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The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
The MeatAxe
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Max Neunhöffer
Chop
Overview
University of St Andrews
GAC 2010, Allahabad
The MeatAxe
Introduction
Max Neunhöffer
Introduction
Let F be a field and Fd×d the set of d × d-matrices.
GAP examples
Definition (F-algebra, matrix algebra)
Linear Algebra
An F-algebra is a ring A with identity together with a ring
homomorphism ι : F → C(A) into the centre of A.
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
The MeatAxe
Introduction
Max Neunhöffer
Introduction
Let F be a field and Fd×d the set of d × d-matrices.
GAP examples
Definition (F-algebra, matrix algebra)
Linear Algebra
An F-algebra is a ring A with identity together with a ring
homomorphism ι : F → C(A) into the centre of A.
An F-subspace A of Fd×d with 1 ∈ A which is closed
under matrix multiplication is called a matrix algebra.
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
The MeatAxe
Introduction
Max Neunhöffer
Introduction
Let F be a field and Fd×d the set of d × d-matrices.
GAP examples
Definition (F-algebra, matrix algebra)
Linear Algebra
An F-algebra is a ring A with identity together with a ring
homomorphism ι : F → C(A) into the centre of A.
An F-subspace A of Fd×d with 1 ∈ A which is closed
under matrix multiplication is called a matrix algebra.
For a subset M ⊆ A we denote by hMiAlg the
intersection of all subalgebras in A containing M, the
algebra generated by M.
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
The MeatAxe
Introduction
Max Neunhöffer
Introduction
Let F be a field and Fd×d the set of d × d-matrices.
GAP examples
Definition (F-algebra, matrix algebra)
Linear Algebra
An F-algebra is a ring A with identity together with a ring
homomorphism ι : F → C(A) into the centre of A.
An F-subspace A of Fd×d with 1 ∈ A which is closed
under matrix multiplication is called a matrix algebra.
For a subset M ⊆ A we denote by hMiAlg the
intersection of all subalgebras in A containing M, the
algebra generated by M.
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Definition (Right A-module)
Let A be an F-algebra. An F-vector space V with a
bilinear map µ : V × A → V is called a right A-module, if
The MeatAxe
Introduction
Max Neunhöffer
Introduction
Let F be a field and Fd×d the set of d × d-matrices.
GAP examples
Definition (F-algebra, matrix algebra)
Linear Algebra
An F-algebra is a ring A with identity together with a ring
homomorphism ι : F → C(A) into the centre of A.
An F-subspace A of Fd×d with 1 ∈ A which is closed
under matrix multiplication is called a matrix algebra.
For a subset M ⊆ A we denote by hMiAlg the
intersection of all subalgebras in A containing M, the
algebra generated by M.
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Definition (Right A-module)
Let A be an F-algebra. An F-vector space V with a
bilinear map µ : V × A → V is called a right A-module, if
µ(v , 1A ) = v for all v ∈ V and
The MeatAxe
Introduction
Max Neunhöffer
Introduction
Let F be a field and Fd×d the set of d × d-matrices.
GAP examples
Definition (F-algebra, matrix algebra)
Linear Algebra
An F-algebra is a ring A with identity together with a ring
homomorphism ι : F → C(A) into the centre of A.
An F-subspace A of Fd×d with 1 ∈ A which is closed
under matrix multiplication is called a matrix algebra.
For a subset M ⊆ A we denote by hMiAlg the
intersection of all subalgebras in A containing M, the
algebra generated by M.
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Definition (Right A-module)
Let A be an F-algebra. An F-vector space V with a
bilinear map µ : V × A → V is called a right A-module, if
µ(v , 1A ) = v for all v ∈ V and
µ(µ(v , X ), Y ) = µ(v , XY ) for all v ∈ V and X , Y ∈ A.
The MeatAxe
A-modules
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Example (Natural module)
If A ≤ Fd×d is a matrix algebra, then V := F1×d is a right
A-module with µ(v , X ) := v · X . It is called the natural
module.
The MeatAxe
A-modules
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Example (Natural module)
If A ≤ Fd×d is a matrix algebra, then V := F1×d is a right
A-module with µ(v , X ) := v · X . It is called the natural
module.
Norton’s Criterion
Chop
Definition (Submodules and quotient modules)
Overview
Let V be an A-module. An A-submodule is an
A-invariant subspace W ≤ V , that is, W A = W .
The MeatAxe
A-modules
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Example (Natural module)
If A ≤ Fd×d is a matrix algebra, then V := F1×d is a right
A-module with µ(v , X ) := v · X . It is called the natural
module.
Norton’s Criterion
Chop
Definition (Submodules and quotient modules)
Overview
Let V be an A-module. An A-submodule is an
A-invariant subspace W ≤ V , that is, W A = W .
If W ≤ V is a submodule, then the quotient space V /W
is an A-module with (v + W )X := vX + W .
The MeatAxe
A-modules
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Example (Natural module)
If A ≤ Fd×d is a matrix algebra, then V := F1×d is a right
A-module with µ(v , X ) := v · X . It is called the natural
module.
Norton’s Criterion
Chop
Definition (Submodules and quotient modules)
Overview
Let V be an A-module. An A-submodule is an
A-invariant subspace W ≤ V , that is, W A = W .
If W ≤ V is a submodule, then the quotient space V /W
is an A-module with (v + W )X := vX + W .
A module V is called irreducible if its only submodules
are {0} and V itself.
The MeatAxe
A-modules
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Example (Natural module)
If A ≤ Fd×d is a matrix algebra, then V := F1×d is a right
A-module with µ(v , X ) := v · X . It is called the natural
module.
Norton’s Criterion
Chop
Definition (Submodules and quotient modules)
Overview
Let V be an A-module. An A-submodule is an
A-invariant subspace W ≤ V , that is, W A = W .
If W ≤ V is a submodule, then the quotient space V /W
is an A-module with (v + W )X := vX + W .
A module V is called irreducible if its only submodules
are {0} and V itself.
A composition series for V is a chain of submodules
{0} = V`+1 < V` < V`−1 < · · · < V1 = V
such that all Vi /Vi+1 are irreducible.
The MeatAxe
A-modules on the computer
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Let V be an A-module for the F-algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
A = hA1 , . . . , Ak iAlg .
The MeatAxe
A-modules on the computer
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Let V be an A-module for the F-algebra
Systems of linear equations
Compressed vectors
Spinning up
A = hA1 , . . . , Ak iAlg .
Norton’s Criterion
Chop
Overview
Then each generator Ai induces a linear map Ai : V → V .
The MeatAxe
A-modules on the computer
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Let V be an A-module for the F-algebra
Systems of linear equations
Compressed vectors
A = hA1 , . . . , Ak iAlg .
Spinning up
Norton’s Criterion
Chop
Overview
Then each generator Ai induces a linear map Ai : V → V .
Fact
To describe this situation to a computer, it is enough to
choose an F-basis (v1 , . . . , vd ) of V and store one
d × d-matrix for each Ai .
The MeatAxe
GAP examples
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
see other window
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
We can efficiently
compute in vector spaces and matrix algebras.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
Chop
Overview
solve systems of linear equations.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
Chop
Overview
solve systems of linear equations.
compute kernels of matrices.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
Chop
Overview
solve systems of linear equations.
compute kernels of matrices.
compute sums and intersections of subspaces given
by bases.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
Chop
Overview
solve systems of linear equations.
compute kernels of matrices.
compute sums and intersections of subspaces given
by bases.
test membership of a vector in a subspace.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
Chop
Overview
solve systems of linear equations.
compute kernels of matrices.
compute sums and intersections of subspaces given
by bases.
test membership of a vector in a subspace.
transpose matrices.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
Chop
Overview
solve systems of linear equations.
compute kernels of matrices.
compute sums and intersections of subspaces given
by bases.
test membership of a vector in a subspace.
transpose matrices.
compute characteristic and minimal polynomials.
The MeatAxe
Available methods from Linear Algebra
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
We can efficiently
compute in vector spaces and matrix algebras.
in particular multiply vectors with matrices and
matrices with matrices.
describe subspaces by bases.
Chop
Overview
solve systems of linear equations.
compute kernels of matrices.
compute sums and intersections of subspaces given
by bases.
test membership of a vector in a subspace.
transpose matrices.
compute characteristic and minimal polynomials.
All these algorithms have time-complexity at most O(d 3 )
in the dimension d.
The MeatAxe
Arithmetic over finite fields
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
For small finite fields we can store a field element using
only a few bits!
The MeatAxe
Arithmetic over finite fields
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
For small finite fields we can store a field element using
only a few bits!
This has several advantages:
The MeatAxe
Arithmetic over finite fields
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
For small finite fields we can store a field element using
only a few bits!
This has several advantages:
We save memory.
The MeatAxe
Arithmetic over finite fields
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
For small finite fields we can store a field element using
only a few bits!
This has several advantages:
We save memory.
Spinning up
Norton’s Criterion
Chop
Overview
Since basic field operations are simple, quite often
the runtime is dominated by memory accesses.
The MeatAxe
Arithmetic over finite fields
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
For small finite fields we can store a field element using
only a few bits!
This has several advantages:
We save memory.
Spinning up
Norton’s Criterion
Chop
Overview
Since basic field operations are simple, quite often
the runtime is dominated by memory accesses.
This saves time as well.
The MeatAxe
Arithmetic over finite fields
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
For small finite fields we can store a field element using
only a few bits!
This has several advantages:
We save memory.
Spinning up
Norton’s Criterion
Chop
Overview
Since basic field operations are simple, quite often
the runtime is dominated by memory accesses.
This saves time as well.
We can execute several field operations using one
processor word operation.
The MeatAxe
Arithmetic over finite fields
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
For small finite fields we can store a field element using
only a few bits!
This has several advantages:
We save memory.
Spinning up
Norton’s Criterion
Chop
Overview
Since basic field operations are simple, quite often
the runtime is dominated by memory accesses.
This saves time as well.
We can execute several field operations using one
processor word operation.
Example time and memory usage:
Operation
Mult. in F24370×4370
Add. in F1×4370
2
Mult. in F500×500
3
Time
Memory
C
U
C
U
320 ms
240 ns
50 ms
1335 s
209 µs
2140 ms
2.3 MB
550 B
78 kB
152 MB
35 kB
2 MB
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
Problem (Module generated by a vector)
Given 0 6= v ∈ V , find a basis for
v A := {vX | X ∈ A}
:= intersection of all A-submodules containing v
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
Problem (Module generated by a vector)
Given 0 6= v ∈ V , find a basis for
v A := {vX | X ∈ A}
Chop
:= intersection of all A-submodules containing v
Overview
Solution: the spinning up procedure
1
Initialise B := [v ] and i := 1
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
Problem (Module generated by a vector)
Given 0 6= v ∈ V , find a basis for
v A := {vX | X ∈ A}
Chop
:= intersection of all A-submodules containing v
Overview
Solution: the spinning up procedure
1
Initialise B := [v ] and i := 1
2
While i ≤ Length(B) do
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
Problem (Module generated by a vector)
Given 0 6= v ∈ V , find a basis for
v A := {vX | X ∈ A}
Chop
:= intersection of all A-submodules containing v
Overview
Solution: the spinning up procedure
1
Initialise B := [v ] and i := 1
2
While i ≤ Length(B) do
3
For j from 1 to k do
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
Problem (Module generated by a vector)
Given 0 6= v ∈ V , find a basis for
v A := {vX | X ∈ A}
Chop
:= intersection of all A-submodules containing v
Overview
Solution: the spinning up procedure
1
Initialise B := [v ] and i := 1
2
While i ≤ Length(B) do
3
4
For j from 1 to k do
If y := B[i] · Aj ∈
/ hBiF then
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
Problem (Module generated by a vector)
Given 0 6= v ∈ V , find a basis for
v A := {vX | X ∈ A}
Chop
:= intersection of all A-submodules containing v
Overview
Solution: the spinning up procedure
1
Initialise B := [v ] and i := 1
2
While i ≤ Length(B) do
3
4
5
For j from 1 to k do
If y := B[i] · Aj ∈
/ hBiF then
Append y to the end of B
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Spinning up
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
Problem (Module generated by a vector)
Given 0 6= v ∈ V , find a basis for
v A := {vX | X ∈ A}
Chop
:= intersection of all A-submodules containing v
Overview
Solution: the spinning up procedure
1
Initialise B := [v ] and i := 1
2
While i ≤ Length(B) do
3
4
5
6
For j from 1 to k do
If y := B[i] · Aj ∈
/ hBiF then
Append y to the end of B
Set i := i + 1
The MeatAxe
Norton’s irreducibility criterion
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Let A = hA1 , . . . , Ak iAlg ≤ Fd×d be a matrix algebra and
B ∈ A a singular element. Let At := At1 , . . . , Atk Alg .
The MeatAxe
Norton’s irreducibility criterion
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Let A = hA1 , . . . , Ak iAlg ≤ Fd×d be a matrix algebra and
B ∈ A a singular element. Let At := At1 , . . . , Atk Alg .
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Theorem (Norton)
At least one of the following holds:
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Overview
The MeatAxe
Norton’s irreducibility criterion
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Let A = hA1 , . . . , Ak iAlg ≤ Fd×d be a matrix algebra and
B ∈ A a singular element. Let At := At1 , . . . , Atk Alg .
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Theorem (Norton)
At least one of the following holds:
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Overview
Proof: Assume that 1 and 3 do not hold, so there is an
invariant subspace 0 < W < V , say of dimension e.
The MeatAxe
Norton’s irreducibility criterion
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Let A = hA1 , . . . , Ak iAlg ≤ Fd×d be a matrix algebra and
B ∈ A a singular element. Let At := At1 , . . . , Atk Alg .
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Theorem (Norton)
At least one of the following holds:
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Overview
Proof: Assume that 1 and 3 do not hold, so there is an
invariant subspace 0 < W < V , say of dimension e.
We can now choose a basis (w1 , . . . , we ) of W and
extend it to a basis (w1 , . . . , we , v1 , . . . , vd−e ) of V and
write all matrices with respect to this basis.
The MeatAxe
Norton’s irreducibility criterion
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Let A = hA1 , . . . , Ak iAlg ≤ Fd×d be a matrix algebra and
B ∈ A a singular element. Let At := At1 , . . . , Atk Alg .
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Theorem (Norton)
At least one of the following holds:
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Overview
Proof: Assume that 1 and 3 do not hold, so there is an
invariant subspace 0 < W < V , say of dimension e.
We can now choose a basis (w1 , . . . , we ) of W and
extend it to a basis (w1 , . . . , we , v1 , . . . , vd−e ) of V and
write all matrices with respect to this basis.
Let T := (w1 , . . . , we , v1 , . . . , vd−e ) and B 0 := TBT −1 .
The MeatAxe
Proof of Norton’s criterion
Max Neunhöffer
Introduction
Theorem (Norton)
GAP examples
At least one of the following holds:
Linear Algebra
Systems of linear equations
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Proof cont’d: Now, B 0 = TBT −1 looks like this:
M 0
0
B =
, where M ∈ Fe×e , N ∈ F(d−e)×(d−e) .
∗ N
The MeatAxe
Proof of Norton’s criterion
Max Neunhöffer
Introduction
Theorem (Norton)
GAP examples
At least one of the following holds:
Linear Algebra
Systems of linear equations
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Proof cont’d: Now, B 0 = TBT −1 looks like this:
M 0
0
B =
, where M ∈ Fe×e , N ∈ F(d−e)×(d−e) .
∗ N
Since
1
does not hold, ker B ∩ W = {0}.
The MeatAxe
Proof of Norton’s criterion
Max Neunhöffer
Introduction
Theorem (Norton)
GAP examples
At least one of the following holds:
Linear Algebra
Systems of linear equations
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Proof cont’d: Now, B 0 = TBT −1 looks like this:
M 0
0
B =
, where M ∈ Fe×e , N ∈ F(d−e)×(d−e) .
∗ N
Since 1 does not hold, ker B ∩ W = {0}.
Thus M has full rank e.
The MeatAxe
Proof of Norton’s criterion
Max Neunhöffer
Introduction
Theorem (Norton)
GAP examples
At least one of the following holds:
Linear Algebra
Systems of linear equations
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Proof cont’d: Now, B 0 = TBT −1 looks like this:
M 0
0
B =
, where M ∈ Fe×e , N ∈ F(d−e)×(d−e) .
∗ N
Since 1 does not hold, ker B ∩ W = {0}.
Thus M has full rank e.
If rank B 0 =: r < d, then rank N = r − e < d − e.
The MeatAxe
Proof of Norton’s criterion
Max Neunhöffer
Introduction
Theorem (Norton)
GAP examples
At least one of the following holds:
Linear Algebra
Systems of linear equations
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Proof cont’d: Now, B 0 = TBT −1 looks like this:
M 0
0
B =
, where M ∈ Fe×e , N ∈ F(d−e)×(d−e) .
∗ N
Since 1 does not hold, ker B ∩ W = {0}.
Thus M has full rank e.
If rank B 0 =: r < d, then rank N = r − e < d − e.
Thus dimF (ker N) = d − r = d − e − rank N.
The MeatAxe
Proof of Norton’s criterion
Max Neunhöffer
Introduction
Theorem (Norton)
GAP examples
At least one of the following holds:
Linear Algebra
Systems of linear equations
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Proof cont’d: Now, B 0 = TBT −1 looks like this:
M 0
0
B =
, where M ∈ Fe×e , N ∈ F(d−e)×(d−e) .
∗ N
Since 1 does not hold, ker B ∩ W = {0}.
Thus M has full rank e.
If rank B 0 =: r < d, then rank N = r − e < d − e.
Thus dimF (ker N) = d − r = d − e − rank N.
Now consider B 0t = (T t )−1 B t T t :
ker B 0t is contained in an (T AT −1 )t -invariant subspace.
The MeatAxe
Proof of Norton’s criterion
Max Neunhöffer
Introduction
Theorem (Norton)
GAP examples
At least one of the following holds:
Linear Algebra
Systems of linear equations
1
There is a 0 6= v ∈ ker B such that v A =
6 V.
2
For all v ∈ ker B t holds v At 6= V .
3
The natural module V := F1×d is irreducible.
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Proof cont’d: Now, B 0 = TBT −1 looks like this:
M 0
0
B =
, where M ∈ Fe×e , N ∈ F(d−e)×(d−e) .
∗ N
Since 1 does not hold, ker B ∩ W = {0}.
Thus M has full rank e.
If rank B 0 =: r < d, then rank N = r − e < d − e.
Thus dimF (ker N) = d − r = d − e − rank N.
Now consider B 0t = (T t )−1 B t T t :
ker B 0t is contained in an (T AT −1 )t -invariant subspace.
Thus ker B t is contained in an At -invariant subspace. The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
A basic step of “Chop”
1
Find an element B ∈ A with small, non-trivial kernel
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
A basic step of “Chop”
1
Find an element B ∈ A with small, non-trivial kernel
2
Compute ker B
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
A basic step of “Chop”
1
Find an element B ∈ A with small, non-trivial kernel
2
Compute ker B
3
Spinup all 0 6= v ∈ ker B
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
A basic step of “Chop”
1
Find an element B ∈ A with small, non-trivial kernel
2
Compute ker B
3
Spinup all 0 6= v ∈ ker B
4
If some v A < V , we found a submodule, goto
7
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
A basic step of “Chop”
1
Find an element B ∈ A with small, non-trivial kernel
2
Compute ker B
3
Spinup all 0 6= v ∈ ker B
4
If some v A < V , we found a submodule, goto
5
Otherwise spinup one 0 6= v ∈ ker B t under At
7
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
A basic step of “Chop”
1
Find an element B ∈ A with small, non-trivial kernel
2
Compute ker B
3
Spinup all 0 6= v ∈ ker B
4
If some v A < V , we found a submodule, goto
5
Otherwise spinup one 0 6= v ∈ ker B t under At
6
If v At = V , we have proved V to be irreducible, stop
7
The MeatAxe
Chopping modules I
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
“Chopping” means computing a composition series.
The MeatAxe basically does the following:
A basic step of “Chop”
1
Find an element B ∈ A with small, non-trivial kernel
2
Compute ker B
3
Spinup all 0 6= v ∈ ker B
4
If some v A < V , we found a submodule, goto
5
Otherwise spinup one 0 6= v ∈ ker B t under At
6
If v At = V , we have proved V to be irreducible, stop
7
7
If 0 < W < V is invariant, compute action on W and
V /W and recurse (with smaller dimensions!)
The MeatAxe
Chopping modules II
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
The result of “Chop” is a composition series
{0} = V`+1 < V` < V`−1 < · · · < V1 = V
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
such that all Vj /Vj+1 are irreducible.
The MeatAxe
Chopping modules II
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
The result of “Chop” is a composition series
{0} = V`+1 < V` < V`−1 < · · · < V1 = V
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
such that all Vj /Vj+1 are irreducible.
Actually, we find a base change T ∈ Fd×d , such that all
matrices TAi T −1 for 1 ≤ i ≤ k look like this:
Overview
TAi T
−1
(i)

M`


=


∗
..
.
M`−1
..
.
···
..
.
..
.
∗
···
∗
(i)
and the matrices Mj
0
(i)
0
..
.
0
(i)
M1






describe the action of A on Vj /Vj+1 .
The MeatAxe
Chopping modules II
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
The result of “Chop” is a composition series
{0} = V`+1 < V` < V`−1 < · · · < V1 = V
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
such that all Vj /Vj+1 are irreducible.
Actually, we find a base change T ∈ Fd×d , such that all
matrices TAi T −1 for 1 ≤ i ≤ k look like this:
Overview
TAi T
−1
(i)

M`


=


∗
..
.
M`−1
..
.
···
..
.
..
.
∗
···
∗
(i)
and the matrices Mj
0
(i)
0
..
.
0
(i)
M1






describe the action of A on Vj /Vj+1 .
A more detailed analysis shows that the MeatAxe can
identify isomorphism types of irreducible modules.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
Find homomorphism spaces from an irreducible
module to another one.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
Find homomorphism spaces from an irreducible
module to another one.
Identify the isomorphism type of irreducible modules.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
Find homomorphism spaces from an irreducible
module to another one.
Identify the isomorphism type of irreducible modules.
Compute the socle and radical series.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
Find homomorphism spaces from an irreducible
module to another one.
Identify the isomorphism type of irreducible modules.
Compute the socle and radical series.
Compute the submodule lattice.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
Find homomorphism spaces from an irreducible
module to another one.
Identify the isomorphism type of irreducible modules.
Compute the socle and radical series.
Compute the submodule lattice.
Compute homomorphism spaces between arbitrary
modules.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
Find homomorphism spaces from an irreducible
module to another one.
Identify the isomorphism type of irreducible modules.
Compute the socle and radical series.
Compute the submodule lattice.
Compute homomorphism spaces between arbitrary
modules.
Compute cohomology groups.
The MeatAxe
Overview over available algorithms
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Assume we are given an A-module V = F1×d by matrices
A1 , . . . , Ak ∈ Fd×d .
The MeatAxe can do the following for you:
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Compute a composition series.
Find homomorphism spaces from an irreducible
module to another one.
Identify the isomorphism type of irreducible modules.
Compute the socle and radical series.
Compute the submodule lattice.
Compute homomorphism spaces between arbitrary
modules.
Compute cohomology groups.
Compute condensed modules.
The MeatAxe
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
The End
The MeatAxe
Bibliography
Max Neunhöffer
Introduction
GAP examples
Linear Algebra
Systems of linear equations
Compressed vectors
Spinning up
Norton’s Criterion
Chop
Overview
Derek F. Holt and Sarah Rees.
Testing modules for irreducibility.
J. Austral. Math. Soc. Ser. A, 57(1):1–16, 1994.
Gábor Ivanyos and Klaus Lux.
Treating the exceptional cases of the MeatAxe.
Experiment. Math., 9(3):373–381, 2000.
R. A. Parker.
The computer calculation of modular characters (the
meat-axe).
In Computational group theory (Durham, 1982),
pages 267–274. Academic Press, London, 1984.