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Modular InvariantTheory
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
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Outline
1 Introduction
2 Dimension 2
3 Dimension 3
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory in general: ingredients
A group G represented on a vector space V over afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].
The ring, F[V ]G , of polynomials fixed by the actionof G .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory in general: ingredients
A group G represented on a vector space V over afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].
The ring, F[V ]G , of polynomials fixed by the actionof G .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory in general: ingredients
A group G represented on a vector space V over afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].
The ring, F[V ]G , of polynomials fixed by the actionof G .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory in general: ingredients
A group G represented on a vector space V over afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].
The ring, F[V ]G , of polynomials fixed by the actionof G .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory in general: ingredients
A group G represented on a vector space V over afield 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 automorphismson F[V ] = F[x1, x2, . . . , xn].
The ring, F[V ]G , of polynomials fixed by the actionof G .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant theory: goal
We seek to understand F[V ]G in terms of its generatorsand relations or in turns of its structure such as theCohen-Macaulay property by relating algebraic propertiesto the geometric properties of the representation.
We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.
For example, in the non-modular case it is a famoustheorem due to Coxeter, Shephard and Todd, Chevalley,Serre that F[V ]G is a polynomial algebra if and only if Gis generated by (pseudo-)reflections.
The modular version of this theorem is still open.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant theory: goal
We seek to understand F[V ]G in terms of its generatorsand relations or in turns of its structure such as theCohen-Macaulay property by relating algebraic propertiesto the geometric properties of the representation.
We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.
For example, in the non-modular case it is a famoustheorem due to Coxeter, Shephard and Todd, Chevalley,Serre that F[V ]G is a polynomial algebra if and only if Gis generated by (pseudo-)reflections.
The modular version of this theorem is still open.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant theory: goal
We seek to understand F[V ]G in terms of its generatorsand relations or in turns of its structure such as theCohen-Macaulay property by relating algebraic propertiesto the geometric properties of the representation.
We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.
For example, in the non-modular case it is a famoustheorem due to Coxeter, Shephard and Todd, Chevalley,Serre that F[V ]G is a polynomial algebra if and only if Gis generated by (pseudo-)reflections.
The modular version of this theorem is still open.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant theory: goal
We seek to understand F[V ]G in terms of its generatorsand relations or in turns of its structure such as theCohen-Macaulay property by relating algebraic propertiesto the geometric properties of the representation.
We refer to the case that the order of G is divisible by pas the modular case, non-modular otherwise. Muchmore is known about the latter case than the former.
For example, in the non-modular case it is a famoustheorem due to Coxeter, Shephard and Todd, Chevalley,Serre that F[V ]G is a polynomial algebra if and only if Gis generated by (pseudo-)reflections.
The modular version of this theorem is still open.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Composition Series
In theory, one can hope to understand the invarianttheory of a p-group by induction on a composition series
1G C G2 C G3 C · · ·C Gr = G withGi+1
Gi= Cp ,
for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes thecyclic group of prime order p.
In practice, the representation theory of G is known tobe wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi isknown not to be Cohen-Macaulay in “most” instances.
Wehlau proved that the invariant ring of anyrepresentation of Cp is generated by norms and tracesand “rational” functions determined by the classicalinvariant theory of SL2(C).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Composition Series
In theory, one can hope to understand the invarianttheory of a p-group by induction on a composition series
1G C G2 C G3 C · · ·C Gr = G withGi+1
Gi= Cp ,
for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes thecyclic group of prime order p.
In practice, the representation theory of G is known tobe wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi isknown not to be Cohen-Macaulay in “most” instances.
Wehlau proved that the invariant ring of anyrepresentation of Cp is generated by norms and tracesand “rational” functions determined by the classicalinvariant theory of SL2(C).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Composition Series
In theory, one can hope to understand the invarianttheory of a p-group by induction on a composition series
1G C G2 C G3 C · · ·C Gr = G withGi+1
Gi= Cp ,
for then F[V ]Gi+1 = (F[V ]Gi )Cp . Here Cp denotes thecyclic group of prime order p.
In practice, the representation theory of G is known tobe wild unless r = 1 or r = 2 and p = 2, and F[V ]Gi isknown not to be Cohen-Macaulay in “most” instances.
Wehlau proved that the invariant ring of anyrepresentation of Cp is generated by norms and tracesand “rational” functions determined by the classicalinvariant theory of SL2(C).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory of (Cp)r in dimenson 2
Suppose we have a representation ρ of (any) p-group Gon a vector space V of dimension 2 over a field ofcharacteristic p. We may assume that
Gρ↪→(
1 ∗0 1
)
Hence G is a subgroup of the additive group (F,+), andso is elementary Abelian, G = (Cp)r for some r . It is nothard to see that
F[V ]G = F[x ,N(y)]
is a polynomial algebra on two generators of degrees 1and |G |.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory of (Cp)r in dimenson 2
Suppose we have a representation ρ of (any) p-group Gon a vector space V of dimension 2 over a field ofcharacteristic p. We may assume that
Gρ↪→(
1 ∗0 1
)
Hence G is a subgroup of the additive group (F,+), andso is elementary Abelian, G = (Cp)r for some r . It is nothard to see that
F[V ]G = F[x ,N(y)]
is a polynomial algebra on two generators of degrees 1and |G |.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Moduli space for (Cp)r , dimension 2
Given {gi}, generators for G we define ci ∈ F by
ρ(gi) =
(1 ci0 1
),
for some ci ∈ F. That is, G is determined by a vectorc = (c1, c2, . . . , cr ) ∈ Fr .
Let C denote the Fp-span of {c1, c2, . . . , cr} ⊂ F. Therepresentation is faithful if
dimFp(C ) = r .
If αb = c , for α ∈ F∗, b, c ∈ Fr then b and c determinethe same representation of G : representations of G areparametrized by P(Fr ).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Moduli space for (Cp)r , dimension 2
Given {gi}, generators for G we define ci ∈ F by
ρ(gi) =
(1 ci0 1
),
for some ci ∈ F. That is, G is determined by a vectorc = (c1, c2, . . . , cr ) ∈ Fr .
Let C denote the Fp-span of {c1, c2, . . . , cr} ⊂ F. Therepresentation is faithful if
dimFp(C ) = r .
If αb = c , for α ∈ F∗, b, c ∈ Fr then b and c determinethe same representation of G : representations of G areparametrized by P(Fr ).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Moduli space for (Cp)r , dimension 2
Given {gi}, generators for G we define ci ∈ F by
ρ(gi) =
(1 ci0 1
),
for some ci ∈ F. That is, G is determined by a vectorc = (c1, c2, . . . , cr ) ∈ Fr .
Let C denote the Fp-span of {c1, c2, . . . , cr} ⊂ F. Therepresentation is faithful if
dimFp(C ) = r .
If αb = c , for α ∈ F∗, b, c ∈ Fr then b and c determinethe same representation of G : representations of G areparametrized by P(Fr ).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
On the action of Aut(G ) = GLr(Fp)
Elements of Aut(G ) act as permutations on the set ofequivalence classes of representations, but preserve thering of invariants. Therefore, the collection of invariantrings is parametrized by P(Fr )//GLr (Fp).
The coordinate ring is therefore given by the Dicksoninvariants
F[c1, c2, . . . , cr ]GLr (Fp) = F[d1, d2, . . . , dr ]
with |di | = pr − pr−i , for 1 ≤ i ≤ r .
The representation is faithful if dr 6= 0.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
On the action of Aut(G ) = GLr(Fp)
Elements of Aut(G ) act as permutations on the set ofequivalence classes of representations, but preserve thering of invariants. Therefore, the collection of invariantrings is parametrized by P(Fr )//GLr (Fp).
The coordinate ring is therefore given by the Dicksoninvariants
F[c1, c2, . . . , cr ]GLr (Fp) = F[d1, d2, . . . , dr ]
with |di | = pr − pr−i , for 1 ≤ i ≤ r .
The representation is faithful if dr 6= 0.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
On the action of Aut(G ) = GLr(Fp)
Elements of Aut(G ) act as permutations on the set ofequivalence classes of representations, but preserve thering of invariants. Therefore, the collection of invariantrings is parametrized by P(Fr )//GLr (Fp).
The coordinate ring is therefore given by the Dicksoninvariants
F[c1, c2, . . . , cr ]GLr (Fp) = F[d1, d2, . . . , dr ]
with |di | = pr − pr−i , for 1 ≤ i ≤ r .
The representation is faithful if dr 6= 0.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
We assume p > 2.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory of (Cp)r in dimension 3
Given a representation ρ of a p-group G in dimension 3we must have
Gρ↪→
1 a b0 1 c0 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.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory of (Cp)r in dimension 3
Given a representation ρ of a p-group G in dimension 3we must have
Gρ↪→
1 a b0 1 c0 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.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory of (Cp)r in dimension 3
Given a representation ρ of a p-group G in dimension 3we must have
Gρ↪→
1 a b0 1 c0 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.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory of (Cp)r in dimension 3
Given a representation ρ of a p-group G in dimension 3we must have
Gρ↪→
1 a b0 1 c0 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.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Invariant Theory of (Cp)r in dimension 3
Given a representation ρ of a p-group G in dimension 3we must have
Gρ↪→
1 a b0 1 c0 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.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Type(2,1)
In this case, G has the form
Gρ↪→
1 0 a0 1 b0 0 1
.
for some finite subgroup A ⊂ F2. The ring of invariantsis a polynomial algebra on {x , y ,N(z)} for {x , y , z} abasis for V ∗3 .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Type(1,2)
In this case, G has the form
Gρ↪→
1 a b0 1 00 0 1
.
for some finite subgroup A ⊂ F2. The ring of invariantsis a polynomial algebra on {x ,N(y),N(z)} for {x , y , z}a basis for V ∗3 .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Type(1,1)
In this case, G has at least one element G whose Jordanform consists of a single block. By choice of basis wemay assume that
g i =
1 i(i2
)0 1 i0 0 1
.
Assuming that G is Abelian, we have that
Gρ↪→
1 a b0 1 a0 0 1
,
for a, b ∈ F.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Given {gi}, generators for G we define bi , ci ∈ F by
ρ(gi) =
1 ai bi0 1 ai0 1 1
,
for ai , bi ∈ F. Therefore, 3-dimensional representationsof G are determined by matrices
M =
(a1 a2 . . . arb1 b2 . . . br
)in F2×r and are of type (1,1) if at least one ai 6= 0.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
A moduli space for dimension 3
Two matrices
M =
(a1 a2 . . . arb1 b2 . . . br
)and M ′ =
(a′1 a′2 . . . a′rb′1 b′2 . . . b′r
)give equivalent representations if and only if there areα, β in F∗ n F such that(
α 0αβ α2
)M = M ′
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
A moduli space for dimension 3, continued
That is, 3-dimensional representations of G areparameterized by the orbits of F2×r under the action ofF n F∗. Here F∗ acts on F by multiplication. We alsohave a right action of GLr (Fp) on G by change of basis,preserving the ring of invariants, and hence an action onF2×r .
Thus the rings of invariants for dimension 3 areparameterized by the F n F∗-orbits acting onF2×r//GLr (Fp).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
A moduli space for dimension 3, continued
That is, 3-dimensional representations of G areparameterized by the orbits of F2×r under the action ofF n F∗. Here F∗ acts on F by multiplication. We alsohave a right action of GLr (Fp) on G by change of basis,preserving the ring of invariants, and hence an action onF2×r .
Thus the rings of invariants for dimension 3 areparameterized by the F n F∗-orbits acting onF2×r//GLr (Fp).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Module spaces for dimension 3, continued
We used elements of F[F 2×r ]GLr (Fp) when r = 2, 3 tostratify F2×r and subsequently provide generators andrelations for the corresponding rings of invariants in thiscases.
Pierron and Shank have extended this work to r = 4, atechnical tour-de-force.
All these rings are complete intersections.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Module spaces for dimension 3, continued
We used elements of F[F 2×r ]GLr (Fp) when r = 2, 3 tostratify F2×r and subsequently provide generators andrelations for the corresponding rings of invariants in thiscases.
Pierron and Shank have extended this work to r = 4, atechnical tour-de-force.
All these rings are complete intersections.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Module spaces for dimension 3, continued
We used elements of F[F 2×r ]GLr (Fp) when r = 2, 3 tostratify F2×r and subsequently provide generators andrelations for the corresponding rings of invariants in thiscases.
Pierron and Shank have extended this work to r = 4, atechnical tour-de-force.
All these rings are complete intersections.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Theorems in dimension 3
Theorem
Let V be a 3-dimensional representation of anelementary Abelian p-group G = (Cp)r . SettingF[V ] = F[x , y , z ], we have
F[V ]G = F[x , f1, f2, . . . , fs ,N(z)]
where LT (f ) = ydi for some {di} ∈ N.
Corollary
There is an efficient algorithm, SAGBI, divide by x , forcomputing generators and relations for F[V ]G .
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
Conjectures
Conjecture 1
Any modular 3-dimensional representation of anelementary Abelian p-group has a complete intersectionas its ring of invariants of embedding dimensions ≤ dr/2e+ 3.
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
The generic conjectures
Conjecture 2, r = 2s even
If the representation above is generic then the ring ofinvariants is a complete intersection of embeddingdimension s + 3, on generators {x , f1, f2, . . . , fs+1,N(z)}of degrees as follows:
1 The case r = 2s:
ps ps + 2ps−1 ps+1 + ps−2 . . . pr−1 + 2 ,
that is, |fi | = ps+i−2 + 2ps−i+1 for 2 ≤ i ≤ s + 1,with relations determined by (f p2 , f
p+21 ), and
(f pi , fi−1f(p2−1)pi−3
1 )
Modular InvariantTheory
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 ofinvariants is a complete intersection of embeddingdimension s + 3, on generators
1 The case r = 2s − 1
2ps−1 ps ps + 2ps−2 ps+1 + 2ps−3 pr−1 + 2 ,
that is, |fi | = ps+i−3 + 2ps−i+1, with relationsdetermined by (f p1 , f
22 ), (f p3 , f1f
p2 ), and
(f pi , fi−1f(p2−1)pi−4
2 ).
Modular InvariantTheory
H E A (Eddy)Campbell
Introduction
Dimension 2
Dimension 3
The generic cases r = 4, 5, 6, 7, 8, s = 2, 3, 4
|f1| |f2| |f3| |f4| |f5|r = 1 2 pr = 2 p p + 2r = 3 2p p2 p2 + 2pr = 4 p2 p2 + 2p p3 + 2r = 5 2p2 p3 p3 + 2p p4 + 2r = 6 p3 p3 + 2p2 p4 + 2p p5 + 2r = 7 2p3 p4 p4 + 2p2 p5 + 2p p6 + 2