Download David Wehlau

Modular Invariant Theory of the Cyclic Group
via Classical Invariant Theory
David Wehlau
June 5, 2010
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 1
Outline
Modular Representation
Theory of Cp
Modular RepresentationTheory of Cp .
Representation Theory of
Representation Theory of SL2 (C)
The Connection
Shank’s Conjecture
Proof of the Conjecture
Some Consequences
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 2
Modular Representation
Theory of Cp
The Modular Group of
Prime Order
Vn
Known Modular
Invariants of Cp
Representation Theory of
SL2 (C)
The Connection
Modular Representation Theory
of Cp
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 3
The Modular Group of Prime Order
Modular Representation
Theory of Cp
The Modular Group of
Prime Order
Let F be any field of characteristic p > 0 and let Cp denote the cyclic
group of order p. We fix a generator σ of Cp .
Vn
Known Modular
Invariants of Cp
Representation Theory of
SL2 (C)
Suppose V is an indecomposable representation of Cp defined over F.
Consider the Jordan Normal Form of the matrix of σ in GL(V ). This is
the matrix:
The Connection
1 1 0
 0 1 1


 0 0 1
 . . .
 .. .. ..

 0 0 0
0 0 0

Shank’s Conjecture
Proof of the conjecture
Some Consequences
where 1 ≤ n ≤ p.
Fredericton, New Brunswick, 5 June 2010
···
···
..
.
..
.
···
···
0
0
0
0 


0 0 
.. 
..
. . 

1 1 
0 1 n×n

CMS Summer Meeting – slide 4
Vn
1. We get p inequivalent indecomposable representations: V1 ( V2 ( · · · ( Vp
C
2. Vn p ∼
= V1 for all n.
3. ∆ := σ − 1
Pp−1 i
p−1
· f = ∆p−1 (f ) where ∆ = σ − 1.
4. Tr(f ) =
i=0 σ · f = (σ − 1)
Cp
Thus a non-zero invariant h is a transfer if and only if h ∈ Vp .
Qp−1 i
5. Norm(f ) = i=0 σ · f .
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 5
Known Modular Invariants of Cp
Modular Representation
Theory of Cp
The Modular Group of
Prime Order
1.
Fp [⊕m V2 ]Cp
2.
Fp [V3 ]Cp
3.
Fp [V4 ]Cp
4.
Fp [V5 ]Cp
5.
Fp [V2 ⊕ V3 ]Cp
6.
Fp [V3 ⊕ V3 ]Cp
7.
Fp [V2 ⊕ V2 ⊕ V3 ]Cp
Vn
Known Modular
Invariants of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 6
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
Representations of
SL2 (C)
The Connection
Shank’s Conjecture
Representation Theory of SL2 (C)
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 7
Representations of SL2 (C)
Modular Representation
Theory of Cp
Representation Theory of
1. R1 denotes the defining two dimensional representation of SL2 (C)
with fixed basis {X, Y }.
SL2 (C)
Representations of
SL2 (C)
2. Rd := Sd (R1 ) = span{X d , X d−1 Y, . . . , Y d }.
The Connection
Shank’s Conjecture
Proof of the conjecture
3. Every irreducible SL2 (C) representation is isomorphic to Rd for
some d.
Some Consequences
4. Classical invariant theorists studied not just
the ring of invariants C[W ]SL2 (C) of a representation W
but also C[R1 ⊕ W ]SL2 (C) , the ring of covariants of W .
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 8
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Robert’s Isomorphism
(1861)
The Integers
Notation
The Connection
From SL2 (C) to Cp
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 9
Robert’s Isomorphism (1861)
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
C[R1 ⊕ W ]SL2 (C) −→ C[W ]U
f (·, ·) 7→ f (X, ·)
Robert’s Isomorphism
(1861)
The Integers
where
Notation
From SL2 (C) to Cp
Shank’s Conjecture
Proof of the conjecture
Some Consequences
U = {τ ∈ SL2 (C) | τ · X = X}
1 u
=
|u∈C .
0 1
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 10
Robert’s Isomorphism (1861)
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
C[R1 ⊕ W ]SL2 (C) −→ C[W ]U
f (·, ·) 7→ f (X, ·)
∼
Robert’s Isomorphism
(1861)
The Integers
where
Notation
From SL2 (C) to Cp
Shank’s Conjecture
Proof of the conjecture
Some Consequences
U = {τ ∈ SL2 (C) | τ · X = X}
1 u
=
|u∈C .
0 1
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 11
The Integers
Modular Representation
Theory of Cp
U contains the subgroup
Representation Theory of
SL2 (C)
The Connection
Robert’s Isomorphism
(1861)
U⊃
1 k
0 1
|k∈Z
The Integers
Notation
From SL2 (C) to Cp
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 12
The Integers
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Robert’s Isomorphism
(1861)
The Integers
Notation
From SL2 (C) to Cp
Shank’s Conjecture
U contains the subgroup
1 k
U⊃
|k∈Z =Z
0 1
The fact that Z is a dense subgroup of U in the Zariski topology implies
C[W ]U = C[W ]Z .
Proof of the conjecture
Some Consequences
Note that the action of Z is generated by the element
Fredericton, New Brunswick, 5 June 2010
1 1
0 1
CMS Summer Meeting – slide 13
Notation
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Robert’s Isomorphism
(1861)
The Integers
Define
• Mn := Rn−1 (Q)
• Ln := Rn−1 (Z) = Mn (Z)
• If W is an SL2 (C)-representation, M = W (Q) and L = W (Z)
If W = ⊕ti=1 Rni −1 then M = ⊕ti=1 Mni and L = ⊕ti=1 Lni .
Notation
From SL2 (C) to Cp
The element 1 ∈ Z ⊂ U acts on Rn−1 , Mn and Ln via the matrix
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
1 1 0
 0 1 1


 0 0 1
 . . .
 .. .. ..

 0 0 0
0 0 0

···
···
..
.
..
.
···
···
0
0
0
0 


0 0 
.. 
..
. . 

1 1 
0 1 n×n

CMS Summer Meeting – slide 14
From SL2 (C) to Cp
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Robert’s Isomorphism
(1861)
The Integers
Notation
From SL2 (C) to Cp
Shank’s Conjecture
C[R1 ⊕ W ]SL2 (C) ∼
= C[W ]U = C[W ]Z
∼
= Q[M ]Z ⊗Q C
∼
= (Z[L]Z ⊗Z Q) ⊗Q C
∼
= Z[L]Z ⊗Z C
S • (L∗ ) = Z[L]
⊃
Z[L]Z
Proof of the conjecture
↓ (mod p)
Some Consequences
S • (V ∗ ) = Fp [V ]
↓ (mod p)
⊃
Fp [V ]Cp
Elements of Fp [V ]Cp which lie in the image of Z[L]Z under reduction
modulo p are called integral invariants.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 15
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Shank’s Conjecture
(1996)
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 16
Shank’s Conjecture (1996)
Modular Representation
Theory of Cp
F[V ]Cp is generated by
Representation Theory of
1.
Certain specific norms (one for each summand of V ).
2.
Certain transfers (finitely many).
3.
A finite set of integral invariants.
SL2 (C)
The Connection
Shank’s Conjecture
Shank’s Conjecture
(1996)
Proof of the conjecture
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 17
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
The Tensor Algebra
Proof of the conjecture
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 18
The Tensor Algebra
Fp [Vn ]d = S d (Vn∗ ) is a summand of ⊗d V ∗ . We consider ⊗d Vn .
(⊗d Mn )
⊂
⊂
S d (Ln )
⊂
(⊗d Ln )
Ln1 ⊕ Ln2 ⊕ · · · ⊕ Lns
↓ (mod p)
S d (Vn )
∼
=
⊂
⊂
S d (Mn )
⊂
↓ (mod p)
(⊗d Vn )
∼
=
Mn1 ⊕ Mn2 ⊕ · · · ⊕ Mns
↓ (mod p)
Vn 1 ⊕ Vn 2 ⊕ · · · ⊕ Vn s
We need to show that the co-kernel of the map on invariants is spanned by transfers for
d < p.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 19
Example (p = 7)
4
S (L5 )
⊂
↓ (mod p)
S 4 (V5 )
⊗ M5
∼
=
5L1 ⊕ 12L3 ⊕ 16L5 ⊕ 17L7
⊕15L9 ⊕ 10L11 ⊕ 6L13 ⊕ 3L15 ⊕ L17
4
⊗ L5
↓ (mod p)
⊂
⊗4 V5
Fredericton, New Brunswick, 5 June 2010
5M1 ⊕ 12M3 ⊕ 16M5 ⊕ 17M7
⊕15M9 ⊕ 10M11 ⊕ 6M13 ⊕ 3M15 ⊕ M17
⊂
⊂
⊂
S (M5 )
4
⊂
4
∼
=
↓ (mod p)
2V1 ⊕ 3V3 ⊕ V5 ⊕ 87V7
CMS Summer Meeting – slide 20
Three Representation Rings
Modular Representation
Theory of Cp
Let m ≤ n. We compare:
Representation Theory of
Rm−1 ⊗ Rn−1
Mm ⊗ Mn
Vm ⊗ Vn
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 21
Clebsch-Gordan Rule
Modular Representation
Theory of Cp
Rm ⊗ Rn ∼
= Rn−m ⊕ Rn−m+2 ⊕ · · · ⊕ Rm+n
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Rm−1 ⊗ Rn−1 ∼
= Rn−m+1 ⊕ Rn−m+3 ⊕ · · · ⊕ Rm+n−1
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 22
Jordan Normal Form
Modular Representation
Theory of Cp
Mm ⊗ Mn ∼
= Mn−m+1 ⊕ Mn−m+3 ⊕ · · · ⊕ Mm+n−1
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Roth (1934)
Aiken (1934)
(Littelwood 1936)
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Markus and Robinson (1975)
Brualdi (1985)
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 23
Representation ring of Cp
If 1 ≤ m ≤ n ≤ p,
Vm ⊗ Vn ∼
=
(
Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ Vm+n−1 ,
if m + n ≤ p + 1;
Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ V2p−m−n−1 ⊕ (⊕m+n−p Vp ), if m + n ≥ p.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 24
Mn−m+1 ⊕ Mn−m+3 ⊕ · · · ⊕ Mm+n−1
⊂
∼
=
⊂
Mm ⊗ Mn
Lm ⊗ Ln
Ln−m+1 ⊕ Ln−m+3 ⊕ · · · ⊕ Lm+n−1
↓ (mod p)
Vm ⊗ Vn
∼
=
↓ (mod p)
Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ Vn+m−1
if m + n ≤ p + 1.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 25
∼
=
⊂
Mn−m+1 ⊕ Mn−m+3 ⊕ · · · ⊕ Mm+n−1
⊂
Mm ⊗ Mn
Lm ⊗ Ln
Ln−m+1 ⊕ Ln−m+3 ⊕ · · · ⊕ Lm+n−1
↓ (mod p)
Vm ⊗ Vn
∼
=
↓ (mod p)
Vn−m+1 ⊕ Vn−m+3 ⊕ · · · ⊕ V2p−m−n−1 ⊕ (⊕m+n−p Vp )
if m + n ≥ p.
It suffices to prove that Lr surjects onto Vr here
for all r ≤ 2p − m − n − 1 when m ≤ n ≤ p.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 26
Modular Representation
Theory of Cp
Consider m = 3 and n = 4.
Representation Theory of
M3 ⊗ M4
SL2 (C)
∼
=
M2 ⊕ M4 ⊕ M6
⊂
Shank’s Conjecture
⊂
The Connection
L3 ⊗ L4
L2 ⊕ L4 ⊕ L6
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
↓ (mod p)
Clebsch-Gordan Rule
↓ (mod p)
Jordan Normal Form
Representation ring of
Cp
V3 ⊗ V4
Explicit Decompositions
∼
=
V2 ⊕ V4 ⊕ V6
Some Consequences
if p ≥ 7.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 27
Modular Representation
Theory of Cp
If m = 3, n = 4 and p = 5.
Representation Theory of
SL2 (C)
M3 ⊗ M4
∼
=
M2 ⊕ M4 ⊕ M6
⊂
Shank’s Conjecture
⊂
The Connection
L3 ⊗ L4
L2 ⊕ L4 ⊕ L6
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
↓ (mod p)
↓ (mod p)
Representation ring of
Cp
Explicit Decompositions
V3 ⊗ V4
∼
=
V2 ⊕ V5 ⊕ V5
Some Consequences
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 28
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
M3 ⊗ M4 ∼
= Q[s]/(s3 ) ⊗ Q[t]/t4 ∼
= Q[s, t]/(s3 , t4 ).
M3 ⊗ M4 ∼
= M2 ⊕ M4 ⊕ M6 .
∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st.
0
1
1
s t
2 s2 st t2
3 s2 t st2 t3
4
s2 t2 st3
5
s2 t3
0
•
1
• •
2 • • •
3 • • •
4
• •
5
•
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 29
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
M3 ⊗ M4 ∼
= Q[s]/(s3 ) ⊗ Q[t]/t4 ∼
= Q[s, t]/(s3 , t4 ).
M3 ⊗ M4 ∼
= M2 ⊕ M4 ⊕ M6 .
∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st.
0
1
1
s t
2 s2 st t2
3 s2 t st2 t3
4
s2 t2 st3
5
s2 t3
•
0
1
• •
2 • • •
3 • • •
4
• •
5
•
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 30
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
M3 ⊗ M4 ∼
= Q[s]/(s3 ) ⊗ Q[t]/t4 ∼
= Q[s, t]/(s3 , t4 ).
M3 ⊗ M4 ∼
= M2 ⊕ M4 ⊕ M6 .
∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st.
0
1
1
s t
2 s2 st t2
3 s2 t st2 t3
4
s2 t2 st3
5
s2 t3
•
0
1
• •
2 • • •
3 • • •
4
• •
5
•
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 31
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
M3 ⊗ M4 ∼
= Q[s]/(s3 ) ⊗ Q[t]/t4 ∼
= Q[s, t]/(s3 , t4 ).
M3 ⊗ M4 ∼
= M2 ⊕ M4 ⊕ M6 .
∆ acts via multiplication by (1 + s)(1 + t) − 1 = s + t + st.
0
1
1
s t
2 s2 st t2
3 s2 t st2 t3
4
s2 t2 st3
5
s2 t3
•
0
1
• •
2 • • •
3 • • •
4
• •
5
•
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 32
Modular Representation
Theory of Cp
M3 ⊗ M4 ∼
= Q[s, t]/(s3 , t4 ) ∼
= M2 ⊕ M4 ⊕ M6 .
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
If p ≥ 7:
V3 ⊗ V4 ∼
= Fp [s, t]/(s3 , t4 ) ∼
= V2 ⊕ V4 ⊕ V6 .
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
Deg M3 ⊗ M4
0
1
2
3
4
5
•
• •
• • •
• • •
• •
•
V3 ⊗ V4
•
• •
• • •
• • •
• •
•
CMS Summer Meeting – slide 33
Modular Representation
Theory of Cp
M3 ⊗ M4 ∼
= Q[s, t]/(s3 , t4 ) ∼
= M2 ⊕ M4 ⊕ M6 .
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
If p = 5:
V3 ⊗ V4 ∼
= Fp [s, t]/(s3 , t4 ) ∼
= V2 ⊕ V5 ⊕ V5 .
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
Deg M3 ⊗ M4
0
1
2
3
4
5
•
• •
• • •
• • •
• •
•
CMS Summer Meeting – slide 34
Modular Representation
Theory of Cp
M3 ⊗ M4 ∼
= Q[s, t]/(s3 , t4 ) ∼
= M2 ⊕ M4 ⊕ M6 .
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
If p = 5:
V3 ⊗ V4 ∼
= Fp [s, t]/(s3 , t4 ) ∼
= V2 ⊕ V5 ⊕ V5 .
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
Deg M3 ⊗ M4
0
1
2
3
4
5
•
• •
• • •
• • •
• •
•
V3 ⊗ V4
•
• •
• • •
• • •
• •
•
CMS Summer Meeting – slide 35
Modular Representation
Theory of Cp
M3 ⊗ M4 ∼
= Q[s, t]/(s3 , t4 ) ∼
= M2 ⊕ M4 ⊕ M6 .
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
If p = 5:
V3 ⊗ V4 ∼
= Fp [s, t]/(s3 , t4 ) ∼
= V2 ⊕ V5 ⊕ V5 .
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
Deg M3 ⊗ M4
0
1
2
3
4
5
•
• •
• • •
• • •
• •
•
V3 ⊗ V4
•
• •
• • •
• • •
• •
•
CMS Summer Meeting – slide 36
Modular Representation
Theory of Cp
M3 ⊗ M4 ∼
= Q[s, t]/(s3 , t4 ) ∼
= M2 ⊕ M4 ⊕ M6 .
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
If p = 5:
V3 ⊗ V4 ∼
= Fp [s, t]/(s3 , t4 ) ∼
= V2 ⊕ V5 ⊕ V5 .
Proof of the conjecture
The Tensor Algebra
Example (p = 7)
Three Representation
Rings
Clebsch-Gordan Rule
Jordan Normal Form
Representation ring of
Cp
Explicit Decompositions
Some Consequences
Fredericton, New Brunswick, 5 June 2010
Deg M3 ⊗ M4
0
1
2
3
4
5
•
• •
• • •
• • •
• •
•
V3 ⊗ V4
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CMS Summer Meeting – slide 37
Explicit Decompositions
If p ≥ 7
M3 ⊗ M4
M6 1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2
7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3
7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 10s2 t3 7→ 0
M4 3s − 2t 7→ 3s2 + st − 2t2 + 3s2 t − 2st2
7→ 4s2 t − st2 − 2t3 + 2s2 t2 − 4st3 − 2s2 t3
7→ 3s2 t2 − 3st3 − 3s2 t3 7→ 0
M2 3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2s2 t2 + 3st3 7→ 0
V6
V3 ⊗ V4
V4
V2
1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2
7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3
7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 10s2 t3 7→ 0
3s − 2t 7→ 3s2 + st − 2t2 + 3s2 t − 2st2
7→ 4s2 t − st2 − 2t3 + 2s2 t2 − 4st3 − 2s2 t3
7→ 3s2 t2 − 3st3 − 3s2 t3 7→ 0
3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2s2 t2 + 3st3 7→ 0
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 38
Explicit Decompositions
If p = 5
M3 ⊗ M4
M6 1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2
7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3
7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 10s2 t3 7→ 0
M4 3s − 2t 7→ 3s2 + st − 2t2 + 3s2 t − 2st2
7→ 4s2 t − st2 − 2t3 + 2s2 t2 − 4st3 − 2s2 t3
7→ 3s2 t2 − 3st3 − 3s2 t3 7→ 0
M2 3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2s2 t2 + 3st3 7→ 0
V5
V3 ⊗ V4
V5
V2
1 7→ s + t + st 7→ s2 + 2st + t2 + 2s2 t + 2st2 + s2 t2
7→ 3s2 t + 3st2 + t3 + 6s2 t2 + 3st3 + 3s2 t3
7→ 6s2 t2 + 4st3 + 12s2 t3 7→ 0= 10s2 t3
s 7→ s2 + st + s2 t 7→ 2s2 t + st2 + 2s2 t2
7→ 3s2 t2 + st3 + 3s2 t3 7→ 4s2 t3 7→ 0
3s2 − 2st + t2 + 2t3 7→ s2 t − st2 + t3 − 2st3 + 3s2 t2 7→ 0
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 39
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Some Consequences
Some New Modular
Invariants
Selected Bibliography
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 40
Some New Modular Invariants
1. F[V2 ⊕ V4 ]Cp
Covariants of R1 ⊕ R3 were computed classically.
2. F[V3 ⊕ V4 ]Cp
Covariants of R2 ⊕ R3 were computed classically.
3. F[⊕m V3 ]Cp
Covariants of ⊕m R2 were computed classically.
4. F[⊕m V4 ]Cp
Covariants of ⊕m R3 were computed by Schwarz(1987).
5. F[V6 ]Cp
Covariants of the quintic were computed by Gordan in 1869.
6. F[V7 ]Cp
Covariants of the sextic were computed by Gordan in 1869.
7. F[V8 ]Cp
Covariants of the septic were computed by Bedratyuk(2009).
8. F[V9 ]Cp
Covariants of the octic were computed by von Gall in 1880.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 41
Selected Bibliography
Modular Representation
Theory of Cp
Representation Theory of
SL2 (C)
The Connection
Shank’s Conjecture
Proof of the conjecture
Some Consequences
Some New Modular
Invariants
Selected Bibliography
[1] Gert Almkvist, Representations of Z/pZ in characteristic p and
reciprocity theorems, J. Algebra 68 (1981), no. 1, 1–27.
[2] G. Almkvist and R. Fossum, Decompositions of exterior and
symmetric powers of indecomposable Z/pZ -modules in
characteristic p, Lecture Notes in Math. 641, 1–114,
Springer-Verlag, 1978.
[3] Marvin Marcus and Herbert Robinson, Elementary divisors of tensor
products, Comm. ACM 18 (1975) 36–39.
[4] Michael Roberts, On the Covariants of a Binary Quantic of the nth
Degree, The Quarterly Journal of Pure and Applied Mathematics 4
(1861) 168178.
[5] R.J. Shank, S.A.G.B.I. bases for rings of formal modular
seminvariants, Commentarii Mathematici Helvetici, 73 (1998) no. 4,
548–565.
Fredericton, New Brunswick, 5 June 2010
CMS Summer Meeting – slide 42