The Erd˝ os-Ko-Rado Theorem for Permutations Karen Meagher (joint work with Bahman Ahmadi, Chris Godsil and Pablo Spiga) Villanova University, June 2014
Erd˝ os-Ko-Rado Theorem Theorem Let F be an intersecting k-set system on an n-set. For n > 2 k
Erd˝ os-Ko-Rado Theorem Theorem Let F be an intersecting k-set system on an n-set. For n > 2 k � n − 1 � 1. |F| ≤ , k − 1
Erd˝ os-Ko-Rado Theorem Theorem Let F be an intersecting k-set system on an n-set. For n > 2 k � n − 1 � 1. |F| ≤ , 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 π − 1 σ ( i ) = i σ ( i ) = π ( i ) or 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 π − 1 σ ( i ) = i σ ( i ) = π ( i ) or 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 π − 1 σ ( i ) = i σ ( i ) = π ( i ) or 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 S i , j = { σ ∈ Sym ( n ) | σ ( i ) = j } .
Canonical Intersecting Permutations Define the set S i , j = { σ ∈ Sym ( n ) | σ ( i ) = j } . This is a set of intersecting permutations of size ( n − 1 )! .
Canonical Intersecting Permutations Define the set S i , j = { σ ∈ Sym ( n ) | σ ( i ) = j } . This is a set of intersecting permutations of size ( n − 1 )! . ◮ The set S i , i is the stabilizer of the point i .
Canonical Intersecting Permutations Define the set S i , j = { σ ∈ Sym ( n ) | σ ( i ) = j } . This is a set of intersecting permutations of size ( n − 1 )! . ◮ The set S i , i is the stabilizer of the point i . ◮ The set S i , j is the coset of the stabilizer of the point i .
Erd˝ os-Ko-Rado Theorem for Permutations Frankl and Deza conjectured the following in 1977:
Erd˝ os-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˝ os-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˝ os-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˝ os-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 of 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 of 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 Γ G 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 Γ G 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 Γ G with vertices elements in G , ◮ and vertices are adjacent if they are not intersecting. ◮ The graph Γ G 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 Γ G with vertices elements in G , ◮ and vertices are adjacent if they are not intersecting. ◮ The graph Γ G is called the derangement graph (vertices σ , π are adjacent if π − 1 σ 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 Γ G with vertices elements in G , ◮ and vertices are adjacent if they are not intersecting. ◮ The graph Γ G is called the derangement graph (vertices σ , π are adjacent if π − 1 σ is a derangement). ◮ A coclique in Γ G 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 Γ G with vertices elements in G , ◮ and vertices are adjacent if they are not intersecting. ◮ The graph Γ G is called the derangement graph (vertices σ , π are adjacent if π − 1 σ is a derangement). ◮ A coclique in Γ G is an intersecting set of permutations. What is are the largest cocliques (independent sets) in Γ G ?
Algebraic Properties of Derangement Graphs Some properties of Γ G :
Algebraic Properties of Derangement Graphs Some properties of Γ G : ◮ Γ G is vertex transitive.
Algebraic Properties of Derangement Graphs Some properties of Γ G : ◮ Γ G is vertex transitive. ◮ Γ G is a normal Cayley graph,
Algebraic Properties of Derangement Graphs Some properties of Γ G : ◮ Γ G is vertex transitive. ◮ Γ G 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 Γ G : ◮ Γ G is vertex transitive. ◮ Γ G is a normal Cayley graph, the connection set, D , is the set of all derangements in G . ◮ The eigenvalues of Γ G are
Algebraic Properties of Derangement Graphs Some properties of Γ G : ◮ Γ G is vertex transitive. ◮ Γ G is a normal Cayley graph, the connection set, D , is the set of all derangements in G . ◮ The eigenvalues of Γ G are 1 � χ ( d ) χ ( 1 ) d ∈ D where χ is an irreducible character of G .
2-Transitive Subgroups 1. If G ≤ Sym ( n ) is a transitive subgroup, then the stabilizer of a point has size | G | / n .
2-Transitive Subgroups 1. If G ≤ Sym ( n ) is a transitive subgroup, then the stabilizer of 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 of 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 of 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 of 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 1 � λ χ = χ ( g ) χ ( 1 ) g ∈ D
2-Transitive Subgroups 1. If G ≤ Sym ( n ) is a transitive subgroup, then the stabilizer of 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 1 χ ( g ) = −| D | � λ χ = χ ( 1 ) n − 1 g ∈ D is an eigenvalue.
Hoffman’s Ratio Bound Ratio Bound If X is a d -regular graph then α ( X ) ≤ | V ( X ) | 1 − d τ
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