Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Modular Invariant Theory of
Elementary Abelian p-groups in
dimensions 2 and 3
H E A (Eddy) Campbell
University of New Brunswick
October 19, 2014
Outline
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
1 Introduction
Dimension 2
Dimension 3
2 Dimension 2
3 Dimension 3
Invariant Theory in general: ingredients
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
A group G represented on a vector space V over a
field F of characteristic p.
Invariant Theory in general: ingredients
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
A group G represented on a vector space V over a
field F of characteristic p.
A basis {x1 , x2 , . . . , xn } for V ∗ .
Invariant Theory in general: ingredients
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
A group G represented on a vector space V over a
field F of characteristic p.
A basis {x1 , x2 , . . . , xn } for V ∗ .
The action of G on V ∗ by σ(f )(v ) = f (σ −1 (v )).
Invariant Theory in general: ingredients
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
A group G represented on a vector space V over a
field F of characteristic p.
A basis {x1 , x2 , . . . , xn } for V ∗ .
The action of G on V ∗ by σ(f )(v ) = f (σ −1 (v )).
The induced action of G by algebra automorphisms
on F[V ] = F[x1 , x2 , . . . , xn ].
Invariant Theory in general: ingredients
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
A group G represented on a vector space V over a
field F of characteristic p.
A basis {x1 , x2 , . . . , xn } for V ∗ .
The action of G on V ∗ by σ(f )(v ) = f (σ −1 (v )).
The induced action of G by algebra automorphisms
on F[V ] = F[x1 , x2 , . . . , xn ].
The ring, F[V ]G , of polynomials fixed by the action
of G .
Invariant theory: goal
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
We seek to understand F[V ]G in terms of its generators
and relations or in turns of its structure such as the
Cohen-Macaulay property by relating algebraic properties
to the geometric properties of the representation.
Invariant theory: goal
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
We seek to understand F[V ]G in terms of its generators
and relations or in turns of its structure such as the
Cohen-Macaulay property by relating algebraic properties
to the geometric properties of the representation.
Dimension 2
Dimension 3
We refer to the case that the order of G is divisible by p
as the modular case, non-modular otherwise. Much
more is known about the latter case than the former.
Invariant theory: goal
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
We seek to understand F[V ]G in terms of its generators
and relations or in turns of its structure such as the
Cohen-Macaulay property by relating algebraic properties
to the geometric properties of the representation.
Dimension 2
Dimension 3
We refer to the case that the order of G is divisible by p
as the modular case, non-modular otherwise. Much
more is known about the latter case than the former.
For example, in the non-modular case it is a famous
theorem due to Coxeter, Shephard and Todd, Chevalley,
Serre that F[V ]G is a polynomial algebra if and only if G
is generated by (pseudo-)reflections.
Invariant theory: goal
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
We seek to understand F[V ]G in terms of its generators
and relations or in turns of its structure such as the
Cohen-Macaulay property by relating algebraic properties
to the geometric properties of the representation.
Dimension 2
Dimension 3
We refer to the case that the order of G is divisible by p
as the modular case, non-modular otherwise. Much
more is known about the latter case than the former.
For example, in the non-modular case it is a famous
theorem due to Coxeter, Shephard and Todd, Chevalley,
Serre that F[V ]G is a polynomial algebra if and only if G
is generated by (pseudo-)reflections.
The modular version of this theorem is still open.
Composition Series
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
In theory, one can hope to understand the invariant
theory of a p-group by induction on a composition series
Gi+1
1G C G2 C G3 C · · · C Gr = G with
= Cp ,
Gi
for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes the
cyclic group of prime order p.
Composition Series
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
In theory, one can hope to understand the invariant
theory of a p-group by induction on a composition series
Gi+1
1G C G2 C G3 C · · · C Gr = G with
= Cp ,
Gi
for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes the
cyclic group of prime order p.
In practice, the representation theory of G is known to
be wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi is
known not to be Cohen-Macaulay in “most” instances.
Composition Series
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
In theory, one can hope to understand the invariant
theory of a p-group by induction on a composition series
Gi+1
1G C G2 C G3 C · · · C Gr = G with
= Cp ,
Gi
for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes the
cyclic group of prime order p.
In practice, the representation theory of G is known to
be wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi is
known not to be Cohen-Macaulay in “most” instances.
Wehlau proved that the invariant ring of any
representation of Cp is generated by norms and traces
and “rational” functions determined by the classical
invariant theory of SL2 (C).
Invariant Theory of (Cp )r in dimenson 2
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Suppose we have a representation ρ of (any) p-group G
on a vector space V of dimension 2 over a field of
characteristic p. We may assume that
ρ
1 ∗
G ,→
0 1
Invariant Theory of (Cp )r in dimenson 2
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Suppose we have a representation ρ of (any) p-group G
on a vector space V of dimension 2 over a field of
characteristic p. We may assume that
ρ
1 ∗
G ,→
0 1
Hence G is a subgroup of the additive group (F, +), and
so is elementary Abelian, G = (Cp )r for some r . It is not
hard to see that
F[V ]G = F[x, N (y )]
is a polynomial algebra on two generators of degrees 1
and |G |.
Moduli space for (Cp )r , dimension 2
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given {gi }, generators for G we define ci ∈ F by
1 ci
ρ(gi ) =
,
0 1
for some ci ∈ F. That is, G is determined by a vector
c = (c1 , c2 , . . . , cr ) ∈ Fr .
Moduli space for (Cp )r , dimension 2
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given {gi }, generators for G we define ci ∈ F by
1 ci
ρ(gi ) =
,
0 1
for some ci ∈ F. That is, G is determined by a vector
c = (c1 , c2 , . . . , cr ) ∈ Fr .
Let C denote the Fp -span of {c1 , c2 , . . . , cr } ⊂ F. The
representation is faithful if
dimFp (C ) = r .
Moduli space for (Cp )r , dimension 2
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given {gi }, generators for G we define ci ∈ F by
1 ci
ρ(gi ) =
,
0 1
for some ci ∈ F. That is, G is determined by a vector
c = (c1 , c2 , . . . , cr ) ∈ Fr .
Let C denote the Fp -span of {c1 , c2 , . . . , cr } ⊂ F. The
representation is faithful if
dimFp (C ) = r .
If αb = c, for α ∈ F∗ , b, c ∈ Fr then b and c determine
the same representation of G : representations of G are
parametrized by P(Fr ).
On the action of Aut(G ) = GLr (Fp )
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Elements of Aut(G ) act as permutations on the set of
equivalence classes of representations, but preserve the
ring of invariants. Therefore, the collection of invariant
rings is parametrized by P(Fr )// GLr (Fp ).
On the action of Aut(G ) = GLr (Fp )
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Elements of Aut(G ) act as permutations on the set of
equivalence classes of representations, but preserve the
ring of invariants. Therefore, the collection of invariant
rings is parametrized by P(Fr )// GLr (Fp ).
The coordinate ring is therefore given by the Dickson
invariants
F[c1 , c2 , . . . , cr ]GLr (Fp ) = F[d1 , d2 , . . . , dr ]
with |di | = p r − p r −i , for 1 ≤ i ≤ r .
On the action of Aut(G ) = GLr (Fp )
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Elements of Aut(G ) act as permutations on the set of
equivalence classes of representations, but preserve the
ring of invariants. Therefore, the collection of invariant
rings is parametrized by P(Fr )// GLr (Fp ).
The coordinate ring is therefore given by the Dickson
invariants
F[c1 , c2 , . . . , cr ]GLr (Fp ) = F[d1 , d2 , . . . , dr ]
with |di | = p r − p r −i , for 1 ≤ i ≤ r .
The representation is faithful if dr =
6 0.
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
We assume p > 2.
Invariant Theory of (Cp )r in dimension 3
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given a representation ρ of a
we must have

1
ρ

G ,→ 0
0
for a, b, c ∈ F.
p-group G in dimension 3

a b
1 c
0 1
Invariant Theory of (Cp )r in dimension 3
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given a representation ρ of a
we must have

1
ρ

G ,→ 0
0
p-group G in dimension 3

a b
1 c
0 1
for a, b, c ∈ F.
These can be classified by means of their socles.
Invariant Theory of (Cp )r in dimension 3
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given a representation ρ of a
we must have

1
ρ

G ,→ 0
0
p-group G in dimension 3

a b
1 c
0 1
for a, b, c ∈ F.
These can be classified by means of their socles.
Type (2,1): dimFp (V G ) = 2, dimFp ((V /V G )G ) = 1.
Invariant Theory of (Cp )r in dimension 3
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given a representation ρ of a
we must have

1
ρ

G ,→ 0
0
p-group G in dimension 3

a b
1 c
0 1
for a, b, c ∈ F.
These can be classified by means of their socles.
Type (2,1): dimFp (V G ) = 2, dimFp ((V /V G )G ) = 1.
Type (1,2): dimFp (V G ) = 1, dimFp ((V /V G )G ) = 2.
Invariant Theory of (Cp )r in dimension 3
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given a representation ρ of a
we must have

1
ρ

G ,→ 0
0
p-group G in dimension 3

a b
1 c
0 1
for a, b, c ∈ F.
These can be classified by means of their socles.
Type (2,1): dimFp (V G ) = 2, dimFp ((V /V G )G ) = 1.
Type (1,2): dimFp (V G ) = 1, dimFp ((V /V G )G ) = 2.
Type (1,1): dimFp (V G ) = 1, dimFp ((V /V G )G ) = 1.
Type(2,1)
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
In this case, G has the form


1 0 a
ρ
G ,→ 0 1 b  .
0 0 1
for some finite subgroup A ⊂ F2 . The ring of invariants
is a polynomial algebra on {x, y , N (z)} for {x, y , z} a
basis for V3∗ .
Type(1,2)
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
In this case, G has the form


1 a b
ρ
G ,→ 0 1 0 .
0 0 1
for some finite subgroup A ⊂ F2 . The ring of invariants
is a polynomial algebra on {x, N (y ), N (z)} for {x, y , z}
a basis for V3∗ .
Type(1,1)
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
In this case, G has at least one element G whose Jordan
form consists of a single block. By choice of basis we
may assume that


1 i 2i
g i = 0 1 i  .
0 0 1
Assuming that G is Abelian,

1
ρ
G ,→ 0
0
for a, b ∈ F.
we have that

a b
1 a ,
0 1
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Given {gi }, generators for G we define bi , ci ∈ F by


1 ai bi
ρ(gi ) = 0 1 ai  ,
0 1 1
for ai , bi ∈ F. Therefore, 3-dimensional representations
of G are determined by matrices
a1 a2 . . . ar
M=
b1 b2 . . . br
in F2×r and are of type (1,1) if at least one ai 6= 0.
A moduli space for dimension 3
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Two matrices
a a ...
M= 1 2
b1 b2 . . .
ar
br
0
and M =
a10 a20 . . .
b10 b20 . . .
ar0
br0
give equivalent representations if and only if there are
α, β in F∗ n F such that
α 0
M = M0
αβ α2
A moduli space for dimension 3, continued
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
That is, 3-dimensional representations of G are
parameterized by the orbits of F2×r under the action of
F n F∗ . Here F∗ acts on F by multiplication. We also
have a right action of GLr (Fp ) on G by change of basis,
preserving the ring of invariants, and hence an action on
F2×r .
A moduli space for dimension 3, continued
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
That is, 3-dimensional representations of G are
parameterized by the orbits of F2×r under the action of
F n F∗ . Here F∗ acts on F by multiplication. We also
have a right action of GLr (Fp ) on G by change of basis,
preserving the ring of invariants, and hence an action on
F2×r .
Thus the rings of invariants for dimension 3 are
parameterized by the F n F∗ -orbits acting on
F2×r // GLr (Fp ).
Module spaces for dimension 3, continued
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
We used elements of F[F 2×r ]GLr (Fp ) when r = 2, 3 to
stratify F2×r and subsequently provide generators and
relations for the corresponding rings of invariants in this
cases.
Module spaces for dimension 3, continued
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
We used elements of F[F 2×r ]GLr (Fp ) when r = 2, 3 to
stratify F2×r and subsequently provide generators and
relations for the corresponding rings of invariants in this
cases.
Pierron and Shank have extended this work to r = 4, a
technical tour-de-force.
Module spaces for dimension 3, continued
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
We used elements of F[F 2×r ]GLr (Fp ) when r = 2, 3 to
stratify F2×r and subsequently provide generators and
relations for the corresponding rings of invariants in this
cases.
Pierron and Shank have extended this work to r = 4, a
technical tour-de-force.
All these rings are complete intersections.
Theorems in dimension 3
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Theorem
Let V be a 3-dimensional representation of an
elementary Abelian p-group G = (Cp )r . Setting
F[V ] = F[x, y , z], we have
Dimension 3
F[V ]G = F[x, f1 , f2 , . . . , fs , N (z)]
where LT (f ) = y di for some {di } ∈ N.
Corollary
There is an efficient algorithm, SAGBI, divide by x, for
computing generators and relations for F[V ]G .
Conjectures
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Conjecture 1
Any modular 3-dimensional representation of an
elementary Abelian p-group has a complete intersection
as its ring of invariants of embedding dimension
s ≤ dr /2e + 3.
The generic conjectures
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Conjecture 2, r = 2s even
If the representation above is generic then the ring of
invariants is a complete intersection of embedding
dimension s + 3, on generators {x, f1 , f2 , . . . , fs+1 , N (z)}
of degrees as follows:
1 The case r = 2s:
p s p s + 2p s−1 p s+1 + p s−2 . . . p r −1 + 2 ,
that is, |fi | = p s+i−2 + 2p s−i+1 for 2 ≤ i ≤ s + 1,
with relations determined by (f2p , f1p+2 ), and
(p 2 −1)p i−3
(fi p , fi−1 f1
)
Modular Invariant
Theory
H E A (Eddy)
Campbell
Introduction
Dimension 2
Dimension 3
Conjecture 2, r = 2s − 1 odd
If the representation above is generic then the ring of
invariants is a complete intersection of embedding
dimension s + 3, on generators
1 The case r = 2s − 1
2p s−1 p s p s + 2p s−2 p s+1 + 2p s−3 p r −1 + 2 ,
that is, |fi | = p s+i−3 + 2p s−i+1 , with relations
determined by (f1p , f22 ), (f3p , f1 f2p ), and
(p 2 −1)p i−4
(fi p , fi−1 f2
).
Modular Invariant
Theory
H E A (Eddy)
Campbell
The generic cases r = 4, 5, 6, 7, 8, s = 2, 3, 4
Introduction
Dimension 2
Dimension 3
r
r
r
r
r
r
r
|f1 |
|f2 |
|f3 |
|f4 |
|f5 |
=1 2
p
=2 p
p+2
= 3 2p
p2
p 2 + 2p
2
2
=4 p
p + 2p
p3 + 2
= 5 2p 2
p3
p 3 + 2p p 4 + 2
3
3
2
=6 p
p + 2p p 4 + 2p p 5 + 2
= 7 2p 3
p4
p 4 + 2p 2 p 5 + 2p p 6 + 2