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Sandpile Groups of Cubes
B. Anzis & R. Prasad
August 1, 2016
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Overview
Introduction
DefinitionsPrevious Results
Grobner Basis Calculations
A Bound on the Largest Cyclic Factor Size
Analogous Bounds on Other Cayley Graphs
Higher Critical Groups
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Overview
Introduction
Definitions
Previous Results
Grobner Basis Calculations
A Bound on the Largest Cyclic Factor Size
Analogous Bounds on Other Cayley Graphs
Higher Critical Groups
Sandpile Groups of Cubes August 1, 2016 2 / 27
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Overview
Introduction
DefinitionsPrevious Results
Grobner Basis Calculations
A Bound on the Largest Cyclic Factor Size
Analogous Bounds on Other Cayley Graphs
Higher Critical Groups
Sandpile Groups of Cubes August 1, 2016 2 / 27
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Overview
Introduction
DefinitionsPrevious Results
Grobner Basis Calculations
A Bound on the Largest Cyclic Factor Size
Analogous Bounds on Other Cayley Graphs
Higher Critical Groups
Sandpile Groups of Cubes August 1, 2016 2 / 27
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Overview
Introduction
DefinitionsPrevious Results
Grobner Basis Calculations
A Bound on the Largest Cyclic Factor Size
Analogous Bounds on Other Cayley Graphs
Higher Critical Groups
Sandpile Groups of Cubes August 1, 2016 2 / 27
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Overview
Introduction
DefinitionsPrevious Results
Grobner Basis Calculations
A Bound on the Largest Cyclic Factor Size
Analogous Bounds on Other Cayley Graphs
Higher Critical Groups
Sandpile Groups of Cubes August 1, 2016 2 / 27
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Overview
Introduction
DefinitionsPrevious Results
Grobner Basis Calculations
A Bound on the Largest Cyclic Factor Size
Analogous Bounds on Other Cayley Graphs
Higher Critical Groups
Sandpile Groups of Cubes August 1, 2016 2 / 27
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Introduction
Definitions
Definition
The n-cube is the graph Qn with V (Qn) = (Z/2Z)n and an edge betweentwo vertices v1, v2 ∈ V (Qn) if v1 and v2 differ in precisely one place.
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Introduction
Definitions
Definition
The Laplacian of a graph G , denoted L(G ), is the matrix
L(G )i ,j =
{deg(vi ) if i = j
−#{edges from vi to vj} if i 6= j
Example
L(Q1) =
(1 −1−1 1
)L(Q2) =
2 −1 −1 0−1 2 0 −1−1 0 2 −10 −1 −1 2
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Introduction
A Final Definition
Definition
Let G be a graph. Since L(G ) is an integer matrix, we may consider it as aZ-linear map L(G ) : Z#V (G) → Z#V (G). The torsion part of the cokernelof this map is the critical group (or sandpile group) of G , denoted K (G ).
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Introduction
Previous Results I
Theorem [Bai]
For every prime p > 2,
Sylp (K (Qn)) ∼= Sylp
(n∏
k=1
(Z/kZ)(nk)
).
Remark
To understand K (Qn), it then remains to understand Syl2(K (Qn)).
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Introduction
Previous Results I
Theorem [Bai]
For every prime p > 2,
Sylp (K (Qn)) ∼= Sylp
(n∏
k=1
(Z/kZ)(nk)
).
Remark
To understand K (Qn), it then remains to understand Syl2(K (Qn)).
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Introduction
Previous Results II
Lemma [Benkart, Klivans, Reiner]
For every u ∈ (Z/2Z)n, let χu ∈ Z2n be the vector with entry in positionv ∈ (Z/2Z)n equal to (−1)u·v . Then χu is an eigenvector of L(Qn) witheigenvalue 2 · wt(u), where wt(u) is the number of non-zero entries in u.
Remark
Thus, we understand L(Qn) entirely as a map Q2n → Q2n . Whenconsidering it as a map Z2n → Z2n , this leaves us with the task ofunderstanding the Z-torsion.
Theorem [Benkart, Klivans, Reiner]
There is an isomorphism of Z-modules
Z⊕ K (Qn) ∼= Z[x1, . . . , xn]/(x21 − 1, . . . , x2n − 1, n −∑
xi ).
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Introduction
Previous Results II
Lemma [Benkart, Klivans, Reiner]
For every u ∈ (Z/2Z)n, let χu ∈ Z2n be the vector with entry in positionv ∈ (Z/2Z)n equal to (−1)u·v . Then χu is an eigenvector of L(Qn) witheigenvalue 2 · wt(u), where wt(u) is the number of non-zero entries in u.
Remark
Thus, we understand L(Qn) entirely as a map Q2n → Q2n . Whenconsidering it as a map Z2n → Z2n , this leaves us with the task ofunderstanding the Z-torsion.
Theorem [Benkart, Klivans, Reiner]
There is an isomorphism of Z-modules
Z⊕ K (Qn) ∼= Z[x1, . . . , xn]/(x21 − 1, . . . , x2n − 1, n −∑
xi ).
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Introduction
Previous Results II
Lemma [Benkart, Klivans, Reiner]
For every u ∈ (Z/2Z)n, let χu ∈ Z2n be the vector with entry in positionv ∈ (Z/2Z)n equal to (−1)u·v . Then χu is an eigenvector of L(Qn) witheigenvalue 2 · wt(u), where wt(u) is the number of non-zero entries in u.
Remark
Thus, we understand L(Qn) entirely as a map Q2n → Q2n . Whenconsidering it as a map Z2n → Z2n , this leaves us with the task ofunderstanding the Z-torsion.
Theorem [Benkart, Klivans, Reiner]
There is an isomorphism of Z-modules
Z⊕ K (Qn) ∼= Z[x1, . . . , xn]/(x21 − 1, . . . , x2n − 1, n −∑
xi ).
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Grobner Basis Calculations
Grobner Basis Background
Definition
Let R = T [x1, . . . , xn], where T is a commutative Noetherian ring. Amonomial order on R is a total order < on the set of monomialsxα11 · · · xαn
n of R. From now on, we implicitly assume a monomial order <on R.
Notation
Let I ⊆ [n]. We write xI :=∏
i∈I xi .
Definition
Let f ∈ R. Then the leading term of f , denoted `t(f ), is the term of fgreatest with respect to <.
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Grobner Basis Calculations
Grobner Basis Background
Definition
Let R = T [x1, . . . , xn], where T is a commutative Noetherian ring. Amonomial order on R is a total order < on the set of monomialsxα11 · · · xαn
n of R. From now on, we implicitly assume a monomial order <on R.
Notation
Let I ⊆ [n]. We write xI :=∏
i∈I xi .
Definition
Let f ∈ R. Then the leading term of f , denoted `t(f ), is the term of fgreatest with respect to <.
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Grobner Basis Calculations
Grobner Basis Background
Definition
Let R = T [x1, . . . , xn], where T is a commutative Noetherian ring. Amonomial order on R is a total order < on the set of monomialsxα11 · · · xαn
n of R. From now on, we implicitly assume a monomial order <on R.
Notation
Let I ⊆ [n]. We write xI :=∏
i∈I xi .
Definition
Let f ∈ R. Then the leading term of f , denoted `t(f ), is the term of fgreatest with respect to <.
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Grobner Basis Calculations
Grobner Basis Background
Definition
Let I / R be an ideal. Then the leading term ideal of I is
LT(I ) = ({`t(f ) | f ∈ I}).
Definition
Let I / R an ideal. A Grobner basis of I is a generating setS = {g1, . . . , gk} of I satisfying either of the following two properties:
For every f ∈ I , we can write `t(f ) = c1 `t(g1) + · · ·+ ck `t(gk) forsome ci ∈ R.
LT (I ) = (`t(g1), . . . , `t(gk)).
Theorem
When T is a PID, every ideal I / R has a Grobner basis.
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Grobner Basis Calculations
Grobner Basis Background
Definition
Let I / R be an ideal. Then the leading term ideal of I is
LT(I ) = ({`t(f ) | f ∈ I}).
Definition
Let I / R an ideal. A Grobner basis of I is a generating setS = {g1, . . . , gk} of I satisfying either of the following two properties:
For every f ∈ I , we can write `t(f ) = c1 `t(g1) + · · ·+ ck `t(gk) forsome ci ∈ R.
LT (I ) = (`t(g1), . . . , `t(gk)).
Theorem
When T is a PID, every ideal I / R has a Grobner basis.
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Grobner Basis Calculations
Grobner Basis Background
Definition
Let I / R be an ideal. Then the leading term ideal of I is
LT(I ) = ({`t(f ) | f ∈ I}).
Definition
Let I / R an ideal. A Grobner basis of I is a generating setS = {g1, . . . , gk} of I satisfying either of the following two properties:
For every f ∈ I , we can write `t(f ) = c1 `t(g1) + · · ·+ ck `t(gk) forsome ci ∈ R.
LT (I ) = (`t(g1), . . . , `t(gk)).
Theorem
When T is a PID, every ideal I / R has a Grobner basis.
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Grobner Basis Calculations
Relevance to Our Situation
Theorem
Let I / R be an ideal. Then, as T -modules,
R/I ∼= R/LT(I ).
Remark
By the isomorphism mentioned previously, to understand K (Qn) it sufficesto understand a Grobner basis for the ideal
In := (x21 − 1, . . . , x2n − 1, n −∑
xi )
in Z[x1, . . . , xn].
However, the Grobner basis is very complicated.
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Grobner Basis Calculations
Relevance to Our Situation
Theorem
Let I / R be an ideal. Then, as T -modules,
R/I ∼= R/LT(I ).
Remark
By the isomorphism mentioned previously, to understand K (Qn) it sufficesto understand a Grobner basis for the ideal
In := (x21 − 1, . . . , x2n − 1, n −∑
xi )
in Z[x1, . . . , xn].
However, the Grobner basis is very complicated.
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Grobner Basis Calculations
Relevance to Our Situation
Theorem
Let I / R be an ideal. Then, as T -modules,
R/I ∼= R/LT(I ).
Remark
By the isomorphism mentioned previously, to understand K (Qn) it sufficesto understand a Grobner basis for the ideal
In := (x21 − 1, . . . , x2n − 1, n −∑
xi )
in Z[x1, . . . , xn]. However, the Grobner basis is very complicated.
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Grobner Basis Calculations
Relevance to Our Situation
Lemma
Let Jn denote the ideal (x21 − 1, . . . , x2n − 1, n−∑
xi ) in Z/2iZ[x1, . . . , xn].Then the factors of Z/2Z, . . . ,Z/2i−1Z in Z[x1, . . . , xn]/In andZ/2iZ[x1, . . . , xn]/Jn are the same.
Goal
Understand a Grobner basis of Jn for i = 2, and thus understand thenumber of Z/2Z-factors in Syl2 K (Qn).
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Grobner Basis Calculations
Relevance to Our Situation
Lemma
Let Jn denote the ideal (x21 − 1, . . . , x2n − 1, n−∑
xi ) in Z/2iZ[x1, . . . , xn].Then the factors of Z/2Z, . . . ,Z/2i−1Z in Z[x1, . . . , xn]/In andZ/2iZ[x1, . . . , xn]/Jn are the same.
Goal
Understand a Grobner basis of Jn for i = 2, and thus understand thenumber of Z/2Z-factors in Syl2 K (Qn).
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Grobner Basis Calculations
The Case i = 2
Conjecture
For every odd integer m, let
Wm = {(2 + ε2, 4 + ε4, . . . ,m − 3 + εm−3,m − 1,m) | εi ∈ {0, 1}}.
ThenLT (Jn) = (x1) + (x22 , . . . , x
2n ) +
∑m≤nm odd
∑I∈Wm
(2xI ).
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Grobner Basis Calculations
The Case i = 2
Conjecture
For every odd integer m, let
Wm = {(2 + ε2, 4 + ε4, . . . ,m − 3 + εm−3,m − 1,m) | εi ∈ {0, 1}}.
ThenLT (Jn) = (x1) + (x22 , . . . , x
2n ) +
∑m≤nm odd
∑I∈Wm
(2xI ).
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A Bound on the Largest Cyclic Factor Size
An Observation
Observation
The highest cyclic factor has size equal to the highest additive order of anelement in
K (Qn) ∼= Z[x1, x2, . . . , xn]/(x21 − 1, . . . , x2n − 1, n − x1 − x2 − . . .− xn)
Lemma
The elements xi − 1 have highest additive order in K (Qn) for alli ∈ {1, . . . , n}.
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A Bound on the Largest Cyclic Factor Size
An Observation
Observation
The highest cyclic factor has size equal to the highest additive order of anelement in
K (Qn) ∼= Z[x1, x2, . . . , xn]/(x21 − 1, . . . , x2n − 1, n − x1 − x2 − . . .− xn)
Lemma
The elements xi − 1 have highest additive order in K (Qn) for alli ∈ {1, . . . , n}.
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A Bound on the Largest Cyclic Factor Size
An Observation
Proof Outline
Show a polynomial has a multiple in In only if it has the form
f (x1, . . . , xn) =∑I⊆[n]
cI (xI − 1)
Show xI − 1 has a multiple in In for every I ⊆ [n].
Show ord(xi − 1) ≥ ord(xI − 1) for any I ⊆ [n].
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A Bound on the Largest Cyclic Factor Size
An Observation
Proof Outline
Show a polynomial has a multiple in In only if it has the form
f (x1, . . . , xn) =∑I⊆[n]
cI (xI − 1)
Show xI − 1 has a multiple in In for every I ⊆ [n].
Show ord(xi − 1) ≥ ord(xI − 1) for any I ⊆ [n].
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A Bound on the Largest Cyclic Factor Size
An Observation
Proof Outline
Show a polynomial has a multiple in In only if it has the form
f (x1, . . . , xn) =∑I⊆[n]
cI (xI − 1)
Show xI − 1 has a multiple in In for every I ⊆ [n].
Show ord(xi − 1) ≥ ord(xI − 1) for any I ⊆ [n].
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A Bound on the Largest Cyclic Factor Size
The Order of x1 − 1
We switch back to Q2n :
x1 − 1 ∼
−110...0
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A Bound on the Largest Cyclic Factor Size
The Order of x1 − 1
We want to find the smallest C such that ∃v ∈ Z2n satisfying
L(Qn) · v =
−CC0...0
Idea: Work in the χu-basis!
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A Bound on the Largest Cyclic Factor Size
The Order of x1 − 1
We want to find the smallest C such that ∃v ∈ Z2n satisfying
L(Qn) · v =
−CC0...0
Idea: Work in the χu-basis!
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A Bound on the Largest Cyclic Factor Size
The Order of x1 − 1
Both terms are nice:
L(Qn) ∼
0 . . . . . . . . . . . . 0... 2
...... 2
...... 4
......
. . ....
0 · · · · · · · · · · · · 2n
−110......0
∼
01
2n−1
01
2n−1
...1
2n−1
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A Bound on the Largest Cyclic Factor Size
The Order of x1 − 1
We now have the χu-coordinates of v:
v ∼(0 1
2n 0 12n+1 . . . 1
n2n
)T
Theorem
The order of x1 − 1 is ≤ 2n · LCM(1, 2, . . . , n)
Corollary
The size of the largest cyclic factor in Syl2(K (Qn)) is ≤ 2n+log2 n
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A Bound on the Largest Cyclic Factor Size
The Order of x1 − 1
We now have the χu-coordinates of v:
v ∼(0 1
2n 0 12n+1 . . . 1
n2n
)TTheorem
The order of x1 − 1 is ≤ 2n · LCM(1, 2, . . . , n)
Corollary
The size of the largest cyclic factor in Syl2(K (Qn)) is ≤ 2n+log2 n
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A Bound on the Largest Cyclic Factor Size
The Order of x1 − 1
We now have the χu-coordinates of v:
v ∼(0 1
2n 0 12n+1 . . . 1
n2n
)TTheorem
The order of x1 − 1 is ≤ 2n · LCM(1, 2, . . . , n)
Corollary
The size of the largest cyclic factor in Syl2(K (Qn)) is ≤ 2n+log2 n
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Analogous Bounds on Other Cayley Graphs
Other Cayley Graphs
Goal
Generalize the technique used for the cube graph to other Cayley graphs.
Key Theorem [Benkart, Klivans, Reiner]
Let G be the n-th power of a directed cycle of size k . Then
K (G ) ∼= Z[x1, . . . , xn]/(xk1 − 1, . . . , xkn − 1, n −∑
xi ).
Lemma [Benkart, Klivans, Reiner]
For every u ∈ (Z/kZ)n, let χu ∈ Zknbe the vector with entry in position
v ∈ (Z/kZ)n equal to ζu·vk . Then χu is an eigenvector of L(G ) witheigenvalue k · wt(u), where wt(u) is the number of non-zero entries in u.
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Analogous Bounds on Other Cayley Graphs
Other Cayley Graphs
Goal
Generalize the technique used for the cube graph to other Cayley graphs.
Key Theorem [Benkart, Klivans, Reiner]
Let G be the n-th power of a directed cycle of size k . Then
K (G ) ∼= Z[x1, . . . , xn]/(xk1 − 1, . . . , xkn − 1, n −∑
xi ).
Lemma [Benkart, Klivans, Reiner]
For every u ∈ (Z/kZ)n, let χu ∈ Zknbe the vector with entry in position
v ∈ (Z/kZ)n equal to ζu·vk . Then χu is an eigenvector of L(G ) witheigenvalue k · wt(u), where wt(u) is the number of non-zero entries in u.
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Analogous Bounds on Other Cayley Graphs
Other Cayley Graphs
Goal
Generalize the technique used for the cube graph to other Cayley graphs.
Key Theorem [Benkart, Klivans, Reiner]
Let G be the n-th power of a directed cycle of size k . Then
K (G ) ∼= Z[x1, . . . , xn]/(xk1 − 1, . . . , xkn − 1, n −∑
xi ).
Lemma [Benkart, Klivans, Reiner]
For every u ∈ (Z/kZ)n, let χu ∈ Zknbe the vector with entry in position
v ∈ (Z/kZ)n equal to ζu·vk . Then χu is an eigenvector of L(G ) witheigenvalue k · wt(u), where wt(u) is the number of non-zero entries in u.
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Analogous Bounds on Other Cayley Graphs
Generalize x1 − 1
Lemma
As before, xi − 1 has maximal order in K (G ) for all i ∈ {1, . . . , n}.
Remark
However, xi − 1 does not have a nice form in the χu-basis. So we mustfind another high-order term with a nice form. One such element is(k − 1)− xi − x2i − · · · − xk−1i .
Lemma
k · ord(
(k − 1)− xi − x2i − · · · − xk−1i
)= ord(xi − 1).
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Analogous Bounds on Other Cayley Graphs
Generalize x1 − 1
Lemma
As before, xi − 1 has maximal order in K (G ) for all i ∈ {1, . . . , n}.
Remark
However, xi − 1 does not have a nice form in the χu-basis. So we mustfind another high-order term with a nice form. One such element is(k − 1)− xi − x2i − · · · − xk−1i .
Lemma
k · ord(
(k − 1)− xi − x2i − · · · − xk−1i
)= ord(xi − 1).
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Analogous Bounds on Other Cayley Graphs
Generalize x1 − 1
Lemma
As before, xi − 1 has maximal order in K (G ) for all i ∈ {1, . . . , n}.
Remark
However, xi − 1 does not have a nice form in the χu-basis. So we mustfind another high-order term with a nice form. One such element is(k − 1)− xi − x2i − · · · − xk−1i .
Lemma
k · ord(
(k − 1)− xi − x2i − · · · − xk−1i
)= ord(xi − 1).
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Analogous Bounds on Other Cayley Graphs
Form in χu-basis
The form for (k − 1)− xi − x2i − · · · − xk−1i in the χu-basis is as follows:
k − 1−1−1
...−10...0
∼
01kn
...1kn
01kn
...1kn
...
.
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Analogous Bounds on Other Cayley Graphs
Bounds for k = 3, 4
Theorem (k = 3)
Let k = 3. Then the size of the largest cyclic factor of Syl3(K (G )) is≤ 3n+1+blog3(n)c.
Theorem (k = 4)
Let k = 4. Then the size of the largest cyclic factor of Syl2(K (G )) is≤ 4n+1+blog4(n)c.
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Higher Critical Groups
A Different Viewpoint
Set C1(G ), C0(G ) to be formal groups of Z-linear combinations of theedges and vertices of G respectively.
There is a chain complex
0→ C1(G )E−→ C0(G )
ε−→ Z→ 0
where E is the incidence matrix of G and ε(∑
nivi ) =∑
ni is theaugmentation map.
Lemma
L(G ) = EET and K (G ) = ker(ε)/Im(L(G )) = ker(ε)/Im(EET )
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Higher Critical Groups
A Different Viewpoint
Set C1(G ), C0(G ) to be formal groups of Z-linear combinations of theedges and vertices of G respectively.There is a chain complex
0→ C1(G )E−→ C0(G )
ε−→ Z→ 0
where E is the incidence matrix of G and ε(∑
nivi ) =∑
ni is theaugmentation map.
Lemma
L(G ) = EET and K (G ) = ker(ε)/Im(L(G )) = ker(ε)/Im(EET )
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Higher Critical Groups
A Different Viewpoint
Set C1(G ), C0(G ) to be formal groups of Z-linear combinations of theedges and vertices of G respectively.There is a chain complex
0→ C1(G )E−→ C0(G )
ε−→ Z→ 0
where E is the incidence matrix of G and ε(∑
nivi ) =∑
ni is theaugmentation map.
Lemma
L(G ) = EET and K (G ) = ker(ε)/Im(L(G )) = ker(ε)/Im(EET )
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Higher Critical Groups
Extension to Cell Complexes
Fix a cell complex X . There is a cellular chain complex
. . .→ Ci (X )∂i−→ Ci−1(X )→ . . .→ C1(X )
∂1−→ C0(X )ε−→ Z→ 0
Definition
The i-th critical group of X is Ki (X ) = ker(∂i )/Im(∂i+1∂Ti+1)
Related to cellular spannng trees, higher-dimensional dynamical systemson X .
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Higher Critical Groups
Extension to Cell Complexes
Fix a cell complex X . There is a cellular chain complex
. . .→ Ci (X )∂i−→ Ci−1(X )→ . . .→ C1(X )
∂1−→ C0(X )ε−→ Z→ 0
Definition
The i-th critical group of X is Ki (X ) = ker(∂i )/Im(∂i+1∂Ti+1)
Related to cellular spannng trees, higher-dimensional dynamical systemson X .
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Higher Critical Groups
Extension to Cell Complexes
Fix a cell complex X . There is a cellular chain complex
. . .→ Ci (X )∂i−→ Ci−1(X )→ . . .→ C1(X )
∂1−→ C0(X )ε−→ Z→ 0
Definition
The i-th critical group of X is Ki (X ) = ker(∂i )/Im(∂i+1∂Ti+1)
Related to cellular spannng trees, higher-dimensional dynamical systemson X .
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Higher Critical Groups
Initial Results
We have an extension of Bai’s Theorem:
Theorem
For any prime p > 2,
Sylp (Ki (Qn)) ' Sylp
n⊕j=i+1
(Z/jZ)(nj)(j−1i )
Proof Outline
Can show ∂i+1∂Ti+1 + ∂Ti ∂i = L(Qn−i )
⊕(ni).
∂i+1∂Ti+1 and ∂Ti ∂i are diagonalizable and commute, so they have the
same eigenvectors.
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Higher Critical Groups
Initial Results
We have an extension of Bai’s Theorem:
Theorem
For any prime p > 2,
Sylp (Ki (Qn)) ' Sylp
n⊕j=i+1
(Z/jZ)(nj)(j−1i )
Proof Outline
Can show ∂i+1∂Ti+1 + ∂Ti ∂i = L(Qn−i )
⊕(ni).
∂i+1∂Ti+1 and ∂Ti ∂i are diagonalizable and commute, so they have the
same eigenvectors.
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Higher Critical Groups
Initial Results
We have an extension of Bai’s Theorem:
Theorem
For any prime p > 2,
Sylp (Ki (Qn)) ' Sylp
n⊕j=i+1
(Z/jZ)(nj)(j−1i )
Proof Outline
Can show ∂i+1∂Ti+1 + ∂Ti ∂i = L(Qn−i )
⊕(ni).
∂i+1∂Ti+1 and ∂Ti ∂i are diagonalizable and commute, so they have the
same eigenvectors.
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Further Directions
Further Directions
Further Directions
A lower bound on the top cyclic factor: Examine minors of L(Qn)?
Top cyclic factor bounds on Ks1 × Ks2 × . . .× Ksn .
Extend the top cyclic factor bound to higher critical groups.
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Conclusion
Acknowledgments
Acknowledgments
We would like to acknowledge support from NSF RTG grantDMS-1148634 as well as the UMN Twin Cities 2016 Math REU. Specialthanks to Vic Reiner for his mentorship and advice, as well as to WillGrodzicki for helpful comments.
Questions?
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Conclusion
Acknowledgments
Acknowledgments
We would like to acknowledge support from NSF RTG grantDMS-1148634 as well as the UMN Twin Cities 2016 Math REU. Specialthanks to Vic Reiner for his mentorship and advice, as well as to WillGrodzicki for helpful comments.
Questions?
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