The Erd os-Ko-Rado Theorem for Permutations Karen Meagher (joint - - PowerPoint PPT Presentation

The Erd˝

• s-Ko-Rado Theorem for

Permutations

Karen Meagher (joint work with Bahman Ahmadi, Chris Godsil and Pablo Spiga) Villanova University, June 2014

Erd˝

• s-Ko-Rado Theorem

Theorem

Let F be an intersecting k-set system on an n-set. For n > 2k

Erd˝

• s-Ko-Rado Theorem

Theorem

Let F be an intersecting k-set system on an n-set. For n > 2k

• 1. |F| ≤

n−1

k−1

• ,

Erd˝

• s-Ko-Rado Theorem

Theorem

Let F be an intersecting k-set system on an n-set. For n > 2k

• 1. |F| ≤

n−1

k−1

• ,
• 2. and F meets this bound if and only if every set in F

contains a common element.

Intersecting Permutations

Two permutations σ and π from the symmetric group on n elements agree at a point i, if σ(i) = π(i)

Intersecting Permutations

Two permutations σ and π from the symmetric group on n elements agree at a point i, if σ(i) = π(i)

• r

π−1σ(i) = i for some i ∈ {1, 2, . . . , n}.

Intersecting Permutations

Two permutations σ and π from the symmetric group on n elements agree at a point i, if σ(i) = π(i)

• r

π−1σ(i) = i for some i ∈ {1, 2, . . . , n}. How big can a set of intersecting permutations be?

Intersecting Permutations

Two permutations σ and π from the symmetric group on n elements agree at a point i, if σ(i) = π(i)

• r

π−1σ(i) = i for some i ∈ {1, 2, . . . , n}. How big can a set of intersecting permutations be? Which systems attain this maximum size?

Canonical Intersecting Permutations

Define the set Si,j = {σ ∈ Sym(n) | σ(i) = j}.

Canonical Intersecting Permutations

Define the set Si,j = {σ ∈ Sym(n) | σ(i) = j}. This is a set of intersecting permutations of size (n − 1)!.

Canonical Intersecting Permutations

Define the set Si,j = {σ ∈ Sym(n) | σ(i) = j}. This is a set of intersecting permutations of size (n − 1)!.

◮ The set Si,i is the stabilizer of the point i.

Canonical Intersecting Permutations

Define the set Si,j = {σ ∈ Sym(n) | σ(i) = j}. This is a set of intersecting permutations of size (n − 1)!.

◮ The set Si,i is the stabilizer of the point i. ◮ The set Si,j is the coset of the stabilizer of the point i.

Erd˝

• s-Ko-Rado Theorem for Permutations

Frankl and Deza conjectured the following in 1977:

Erd˝

• s-Ko-Rado Theorem for Permutations

Frankl and Deza conjectured the following in 1977:

Theorem

Let S be a intersecting set of permutations from Sym(n), then |S| ≤ (n − 1)!

Erd˝

• s-Ko-Rado Theorem for Permutations

Frankl and Deza conjectured the following in 1977:

Theorem

Let S be a intersecting set of permutations from Sym(n), then |S| ≤ (n − 1)! and equality holds if and only if S is the coset of the stabilizer of a point in G.

Erd˝

• s-Ko-Rado Theorem for Permutations

Frankl and Deza conjectured the following in 1977:

Theorem

Let S be a intersecting set of permutations from Sym(n), then |S| ≤ (n − 1)! and equality holds if and only if S is the coset of the stabilizer of a point in G. Several recent proofs:

Erd˝

• s-Ko-Rado Theorem for Permutations

Frankl and Deza conjectured the following in 1977:

Theorem

Let S be a intersecting set of permutations from Sym(n), then |S| ≤ (n − 1)! and equality holds if and only if S is the coset of the stabilizer of a point in G. Several recent proofs:

◮ 2003 - Cameron and Ku ◮ 2004 - Larose and Malvenuto ◮ 2008 - Wang and Zhang ◮ 2009 - Godsil and Meagher

Subgroups of the Symmetric Group

◮ Let G ≤ Sym(n),

Subgroups of the Symmetric Group

◮ Let G ≤ Sym(n), then G acts on {1, 2, . . . , n}.

Subgroups of the Symmetric Group

◮ Let G ≤ Sym(n), then G acts on {1, 2, . . . , n}. ◮ G has the EKR property if the size of largest set of

intersecting permutations in G is the size of a stabilizer of a point.

Subgroups of the Symmetric Group

◮ Let G ≤ Sym(n), then G acts on {1, 2, . . . , n}. ◮ G has the EKR property if the size of largest set of

intersecting permutations in G is the size of a stabilizer of a point.

◮ G has the strict EKR property if the only intersecting set

• f maximum size are the cosets of the stabilizer of a point.

Subgroups of the Symmetric Group

◮ Let G ≤ Sym(n), then G acts on {1, 2, . . . , n}. ◮ G has the EKR property if the size of largest set of

intersecting permutations in G is the size of a stabilizer of a point.

◮ G has the strict EKR property if the only intersecting set

• f maximum size are the cosets of the stabilizer of a point.

Which groups have the (strict) EKR property?

Derangement Graph for a Group

Let G be a permutation group acting on a set X of size n.

Derangement Graph for a Group

Let G be a permutation group acting on a set X of size n.

◮ Build a graph ΓG with vertices elements in G,

Derangement Graph for a Group

Let G be a permutation group acting on a set X of size n.

◮ Build a graph ΓG with vertices elements in G, ◮ and vertices are adjacent if they are not intersecting.

Derangement Graph for a Group

Let G be a permutation group acting on a set X of size n.

◮ Build a graph ΓG with vertices elements in G, ◮ and vertices are adjacent if they are not intersecting. ◮ The graph ΓG is called the derangement graph

Derangement Graph for a Group

Let G be a permutation group acting on a set X of size n.

◮ Build a graph ΓG with vertices elements in G, ◮ and vertices are adjacent if they are not intersecting. ◮ The graph ΓG is called the derangement graph

(vertices σ, π are adjacent if π−1σ is a derangement).

Derangement Graph for a Group

Let G be a permutation group acting on a set X of size n.

◮ Build a graph ΓG with vertices elements in G, ◮ and vertices are adjacent if they are not intersecting. ◮ The graph ΓG is called the derangement graph

(vertices σ, π are adjacent if π−1σ is a derangement).

◮ A coclique in ΓG is an intersecting set of permutations.

Derangement Graph for a Group

Let G be a permutation group acting on a set X of size n.

◮ Build a graph ΓG with vertices elements in G, ◮ and vertices are adjacent if they are not intersecting. ◮ The graph ΓG is called the derangement graph

(vertices σ, π are adjacent if π−1σ is a derangement).

◮ A coclique in ΓG is an intersecting set of permutations.

What is are the largest cocliques (independent sets) in ΓG?

Algebraic Properties of Derangement Graphs

Some properties of ΓG:

Algebraic Properties of Derangement Graphs

Some properties of ΓG:

◮ ΓG is vertex transitive.

Algebraic Properties of Derangement Graphs

Some properties of ΓG:

◮ ΓG is vertex transitive. ◮ ΓG is a normal Cayley graph,

Algebraic Properties of Derangement Graphs

Some properties of ΓG:

◮ ΓG is vertex transitive. ◮ ΓG is a normal Cayley graph, the connection set, D, is the

set of all derangements in G.

Algebraic Properties of Derangement Graphs

Some properties of ΓG:

◮ ΓG is vertex transitive. ◮ ΓG is a normal Cayley graph, the connection set, D, is the

set of all derangements in G.

◮ The eigenvalues of ΓG are

Algebraic Properties of Derangement Graphs

Some properties of ΓG:

◮ ΓG is vertex transitive. ◮ ΓG is a normal Cayley graph, the connection set, D, is the

set of all derangements in G.

◮ The eigenvalues of ΓG are

1 χ(1)

• d∈D

χ(d) where χ is an irreducible character of G.

2-Transitive Subgroups

• 1. If G ≤ Sym(n) is a transitive subgroup, then the stabilizer
• f a point has size |G|/n.

2-Transitive Subgroups

• 1. If G ≤ Sym(n) is a transitive subgroup, then the stabilizer
• f a point has size |G|/n.
• 2. The value of the standard character χ on an element

g ∈ G is fix(g) − 1.

2-Transitive Subgroups

• 1. If G ≤ Sym(n) is a transitive subgroup, then the stabilizer
• f a point has size |G|/n.
• 2. The value of the standard character χ on an element

g ∈ G is fix(g) − 1.

• 3. If G is 2-transitive, the standard character is irreducible.

2-Transitive Subgroups

• 1. If G ≤ Sym(n) is a transitive subgroup, then the stabilizer
• f a point has size |G|/n.
• 2. The value of the standard character χ on an element

g ∈ G is fix(g) − 1.

• 3. If G is 2-transitive, the standard character is irreducible.
• 4. If D is the set of derangements in G, then

2-Transitive Subgroups

• 1. If G ≤ Sym(n) is a transitive subgroup, then the stabilizer
• f a point has size |G|/n.
• 2. The value of the standard character χ on an element

g ∈ G is fix(g) − 1.

• 3. If G is 2-transitive, the standard character is irreducible.
• 4. If D is the set of derangements in G, then

λχ = 1 χ(1)

• g∈D

χ(g)

2-Transitive Subgroups

• 1. If G ≤ Sym(n) is a transitive subgroup, then the stabilizer
• f a point has size |G|/n.
• 2. The value of the standard character χ on an element

g ∈ G is fix(g) − 1.

• 3. If G is 2-transitive, the standard character is irreducible.
• 4. If D is the set of derangements in G, then

λχ = 1 χ(1)

• g∈D

χ(g) = −|D| n − 1 is an eigenvalue.

Hoffman’s Ratio Bound

Ratio Bound If X is a d-regular graph then α(X) ≤ |V(X)| 1 − d

τ

Hoffman’s Ratio Bound

Ratio Bound If X is a d-regular graph then α(X) ≤ |V(X)| 1 − d

τ

where d is the degree and τ is the least eigenvalue for the adja- cency matrix for X.

Hoffman’s Ratio Bound

Ratio Bound If X is a d-regular graph then α(X) ≤ |V(X)| 1 − d

τ

where d is the degree and τ is the least eigenvalue for the adja- cency matrix for X. If

◮ equality holds in the ratio bound

Hoffman’s Ratio Bound

Ratio Bound If X is a d-regular graph then α(X) ≤ |V(X)| 1 − d

τ

where d is the degree and τ is the least eigenvalue for the adja- cency matrix for X. If

◮ equality holds in the ratio bound ◮ and y is a characteristic vector for a maximum coclique,

Hoffman’s Ratio Bound

Ratio Bound If X is a d-regular graph then α(X) ≤ |V(X)| 1 − d

τ

where d is the degree and τ is the least eigenvalue for the adja- cency matrix for X. If

◮ equality holds in the ratio bound ◮ and y is a characteristic vector for a maximum coclique,

then y − α(X) |V(X)|1 is an eigenvector for τ.

Least Eigenvalues of Derangement Graph

If −|D|

n−1 is the least eigenvalue for ΓG then

Least Eigenvalues of Derangement Graph

If −|D|

n−1 is the least eigenvalue for ΓG then

α(ΓG) ≤ |G| 1 −

|D|

−|D| n−1

= |G| n .

Least Eigenvalues of Derangement Graph

If −|D|

n−1 is the least eigenvalue for ΓG then

α(ΓG) ≤ |G| 1 −

|D|

−|D| n−1

= |G| n . This implies that G has the EKR property!

Least Eigenvalues of Derangement Graph

If −|D|

n−1 is the least eigenvalue for ΓG then

α(ΓG) ≤ |G| 1 −

|D|

−|D| n−1

= |G| n . This implies that G has the EKR property! For which 2-transitive groups does the standard character give the least eigenvalue of the derangement graph?

The standard module

Let χ be an irreducible representation of G.

The standard module

Let χ be an irreducible representation of G. Define a |G| × |G| matrix (Eχ)π,σ = χ(1) |G| χ(π−1σ).

The standard module

Let χ be an irreducible representation of G. Define a |G| × |G| matrix (Eχ)π,σ = χ(1) |G| χ(π−1σ). The vector space generated by the columns of Eχ is called the χ-module of ΓG.

The standard module

Let χ be an irreducible representation of G. Define a |G| × |G| matrix (Eχ)π,σ = χ(1) |G| χ(π−1σ). The vector space generated by the columns of Eχ is called the χ-module of ΓG.

◮ If only the standard character gives the least eigenvalue,

then the shifted characteristic vector for every maximum coclique will be in the standard module.

A basis for the standard module

Define Si,j to be the set of all permutations in G that map i to j,

A basis for the standard module

Define Si,j to be the set of all permutations in G that map i to j, and let vi,j be the characteristic vector of Si,j.

A basis for the standard module

Define Si,j to be the set of all permutations in G that map i to j, and let vi,j be the characteristic vector of Si,j. Let G be a 2-transitive group, then vi,j − 1

n1 lies in the standard

module.

A basis for the standard module

Define Si,j to be the set of all permutations in G that map i to j, and let vi,j be the characteristic vector of Si,j. Let G be a 2-transitive group, then vi,j − 1

n1 lies in the standard

module. If G is a 2-transitive group,

A basis for the standard module

Define Si,j to be the set of all permutations in G that map i to j, and let vi,j be the characteristic vector of Si,j. Let G be a 2-transitive group, then vi,j − 1

n1 lies in the standard

module. If G is a 2-transitive group, then the set B := {vi,j − 1 n1 | i, j ∈ [n − 1]}

A basis for the standard module

Define Si,j to be the set of all permutations in G that map i to j, and let vi,j be the characteristic vector of Si,j. Let G be a 2-transitive group, then vi,j − 1

n1 lies in the standard

module. If G is a 2-transitive group, then the set B := {vi,j − 1 n1 | i, j ∈ [n − 1]} is a basis for the standard module of G.

Characterizing the Maximum Cocliques

◮ Assume G ≤ Sym(n) is 2-transitive and has the EKR

property.

Characterizing the Maximum Cocliques

◮ Assume G ≤ Sym(n) is 2-transitive and has the EKR

property.

◮ Let S be a maximum coclique in ΓG and y be the

characteristic vector for S.

Characterizing the Maximum Cocliques

◮ Assume G ≤ Sym(n) is 2-transitive and has the EKR

property.

◮ Let S be a maximum coclique in ΓG and y be the

characteristic vector for S.

◮ If y − 1 n1 is in the standard module,

Characterizing the Maximum Cocliques

◮ Assume G ≤ Sym(n) is 2-transitive and has the EKR

property.

◮ Let S be a maximum coclique in ΓG and y be the

characteristic vector for S.

◮ If y − 1 n1 is in the standard module, then it is in the span of

{vi,j − 1

n1|i, j ∈ [n − 1]}.

Characterizing the Maximum Cocliques

◮ Assume G ≤ Sym(n) is 2-transitive and has the EKR

property.

◮ Let S be a maximum coclique in ΓG and y be the

characteristic vector for S.

◮ If y − 1 n1 is in the standard module, then it is in the span of

{vi,j − 1

n1|i, j ∈ [n − 1]}. ◮ Let L be the matrix whose columns are the vectors vi,j.

Characterizing the Maximum Cocliques

◮ Assume G ≤ Sym(n) is 2-transitive and has the EKR

property.

◮ Let S be a maximum coclique in ΓG and y be the

characteristic vector for S.

◮ If y − 1 n1 is in the standard module, then it is in the span of

{vi,j − 1

n1|i, j ∈ [n − 1]}. ◮ Let L be the matrix whose columns are the vectors vi,j. ◮ If y is the characteristic vector for a maximum independent

set, then there exists a vector x with Lx = y

Proof for Sym(n)

L

• x

=

• y

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

• 1. First, the matrix M has full rank

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

• 1. First, the matrix M has full rank HARD!,

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

• 1. First, the matrix M has full rank HARD!, so x2 = 0.

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

• 1. First, the matrix M has full rank HARD!, so x2 = 0.
• 2. Second the matrix X contains a n × n identity matrix,

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

• 1. First, the matrix M has full rank HARD!, so x2 = 0.
• 2. Second the matrix X contains a n × n identity matrix,

so x1 is a 01-vector.

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

• 1. First, the matrix M has full rank HARD!, so x2 = 0.
• 2. Second the matrix X contains a n × n identity matrix,

so x1 is a 01-vector.

• 3. Next, the sum of the entries of x1 is 1, so x1 contains

exactly one 1.

Proof for Sym(n)

L

• x

=

• y

i →i i →j identity 1 derangements M

• thers

X Y x1 x2

• =

  1 y′  

• 1. First, the matrix M has full rank HARD!, so x2 = 0.
• 2. Second the matrix X contains a n × n identity matrix,

so x1 is a 01-vector.

• 3. Next, the sum of the entries of x1 is 1, so x1 contains

exactly one 1.

• 4. Finally, the vector y is one of the columns of L.

Groups with Strict EKR Property

This works for

◮ Sym(n)

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n)

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013)

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013) ◮ PGL(2, q)

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013) ◮ PGL(2, q) (M. and Spiga, 2009)

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013) ◮ PGL(2, q) (M. and Spiga, 2009) ◮ PGL(3, q)

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013) ◮ PGL(2, q) (M. and Spiga, 2009) ◮ PGL(3, q) this group does not have the strict EKR property

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013) ◮ PGL(2, q) (M. and Spiga, 2009) ◮ PGL(3, q) this group does not have the strict EKR property

(M. and Spiga, 2013)

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013) ◮ PGL(2, q) (M. and Spiga, 2009) ◮ PGL(3, q) this group does not have the strict EKR property

(M. and Spiga, 2013)

◮ the Mathieu groups M11, M12, M22, M23, M24

Groups with Strict EKR Property

This works for

◮ Sym(n)

• 1. Can show this works without knowing the least eigenvalue
• f the derangement graph.
• 2. Renteln (2008) showed that − |D|

n−1 is the least eigenvalue.

◮ Alt(n) (Ahmadi, 2013) ◮ PGL(2, q) (M. and Spiga, 2009) ◮ PGL(3, q) this group does not have the strict EKR property

(M. and Spiga, 2013)

◮ the Mathieu groups M11, M12, M22, M23, M24 (Ahmadi,

2013)

Future Work

• 1. Determine for which 2-transtitive groups the standard

representation minus gives the least eigenvalue of the derangement graph.

Future Work

• 1. Determine for which 2-transtitive groups the standard

representation minus gives the least eigenvalue of the derangement graph.

• 2. Decide which 2-transitive groups have the EKR property.

Future Work

• 1. Determine for which 2-transtitive groups the standard

representation minus gives the least eigenvalue of the derangement graph.

• 2. Decide which 2-transitive groups have the EKR property.
• 3. If G is 2-transitive, can the characteristic vector of a

maximum coclique lie in a module other than the standard and trivial module?

Future Work

• 1. Determine for which 2-transtitive groups the standard

representation minus gives the least eigenvalue of the derangement graph.

• 2. Decide which 2-transitive groups have the EKR property.
• 3. If G is 2-transitive, can the characteristic vector of a

maximum coclique lie in a module other than the standard and trivial module?

• 4. Computationally check which small 2-transitive groups

have (strict) EKR property.