a > 0 and p - m.
Let P � G be a subgroup such that jP j = pa (“Sylow subgroup”).
Then the normaliser NG (P) is called a “p-local subgroup of G”.
Simplest example of an unsolved local-global problem:
McKay Conjecture (1971)
jfV 2 IrrC(G ) j p - dim V gj| {z }global
= jfV 0 2 IrrC(NG (P)) j p - dim V 0gj| {z }local
.
I. M. Isaacs, G. Malle, G. Navarro: A reduction theorem for the McKay
conjecture, Invent. Math. 170 (2007), 33–101.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 5 / 10
Local-global problems
Let G be a finite group, p be a prime number.
Guiding Principle (Brauer, : : :): Global representation-theoretic
invariants of G should be determined “p-locally”.
Write jG j = pam where a > 0 and p - m.
Let P � G be a subgroup such that jP j = pa (“Sylow subgroup”).
Then the normaliser NG (P) is called a “p-local subgroup of G”.
Simplest example of an unsolved local-global problem:
McKay Conjecture (1971)
jfV 2 IrrC(G ) j p - dim V gj| {z }global
= jfV 0 2 IrrC(NG (P)) j p - dim V 0gj| {z }local
.
I. M. Isaacs, G. Malle, G. Navarro: A reduction theorem for the McKay
conjecture, Invent. Math. 170 (2007), 33–101.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 5 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C.
Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`.
Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V :
fg 2 G j g has order prime to ` g
! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Character tables
Let G be a finite group, V a G -module.
Frobenius character: char(k) = 0, k = C. Define �V : G ! C by
�V (g) = trace(g ;V ) = sum of the eigenvalues of g on V .
Brauer character: char(k) = ` > 0, k = F`. Define
�V : fg 2 G j g has order prime to ` g ! C
by �V (g) =P
i !i where !i 2 C correspond to the eigenvalues of g
via a fixed isomorphism f! 2 C j !n = 1; gcd(n; `) = 1g��! F
�` .
Example: G = A5�= SL2(F4).
() (12)(34) (123) (12345) (13524)
�1 1 1 1 1 1
�2 3 �1 0 12(1+
p5) 1
2(1�
p5)
�3 3 �1 0 12(1�
p5) 1
2(1+
p5)
�4 4 0 1 �1 �1
�5 5 1 �1 0 0
Frobenius character table, char(k)=0
() (123) (12345) (13524)
�1 1 1 1 1
�2 2 �1 12(�1+
p5) 1
2(�1�
p5)
�3 2 �1 12(�1�
p5) 1
2(�1+
p5)
�4 4 1 �1 �1
Brauer character table, char(k)=2
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 6 / 10
Groups of Lie type
Let p be a prime number and q = pf for some f > 1. Then
SL2(Fq) =n "
a bc d
# ��� a; b; c ; d 2 Fq; ad � bc = 1o
=f�I2g
is an example of a finite group of Lie type
(simple unless q 2 f2; 3g).
Try to study all these groups simultaneously, as q varies and the
“type” (i.e., “(P)SL2” in this case, or A1 in Lie notation) is fixed.
In general, work in the context of the theory of algebraic groups and
use algebraic geometry over Fp, an algebraic closure of Fp = Z=pZ.
Linear algebraic
group G over Fp
=Affine algebraic
variety over Fp
+compatible group
structure
Chevalley (1950’s): The simple G (possibly finite centre) are
classified in terms of Dynkin types An, Bn, Cn, Dn, G2, F4, E6, E7, E8.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 7 / 10
Groups of Lie type
Let p be a prime number and q = pf for some f > 1. Then
PSL2(Fq) =n "
a bc d
# ��� a; b; c ; d 2 Fq; ad � bc = 1o=f�I2g
is an example of a finite group of Lie type (simple unless q 2 f2; 3g).
Try to study all these groups simultaneously, as q varies and the
“type” (i.e., “(P)SL2” in this case, or A1 in Lie notation) is fixed.
In general, work in the context of the theory of algebraic groups and
use algebraic geometry over Fp, an algebraic closure of Fp = Z=pZ.
Linear algebraic
group G over Fp
=Affine algebraic
variety over Fp
+compatible group
structure
Chevalley (1950’s): The simple G (possibly finite centre) are
classified in terms of Dynkin types An, Bn, Cn, Dn, G2, F4, E6, E7, E8.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 7 / 10
Groups of Lie type
Let p be a prime number and q = pf for some f > 1. Then
PSL2(Fq) =n "
a bc d
# ��� a; b; c ; d 2 Fq; ad � bc = 1o=f�I2g
is an example of a finite group of Lie type (simple unless q 2 f2; 3g).
Try to study all these groups simultaneously, as q varies and the
“type” (i.e., “(P)SL2” in this case, or A1 in Lie notation) is fixed.
In general, work in the context of the theory of algebraic groups and
use algebraic geometry over Fp, an algebraic closure of Fp = Z=pZ.
Linear algebraic
group G over Fp
=Affine algebraic
variety over Fp
+compatible group
structure
Chevalley (1950’s): The simple G (possibly finite centre) are
classified in terms of Dynkin types An, Bn, Cn, Dn, G2, F4, E6, E7, E8.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 7 / 10
Groups of Lie type
Let p be a prime number and q = pf for some f > 1. Then
PSL2(Fq) =n "
a bc d
# ��� a; b; c ; d 2 Fq; ad � bc = 1o=f�I2g
is an example of a finite group of Lie type (simple unless q 2 f2; 3g).
Try to study all these groups simultaneously, as q varies and the
“type” (i.e., “(P)SL2” in this case, or A1 in Lie notation) is fixed.
In general, work in the context of the theory of algebraic groups and
use algebraic geometry over Fp, an algebraic closure of Fp = Z=pZ.
Linear algebraic
group G over Fp
=Affine algebraic
variety over Fp
+compatible group
structure
Chevalley (1950’s): The simple G (possibly finite centre) are
classified in terms of Dynkin types An, Bn, Cn, Dn, G2, F4, E6, E7, E8.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 7 / 10
Groups of Lie type
Let p be a prime number and q = pf for some f > 1. Then
PSL2(Fq) =n "
a bc d
# ��� a; b; c ; d 2 Fq; ad � bc = 1o=f�I2g
is an example of a finite group of Lie type (simple unless q 2 f2; 3g).
Try to study all these groups simultaneously, as q varies and the
“type” (i.e., “(P)SL2” in this case, or A1 in Lie notation) is fixed.
In general, work in the context of the theory of algebraic groups and
use algebraic geometry over Fp, an algebraic closure of Fp = Z=pZ.
Linear algebraic
group G over Fp
=Affine algebraic
variety over Fp
+compatible group
structure
Chevalley (1950’s): The simple G (possibly finite centre) are
classified in terms of Dynkin types An, Bn, Cn, Dn, G2, F4, E6, E7, E8.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 7 / 10
Groups of Lie type
Let p be a prime number and q = pf for some f > 1. Then
PSL2(Fq) =n "
a bc d
# ��� a; b; c ; d 2 Fq; ad � bc = 1o=f�I2g
is an example of a finite group of Lie type (simple unless q 2 f2; 3g).
Try to study all these groups simultaneously, as q varies and the
“type” (i.e., “(P)SL2” in this case, or A1 in Lie notation) is fixed.
In general, work in the context of the theory of algebraic groups and
use algebraic geometry over Fp, an algebraic closure of Fp = Z=pZ.
Linear algebraic
group G over Fp
=Affine algebraic
variety over Fp
+compatible group
structure
Chevalley (1950’s): The simple G (possibly finite centre) are
classified in terms of Dynkin types An, Bn, Cn, Dn, G2, F4, E6, E7, E8.Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 7 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Groups of Lie type
Examples: An�1 $ SLn(Fp), Bn $ SO2n+1(Fp), : : :
If G is defined over Fq � Fp (where q = pf for some f > 1), then
G(Fq) := group of Fq-rational points (“Fq-form of G ”)
This leads to a series of finite groups of Lie type:
S = fG (q) := G(Fq) j q prime power; G fixed Dynkin typeg.
Order formula (Chevalley, Solomon, Steinberg).
There exists a polynomial f 2 Z[X ], which only depends on the
“type” of the series S, such that jG (q)j = f (q) for any q.
jSL2(q)j = q(q2 � 1)
jSLn(q)j = q12 n(n�1)(q2 � 1)(q3 � 1) � � � (qn � 1)
jE8(q)j = q120(q2 � 1)(q8 � 1)(q12 � 1)(q14 � 1)
�(q18 � 1)(q20 � 1)(q24 � 1)(q30 � 1)
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 8 / 10
Genericity of characters
Theorem (Lusztig 1980’s)
/ Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 Irr(G (q)) g � f f (q) j f 2 P g
for any q and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s)
/ Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 IrrC(G (q)) g
� f f (q) j f 2 P g
for any q and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s)
/ Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 IrrC(G (q)) g � f f (q) j f 2 P g
for any q
and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s)
/ Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 IrrC(G (q)) g � f f (q) j f 2 P g
for any q
and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s)
/ Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 IrrC(G (q)) g � f f (q) j f 2 P g
for any q
and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s)
/ Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 IrrC(G (q)) g � f f (q) j f 2 P g
for any q
and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s)
/ Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 IrrC(G (q)) g � f f (q) j f 2 P g
for any q
and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s) / Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 Irrk(G (q)) g � f f (q) j f 2 P g
for any q and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s) / Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 Irrk(G (q)) g � f f (q) j f 2 P g
for any q and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Genericity of characters
Theorem (Lusztig 1980’s) / Conjecture
There is a finite set of polynomials P � Q[X ], depending only on the
“type” of the series S, such that
f dim V j V 2 Irrk(G (q)) g � f f (q) j f 2 P g
for any q and any k such that char(k) 6= p (where q = pf ).
Example: G (q) = SL2(Fq), q = pf , k = C.
q fdim V g
2 1; 1; 2
3 1; 1; 1; 2; 2; 2; 3
4 1; 3; 3; 4; 5
5 1; 2; 2; 3; 3; 4; 4; 5; 6
7 1; 3; 3; 4; 4; 6; 6; 6; 7; 8; 8
8 1; 7; 7; 7; 7; 8; 9; 9; 9
P = f 1; X ; X � 1; 12 (X � 1) g
k = F`, ` 6= p: Same P works here !
Other series: enlarge Lusztig’s P by
finitely many new polynomials.
Note: Contrary to this, if k = Fp, then f1; p; p2; p3; : : : ; pf g � fdim V g.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 9 / 10
Research areas
Finite simple groups: Subgroup structure, construction
(computational methods), character tables, : : :;
Modular representations: Local–global problems, block theory,
fusion systems, : : :;
Related structures in Lie theory: Quantum groups, canonical
bases, Hecke algebras, Cherednik algebras, : : :;
Geometric representation theory: Connections with algebraic
topology, derived categories, Langlands programme, : : :.
Active research in all major UK universities.
Meinolf Geck (Aberdeen University) Representations of finite groups Prospects Edinburgh 2010 10 / 10