Upper and Lower Semimodularity of the Supercharacter...

Upper and Lower Semimodularity of theSupercharacter Theory Lattices of Cyclic Groups

Samuel Benidt, William Hall, & Anders Hendrickson

ΠME ConferenceSt. John’s University

April 17, 2010

DefinitionsOur Work

Two main theorems

Outline

1 DefinitionsLatticesSupercharacter Theories

2 Our WorkGoals and StrategyAnalysis of Specific Cases

3 Two main theoremsUpper SemimodularityLower Semimodularity

Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

DefinitionsOur Work

Two main theorems

LatticesSupercharacter Theories

Lattices

As you may know, a partially ordered set is a set with an order thatrequires the following for elements a, b in the set:

either a ≤ b

or a ≥ b

or a and b are incomparable

A good example of a poset is set inclusion: either A ⊆ B, or B ⊆ A, orneither.

Definition

A lattice is a partially ordered set in which any two elements a and bhave a unique least upper bound, a ∨ b, and greatest lower bound, a ∧ b.We read a ∨ b as “a join b” and a ∧ b as “a meet b”.


DefinitionsOur Work

Two main theorems


Examples

e

d

cb

a

e

d

cb

a

A Lattice

Not a lattice

Definition

We say an element b of a lattice covers another element d if b ≥ d andthere are no elements between b and d . We write d � b.


DefinitionsOur Work

Two main theorems


Examples

e

d

cb

a

e

d

cb

a

A Lattice Not a lattice

Definition

We say an element b of a lattice covers another element d if b ≥ d andthere are no elements between b and d . We write d � b.


DefinitionsOur Work

Two main theorems


Upper Semimodularity

Definition

A lattice L of finite length, is said to be upper semimodular if thefollowing condition is satisfied:

if a ∧ b � a, b then a, b � a ∨ b.

⇒


DefinitionsOur Work

Two main theorems


Examples of some USM Lattices


DefinitionsOur Work

Two main theorems


Lower Semimodularity

Definition

Let L be a lattice of finite length. Then L is lower semimodular if for alla, b ∈ L,

if a, b � a ∨ b then a ∧ b � a, b

⇒


DefinitionsOur Work

Two main theorems


Examples of some LSM Lattices


DefinitionsOur Work

Two main theorems


Notes on Groups

In our presentation, we’ll be focusing exclusively on finite cyclicgroups and their subgroups.

Note that since every group we’ll be discussing is Abelian, eachsubgroup N is normal.

This means that the quotient group G/N exists.


DefinitionsOur Work

Two main theorems


Supercharacter Theory

Definition

Let G be a finite group and let K be a partition of G .Then we say K is a superchacter theory if the following three conditionshold:

{1} is a part of K .

A product of sums of parts of K is a linear combination of sums ofparts of K .

Each part of K is a union of some conjugacy classes of G .

Sup(G ) denotes the set of all supercharacter theories of the group G ; itforms a lattice.

Supercharacter theories were defined in a 2008 paper by P. Diaconisand I.M. Isaacs.

They are the subject of very active study and the topic of anAmerican Institute of Mathematics workshop this May.


DefinitionsOur Work

Two main theorems


Sup(G ) as a Lattice

Given X , Y ∈ Sup(G ), we can find their join and meet, so Sup(G ) is alattice.

a1

��a2, a3

��a4 a1, a2

��a3

��a4

X ∧ Y

X ∨ Y

X Y


DefinitionsOur Work

Two main theorems




a1

��a2, a3

��a4 a1, a2

��a3

��a4

a1, a2, a3

��a4

X ∧ Y

X ∨ Y

X Y


DefinitionsOur Work

Two main theorems




a1

��a2, a3

��a4 a1, a2

��a3

��a4

a1, a2, a3

��a4

a1

��a2

��a3

��a4

X ∧ Y

X ∨ Y

X Y


DefinitionsOur Work

Two main theorems


Some Examples: Sup(Z3)

Figure: The graph of Sup(Z3)


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Two main theorems



Figure: The graph of Sup(Z17)


DefinitionsOur Work

Two main theorems



Figure: The graph of Sup(Z51)Benidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

DefinitionsOur Work

Two main theorems


Finding Supercharacter Theories

There are several methods of constructing supercharacter theories of agroup, including the following:

minimal supercharacter theory: m(Zn)m(Zn) is formed by placing each group element in its own part.This partition will always be the least element in the lattice.

maximal supercharacter theory: M(Zn)M(Zn) is formed by placing all group elements in two parts:{1}, and all other group elements.This partition will always be the greatest element in the lattice.

inverse supercharacter theory: Inv(Zn) = {1/g , g−1/g2, g−2/...}We define the inverse supercharacter theory by placing each groupelement and its inverse together in their own part.

partition the elements by their orders: A(Zn)

∗-product supercharacter theory: X ∗ Y ,where X ∈ Sup(N) and Y ∈ Sup(G/N).


DefinitionsOur Work

Two main theorems


∗-product

As mentioned on the previous slide, one way to find supercharactertheories is a method called the ∗-product.

Let N be normal in G with X ∈ Sup(N), Y ∈ Sup(G/N).Then X ∗ Y ∈ Sup(G ).

X Y

X ∗ Y


DefinitionsOur Work

Two main theorems


∗-product

As mentioned on the previous slide, one way to find supercharactertheories is a method called the ∗-product.

Let N be normal in G with X ∈ Sup(N), Y ∈ Sup(G/N).Then X ∗ Y ∈ Sup(G ).

X Y

X ∗ YBenidt, Hall, Hendrickson Modularity Conditions of Lattices of Cyclic Groups

DefinitionsOur Work

Two main theorems

Goals and StrategyAnalysis of Specific Cases

Research Goal

Research Goal

We seek necessary and sufficient conditions for when Sup(G )is upper- and lower-semimodular.


DefinitionsOur Work

Two main theorems


Inheritance Lemma

Lemma (Inheritance)

Let N be a subgroup of cyclic group G. Then

1 If Sup(N) is not upper semimodular, then Sup(G ) is not uppersemimodular.

2 If Sup(N) is not lower semimodular, then Sup(G ) is not lowersemimodular.


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p

p2 pq pqr 4p 4

USM

Yes No∗ No No No Yes

LSM

Yes No∗ Yes No No Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Case where |G | = p

Lemma

Given a group G of order p, Sup(G ) is isomorphic to the subgroup latticeof the automorphism group of G.

Lemma

It is a well-known fact in lattice theory that the subgroup lattice of anAbelian group is both USM and LSM.

Therefore, Sup(G ) is both USM and LSM.


DefinitionsOur Work

Two main theorems


Example of |G | = p

〈[1]〉

〈[18]〉〈[7]〉

〈[8]〉〈[4]〉

〈[2]〉

Lattice of Sup(Z19) Lattice of subgroupsof Aut(Z19)


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p

p2 pq pqr 4p 4

USM

Yes No∗ No No No Yes

LSM

Yes No∗ Yes No No Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p

p2 pq pqr 4p 4

USM Yes

No∗ No No No Yes

LSM Yes

No∗ Yes No No Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2

pq pqr 4p 4

USM Yes

No∗ No No No Yes

LSM Yes

No∗ Yes No No Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Case where |G | = p2, if p is odd

If G is cyclic of order p2, p an odd prime,then Sup(G ) contains the following sublattice.

m(Zp2 )

m(Zp) ∗m(Zp) Inv(Zp2 )

Inv(Zp) ∗m(Zp)m(Zp) ∗ Inv(Zp)

Inv(Zp) ∗ Inv(Zp)

Note that each line drawn in on this particular diagram indicates acovering relation, and each element listed is distinct from the others.


DefinitionsOur Work

Two main theorems


Case where |G | = 4

m(Z4)

M(Z2) ∗M(Z2)

M(Z4)


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2

pq pqr 4p 4

USM Yes

No∗ No No No Yes

LSM Yes

No∗ Yes No No Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2

pq pqr 4p

4USM Yes No∗

No No No

YesLSM Yes No∗

Yes No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq

pqr 4p

4USM Yes No∗

No No No

YesLSM Yes No∗

Yes No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


USM Case for |G | = pq

m(Zpq)

m(Zp) ∗m(Zq) m(Zq) ∗m(Zp)

M(Zp) ∗M(Zq)

M(Zpq)


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq

pqr 4p

4USM Yes No∗

No No No

YesLSM Yes No∗

Yes No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq

pqr 4p

4USM Yes No∗ No

No No

YesLSM Yes No∗

Yes No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


LSM Case for |G | = pq

Lemma

Let G be a cyclic group of order pq. Then Sup(G ) is lower semimodular.


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq

pqr 4p

4USM Yes No∗ No

No No

YesLSM Yes No∗

Yes No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq

pqr 4p

4USM Yes No∗ No

No No

YesLSM Yes No∗ Yes

No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq pqr 4p 4USM Yes No∗ No

No No

YesLSM Yes No∗ Yes

No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes

No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Case where |G | = pqr

Aut(Zpq) ∗M(Zr )

M(Zp) ∗M(Zq) ∗M(Zr )

M(Zq) ∗M(Zpr )M(Zp) ∗M(Zqr )

M(Zpqr )


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes

No No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes No

No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Case where |G | = 4p, if p is odd

A(Z2p) ∗M(Z2)

M(Z2) ∗M(Zp) ∗M(Z2) M(Zp) ∗M(Z2) ∗M(Z2)

M(Z2) ∗M(Z2p) M(Zp) ∗M(Z4)

M(Z4p)


DefinitionsOur Work

Two main theorems


Case where |G | = 4p, if p is even

G ∼= Z8


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes No

No

Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems


Summary Table

Summary Table

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes No No Yes

∗for odd primes p


DefinitionsOur Work

Two main theorems

Upper SemimodularityLower Semimodularity

Upper Semimodularity Theorem

Theorem (USM)

Let G be a cyclic group. Then Sup(G ) is upper semimodular if and onlyif the order of G is prime or four.


DefinitionsOur Work

Two main theorems


USM Proof


Cases where cyclic group G is USM:

If |G | = p, then we proved that Sup(G ) is USM.

If |G | = 4, we verified that this supercharacter theory lattice is USM.

What happens for other orders of G?


DefinitionsOur Work

Two main theorems


USM Proof

p p2 pq pqr 4p 4USM YesYes No∗ No No No YesLSM Yes No∗ Yes No No Yes






DefinitionsOur Work

Two main theorems


USM Proof

p p2 pq pqr 4p 4USM Yes No∗ No No No YesYesLSM Yes No∗ Yes No No Yes






DefinitionsOur Work

Two main theorems


USM Proof







DefinitionsOur Work

Two main theorems


USM Proof cont.


If |G | is not prime or four, then it is either a multiple of two distinctprimes or a prime power.

For |G | = k · pq, where p 6= q, then we showed that Sup(Zpq) is notUSM. By the Inheritance Lemma, Sup(G ) is not USM either.

For |G | = pn, where p 6= 2 and n ≥ 2, we again showed thatSup(Zp2) is not USM. Then, by the Inheritance Lemma, Sup(G ) isnot USM.

This leaves |G | = 2n with n ≥ 3. We verified that Sup(Z8) is notUSM, and thus, for all higher 2-powers, Sup(G ) is not USM. 2


DefinitionsOur Work

Two main theorems


USM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗ NoNo No No YesLSM Yes No∗ Yes No No Yes






DefinitionsOur Work

Two main theorems


USM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗No∗ No No No YesLSM Yes No∗ Yes No No Yes






DefinitionsOur Work

Two main theorems


USM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗ No No NoNo YesLSM Yes No∗ Yes No No Yes






DefinitionsOur Work

Two main theorems


USM Proof cont.







DefinitionsOur Work

Two main theorems


Lower Semimodularity Theorem

Theorem (LSM)

Let G be a cyclic group. Then Sup(G ) is lower semimodular if and only ifthe order of G is prime, the product of two distinct primes, or four.


DefinitionsOur Work

Two main theorems


LSM Proof


Cases where cyclic group G is LSM:

When |G | = p, Sup(G ) is lattice-isomorphic to the lattice C ofsubgroups of Aut(G ). Since C is LSM, Sup(G ) is also.

When |G | = pq, where p 6= q, a previously-stated lemma provedthat Sup(G ) is LSM.

When |G | = 4, we verified that this supercharacter theory lattice isLSM.



DefinitionsOur Work

Two main theorems


LSM Proof

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM YesYes No∗ Yes No No Yes







DefinitionsOur Work

Two main theorems


LSM Proof

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ YesYes No No Yes







DefinitionsOur Work

Two main theorems


LSM Proof

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes No No YesYes







DefinitionsOur Work

Two main theorems


LSM Proof








DefinitionsOur Work

Two main theorems


LSM Proof cont.


Suppose |G | is not p, pq, or 4.Consider the number of distinct prime factors of |G |.

1: If |G | = pa, then

If p is odd, then p2��|G |. So Sup(G) is not LSM.

If p is even, then 4p = 8��|G |, so Sup(G) is not LSM.

2: If |G | = paqb, then wlog a ≥ 2.

If p is odd, then p2��|G |, so Sup(G) is not LSM.

If p = 2, then 4q��|G | so Sup(G) is not LSM.

≥ 3: If |G | has at least 3 distinct prime factors,then pqr

��|G |, so Sup(G ) is not LSM. 2


DefinitionsOur Work

Two main theorems


LSM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗No∗ Yes No No Yes











DefinitionsOur Work

Two main theorems


LSM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes No NoNo Yes











DefinitionsOur Work

Two main theorems


LSM Proof cont.












DefinitionsOur Work

Two main theorems


LSM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗No∗ Yes No No Yes











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Two main theorems


LSM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes No NoNo Yes











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Two main theorems


LSM Proof cont.

p p2 pq pqr 4p 4USM Yes No∗ No No No YesLSM Yes No∗ Yes NoNo No Yes











DefinitionsOur Work

Two main theorems


LSM Proof cont.












Acknowledgments

We’d like to thank Dr. Hendrickson for inviting us to research withhim and guiding us through the project.

Also, thanks to Concordia College for the financial support.

Lastly, we’d like to thank the organizers of the St. John’s Pi MuEpsilon conference.


Questions?


Upper and Lower Semimodularity of the Supercharacter...

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