Quotients of Lower Central Series With Multiple Relations Mentor: Yael Fregier Isaac Xia
Quotients of Lower Central Series With Multiple
Relations
Mentor: Yael Fregier
Isaac Xia
MIT PRIMES
Saturday, May 18 2013
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Outline
1
Abelian Groups: Rank and Torsion
2
3
4
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
What is a Non-commutative Algebra?
Take 2 letters: x , y
Words: xyx , yyxyx , xyxyxyx
Sentences: 2 xyx + 5 yyxyx
This describes A
2
Non-commutative letters: logarithm = algorithm
Why study Non-commutative algebras?
Classical physics ⇐⇒ Quantum physics
Commutative algebras ⇐⇒ Non-commutative algebras
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
What is a Non-commutative Algebra?
Take 2 letters: x , y
Words: xyx , yyxyx , xyxyxyx
Sentences: 2 xyx + 5 yyxyx
This describes A
2
Non-commutative letters: logarithm = algorithm
Why study Non-commutative algebras?
Classical physics ⇐⇒ Quantum physics
Commutative algebras ⇐⇒ Non-commutative algebras
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
What is a Non-commutative Algebra?
Take 2 letters: x , y
Words: xyx , yyxyx , xyxyxyx
Sentences: 2 xyx + 5 yyxyx
This describes A
2
Non-commutative letters: logarithm = algorithm
Why study Non-commutative algebras?
Classical physics ⇐⇒ Quantum physics
Commutative algebras ⇐⇒ Non-commutative algebras
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
What is a Non-commutative Algebra?
Take 2 letters: x , y
Words: xyx , yyxyx , xyxyxyx
Sentences: 2 xyx + 5 yyxyx
This describes A
2
Non-commutative letters: logarithm = algorithm
Why study Non-commutative algebras?
Classical physics ⇐⇒ Quantum physics
Commutative algebras ⇐⇒ Non-commutative algebras
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
What is a Non-commutative Algebra?
Take 2 letters: x , y
Words: xyx , yyxyx , xyxyxyx
Sentences: 2 xyx + 5 yyxyx
This describes A
2
Non-commutative letters: logarithm = algorithm
Why study Non-commutative algebras?
Classical physics ⇐⇒ Quantum physics
Commutative algebras ⇐⇒ Non-commutative algebras
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
What is a Non-commutative Algebra?
Take 2 letters: x , y
Words: xyx , yyxyx , xyxyxyx
Sentences: 2 xyx + 5 yyxyx
This describes A
2
Non-commutative letters: logarithm = algorithm
Why study Non-commutative algebras?
Classical physics ⇐⇒ Quantum physics
Commutative algebras ⇐⇒ Non-commutative algebras
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
What is a Non-commutative Algebra?
Take 2 letters: x , y
Words: xyx , yyxyx , xyxyxyx
Sentences: 2 xyx + 5 yyxyx
This describes A
2
Non-commutative letters: logarithm = algorithm
Why study Non-commutative algebras?
Classical physics ⇐⇒ Quantum physics
Commutative algebras ⇐⇒ Non-commutative algebras
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Free Algebra A n
Definition
Free algebra :
A n
:= k h x
1
, x
2
, x
3
, . . . , x n i and
{ 1 , x
1
, x
2
, x
3
, . . . , x
1 x
1
, x
1 x
2
, x
1 x
3
, x
2 x
1
, . . .
}
Free algebra is the most “non-commutative”
A n generated by generators x
1
, x
2
, . . . , x n
Example: A
3
:
1 x
2
1 x
1
, x
2
, x
3
, x
1 x
2
, x
1 x
3
, x
2 x
1
, x
2
2
, x
2 x
3
, x
3 x
1
, x
3 x
2
, x
2
3
Grading keeps track of data better: A n
[ d ]
A n
[ d ] spanned by n d basis elements
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Relations
A n already studied
All finitely generated non-commutative algebras are quotients of A n by relations
Relations: e.g. ( xy = 0 , y
2
= 0)
A
2
/ ( xy , y
2
)
1 x , y x x
3
2
, yx
, yx
2
Restrict to homogeneous relations to preserve grading
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Commutator
Abelian Groups: Rank and Torsion
Definition
Commutator : Let A be an algebra. The commutator of x and y is [ x , y ] = xy − yx . If H and K are subspaces of A , then
[ H , K ] = span { [ h , k ] | h ∈ H and k ∈ K } ).
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Lower Central Series
Abelian Groups: Rank and Torsion
Study A n by Lower Central Series Filtration
Definition
Lower Central Series Filtration:
A = L
1
⊇ L
2
Where L
1
:=
⊇
A
L
3 and
⊇
L
. . .
k +1
:= [ L k
, A ] for k ≥ 1.
L i +1 all ` is the smallest subspace such that a ` = ` a mod L i +1
∈ L i and a ∈ A .
for
Provides a measure of non-commutativity for the original algebra
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Lower Central Series
Abelian Groups: Rank and Torsion
Study A n by Lower Central Series Filtration
Definition
Lower Central Series Filtration:
A = L
1
⊇ L
2
Where L
1
:=
⊇
A
L
3 and
⊇
L
. . .
k +1
:= [ L k
, A ] for k ≥ 1.
L i +1 all ` is the smallest subspace such that a ` = ` a mod L i +1
∈ L i and a ∈ A .
for
Provides a measure of non-commutativity for the original algebra
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Lower Central Series
Abelian Groups: Rank and Torsion
Study A n by Lower Central Series Filtration
Definition
Lower Central Series Filtration:
A = L
1
⊇ L
2
Where L
1
:=
⊇
A
L
3 and
⊇
L
. . .
k +1
:= [ L k
, A ] for k ≥ 1.
L i +1 all ` is the smallest subspace such that a ` = ` a mod L i +1
∈ L i and a ∈ A .
for
Provides a measure of non-commutativity for the original algebra
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Ideal
Study ideals M k in order to preserve structures
Sets and element: Cd := { cd | c ∈ C }
Definition
Ideal M k
:
M k
= AL k
= L k
A = span { ` a | ` ∈ L k
, a ∈ A } .
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Ideal
Study ideals M k in order to preserve structures
Sets and element: Cd := { cd | c ∈ C }
Definition
Ideal M k
:
M k
= AL k
= L k
A = span { ` a | ` ∈ L k
, a ∈ A } .
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups: Rank and Torsion
Ideal
Study ideals M k in order to preserve structures
Sets and element: Cd := { cd | c ∈ C }
Definition
Ideal M k
:
M k
= AL k
= L k
A = span { ` a | ` ∈ L k
, a ∈ A } .
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Question
Abelian Groups: Rank and Torsion
Problem
1. How many basis elements are in each M k in each degree?
Problem
2. How fast does this number shrink as k grows?
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Question
Abelian Groups: Rank and Torsion
Problem
1. How many basis elements are in each M k in each degree?
Problem
2. How fast does this number shrink as k grows?
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Quotient of the Ideals
Abelian Groups: Rank and Torsion
Shrinking is essentially “Difference”
We take the quotient of these ideals:
Definition
Quotient N k
:= M k
/ M k +1
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Quotient of the Ideals
Abelian Groups: Rank and Torsion
Shrinking is essentially “Difference”
We take the quotient of these ideals:
Definition
Quotient N k
:= M k
/ M k +1
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Quotient of the Ideals
Abelian Groups: Rank and Torsion
Shrinking is essentially “Difference”
We take the quotient of these ideals:
Definition
Quotient N k
:= M k
/ M k +1
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Coefficients
Abelian Groups: Rank and Torsion
What coefficients do we work over for A
2
= k h x
1
, x
2 i ?
Option 1: Work over k =
Z make N k
Abelian groups (
Z not field)
Option 2: k = GF ( p ) or
Q make N k vector spaces
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Coefficients
Abelian Groups: Rank and Torsion
What coefficients do we work over for A
2
= k h x
1
, x
2 i ?
Option 1: Work over k =
Z make N k
Abelian groups (
Z not field)
Option 2: k = GF ( p ) or
Q make N k vector spaces
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Coefficients
Abelian Groups: Rank and Torsion
What coefficients do we work over for A
2
= k h x
1
, x
2 i ?
Option 1: Work over k =
Z make N k
Abelian groups (
Z not field)
Option 2: k = GF ( p ) or
Q make N k vector spaces
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Abelian Groups
Abelian Groups: Rank and Torsion
Working with coefficients over
Z yield Abelian groups
Finitely generated Abelian groups decomposable:
A
=
Z r
⊕
M
Z i
α i for prime powers i .
i
Data presented as r ( Q ( i
α i ))
Z
3 ⊕
Z
4
2
⊕
Z 4
⊕
Z 5
⇐⇒ 3(2
4 · 4 · 5)
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Data on N k
[ d ]
MAGMA calculations/data reconfiguration
Previous work exists, modified to automate data collection
Example:
Z h x , y i / ( x
5
, y
7
)
N i
[ d ]
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
2
1
0
0
0
0
0
0
0
3
0
0
0
0
1(2)
2
0
0
4
0
0
0
0
1(2)
3(2)
2
0
5
0
0
0
0
1(2)
3(2
2
)
3(2
2
)
4
Table:
Z h x , y i / ( x 2 , y 5 )
6
0
0
5
0
0(2 · 5)
3(2
2
)
3(2
4
)
7(2
3
)
7
0
0
0
1(2
2
2(2
6
)
· 5
2
)
7(2
7
)
9(2
5
)
9
8
0
0(2 · 5)
0(2
6
)
5(2
10
)
8(2
12
)
18(2
7
)
12
0
9
0
0
0(2
3
)
1(2
11
)
4(2
17
)
17(2
19
)
24(2
13
)
20
10
0
0
0(2)
0(2
7
)
0(2
18
)
10(2
27
)
20(2
33
43(2
18
)
· 3 · 4 · 5 · 7)
Only 2 or 5 torsion appears except in N
8
[10]
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Data on N k
[ d ]
MAGMA calculations/data reconfiguration
Previous work exists, modified to automate data collection
Example:
Z h x , y i / ( x
5
, y
7
)
N i
[ d ]
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
2
1
0
0
0
0
0
0
0
3
0
0
0
0
1(2)
2
0
0
4
0
0
0
0
1(2)
3(2)
2
0
5
0
0
0
0
1(2)
3(2
2
)
3(2
2
)
4
Table:
Z h x , y i / ( x 2 , y 5 )
6
0
0
5
0
0(2 · 5)
3(2
2
)
3(2
4
)
7(2
3
)
7
0
0
0
1(2
2
2(2
6
)
· 5
2
)
7(2
7
)
9(2
5
)
9
8
0
0(2 · 5)
0(2
6
)
5(2
10
)
8(2
12
)
18(2
7
)
12
0
9
0
0
0(2
3
)
1(2
11
)
4(2
17
)
17(2
19
)
24(2
13
)
20
10
0
0
0(2)
0(2
7
)
0(2
18
)
10(2
27
)
20(2
33
43(2
18
)
· 3 · 4 · 5 · 7)
Only 2 or 5 torsion appears except in N
8
[10]
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Data on N k
[ d ]
MAGMA calculations/data reconfiguration
Previous work exists, modified to automate data collection
Example:
Z h x , y i / ( x
5
, y
7
)
N i
[ d ]
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
2
1
0
0
0
0
0
0
0
3
0
0
0
0
1(2)
2
0
0
4
0
0
0
0
1(2)
3(2)
2
0
5
0
0
0
0
1(2)
3(2
2
)
3(2
2
)
4
Table:
Z h x , y i / ( x 2 , y 5 )
6
0
0
5
0
0(2 · 5)
3(2
2
)
3(2
4
)
7(2
3
)
7
0
0
0
1(2
2
2(2
6
)
· 5
2
)
7(2
7
)
9(2
5
)
9
8
0
0(2 · 5)
0(2
6
)
5(2
10
)
8(2
12
)
18(2
7
)
12
0
9
0
0
0(2
3
)
1(2
11
)
4(2
17
)
17(2
19
)
24(2
13
)
20
10
0
0
0(2)
0(2
7
)
0(2
18
)
10(2
27
)
20(2
33
43(2
18
)
· 3 · 4 · 5 · 7)
Only 2 or 5 torsion appears except in N
8
[10]
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Data on N k
[ d ]
MAGMA calculations/data reconfiguration
Previous work exists, modified to automate data collection
Example:
Z h x , y i / ( x
5
, y
7
)
N i
[ d ]
N
2
N
3
N
4
N
5
N
6
N
7
N
8
N
9
2
1
0
0
0
0
0
0
0
3
0
0
0
0
1(2)
2
0
0
4
0
0
0
0
1(2)
3(2)
2
0
5
0
0
0
0
1(2)
3(2
2
)
3(2
2
)
4
Table:
Z h x , y i / ( x 2 , y 5 )
6
0
0
5
0
0(2 · 5)
3(2
2
)
3(2
4
)
7(2
3
)
7
0
0
0
1(2
2
2(2
6
)
· 5
2
)
7(2
7
)
9(2
5
)
9
8
0
0(2 · 5)
0(2
6
)
5(2
10
)
8(2
12
)
18(2
7
)
12
0
9
0
0
0(2
3
)
1(2
11
)
4(2
17
)
17(2
19
)
24(2
13
)
20
10
0
0
0(2)
0(2
7
)
0(2
18
)
10(2
27
)
20(2
33
43(2
18
)
· 3 · 4 · 5 · 7)
Only 2 or 5 torsion appears except in N
8
[10]
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Row Sums Over GF ( p )
Proposition 1
Take A
2
/ ( x m , y n ), where m is divisible by a prime p . If N i is computed over GF ( p ), its total dimension is divisible by p . In fact, if m and n are both divisible by p , then the total dimension is divisible by p
2
.
Table:
Z 5 h x , y i / ( x 5 , y 4 )
N i
[ d ] 0 1 2 3 4 5 6 7 8 9 10 11
N
2
N
3
N
4
0 0 1 2 3 3 3 2 1 0 0 0
0 0 0 2 5 8 9 9 7 4 1 0
0 0 0 0 3 8 14 16 16 13 8 2
Sums: 15, 45
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Row Sums Over GF ( p )
Proposition 1
Take A
2
/ ( x m , y n ), where m is divisible by a prime p . If N i is computed over GF ( p ), its total dimension is divisible by p . In fact, if m and n are both divisible by p , then the total dimension is divisible by p
2
.
Table:
Z 5 h x , y i / ( x 5 , y 4 )
N i
[ d ] 0 1 2 3 4 5 6 7 8 9 10 11
N
2
N
3
N
4
0 0 1 2 3 3 3 2 1 0 0 0
0 0 0 2 5 8 9 9 7 4 1 0
0 0 0 0 3 8 14 16 16 13 8 2
Sums: 15, 45
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Another Example of Proposition 1
Table:
Z 3 h x , y i / ( x
3
, y
3
)
N i
[ d ] 0 1 2 3 4 5 6 7 8 9
N
2
N
3
0
0
0
0
1
0
2
2
3
5
2
8
1
7
0
4
0
1
0
0
Sums: 9, 27
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Proving Proposition 1
Theorem 1
If all relations are functions of x
1 p
, then all N i
( A ) carry an action of the Weyl algebra D ( k ) with generators D , x and relations
[ D , x ] = 1.
Proof sketch: Define D acting on an element a as d
D ( a ) = a and xa = x
1 a . We can verify that [ D , x ] = 1.
dx
1
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Proving Proposition 1
Theorem 1
If all relations are functions of x
1 p
, then all N i
( A ) carry an action of the Weyl algebra D ( k ) with generators D , x and relations
[ D , x ] = 1.
Proof sketch: Define D acting on an element a as d
D ( a ) = a and xa = x
1 a . We can verify that [ D , x ] = 1.
dx
1
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Basic Corollary of Theorem 1
Corollary 2
If N i
( A ) is finite dimensional, then dim( N i
( A )) is divisible by p . In general, if the relations are non-commutative polynomials of p -th powers of the first divisible by p r
.
r variables x p
1
, x p
2
, . . . , x r p
, then dim( N i
( A )) is
Proof sketch: We use k = GF ( p ).
0 = Tr ([ D , x ]) = Tr (1) = dim ( V ) in GF ( p ) where V is a representation of D ( k ), so every representation of D ( k ) has dimension divisible by p . For relations which are functions of x p
1
, x
D ( k ) p
2
, . . . ,
⊗ r x r p
, we have an action of the tensor power algebra whose representation dimensions are divisible by p r
.
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Basic Corollary of Theorem 1
Corollary 2
If N i
( A ) is finite dimensional, then dim( N i
( A )) is divisible by p . In general, if the relations are non-commutative polynomials of p -th powers of the first divisible by p r
.
r variables x p
1
, x p
2
, . . . , x r p
, then dim( N i
( A )) is
Proof sketch: We use k = GF ( p ).
0 = Tr ([ D , x ]) = Tr (1) = dim ( V ) in GF ( p ) where V is a representation of D ( k ), so every representation of D ( k ) has dimension divisible by p . For relations which are functions of x p
1
, x
D ( k ) p
2
, . . . ,
⊗ r x r p
, we have an action of the tensor power algebra whose representation dimensions are divisible by p r
.
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Further Corollary of Theorem 1
Corollary 3
Suppose that the relations are homogeneous in x i
. Then, the
Hilbert series
P r
:= (1 + X
1
H of N i
( A ) with respect to X
1
, . . . , X r is divisible by
+ . . .
+ X p − 1
1
) . . .
(1 + X r
+ . . .
+ X power series with non-negative coefficients.
r p − 1
), i.e.
H
P r is a
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Further Research Options
Bigrading: N k
[ d x
, d y
] instead of N k
[ d ]
Example: xy
2 x
3 has bidegree (4 , 2), yxyx has bidegree (2 , 2)
Instead of N k
, find B k
(Quotients of L k
/ L k +1
)
Use A
3 or even A
4
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Further Research Options
Bigrading: N k
[ d x
, d y
] instead of N k
[ d ]
Example: xy
2 x
3 has bidegree (4 , 2), yxyx has bidegree (2 , 2)
Instead of N k
, find B k
(Quotients of L k
/ L k +1
)
Use A
3 or even A
4
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Further Research Options
Bigrading: N k
[ d x
, d y
] instead of N k
[ d ]
Example: xy
2 x
3 has bidegree (4 , 2), yxyx has bidegree (2 , 2)
Instead of N k
, find B k
(Quotients of L k
/ L k +1
)
Use A
3 or even A
4
Isaac Xia
Quotients of Lower Central Series With Multiple Relations
Acknowledgements
Thanks to Pavel Etingof for suggesting the project and conjectures
Thanks to Teng Fei for providing assistance with MAGMA
Thanks to Tanya Khovanova for her guidance and invaluable help in looking over the project
Many thanks to my mentor Yael Fregier for his uncountable hours spent explaining concepts and closely monitored guidance!
Thanks to PRIMES for this opportunity!
Isaac Xia
