Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x ( v ) times, ending up with configuration s ′ . s ′ ( v ) = s ( v ) − x ( v ) deg ( v ) + � ( v , w ) ∈ E x ( w )
Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x ( v ) times, ending up with configuration s ′ . s ′ ( v ) = s ( v ) − x ( v ) deg ( v ) + � ( v , w ) ∈ E x ( w ) s ′ = s − Lx
Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x ( v ) times, ending up with configuration s ′ . s ′ ( v ) = s ( v ) − x ( v ) deg ( v ) + � ( v , w ) ∈ E x ( w ) s ′ = s − Lx Theorem (Biggs) Let s be a configuration in the chip-firing game on a connected graph G. Then there is a unique critical configuration which can be reached from s.
Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x ( v ) times, ending up with configuration s ′ . s ′ ( v ) = s ( v ) − x ( v ) deg ( v ) + � ( v , w ) ∈ E x ( w ) s ′ = s − Lx Theorem (Biggs) Let s be a configuration in the chip-firing game on a connected graph G. Then there is a unique critical configuration which can be reached from s. Theorem (Biggs) The set of critical configurations has a natural group operation making it isomorphic to the critical group K (Γ) .
Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Smith group of Paley graphs Critical group of Paley graphs
Smith normal form Two integer matrices X and Y are equivalent iff there exist unimodular integer matrices P and Q such that PXQ = Y
Smith normal form Two integer matrices X and Y are equivalent iff there exist unimodular integer matrices P and Q such that PXQ = Y Each equivalence class contains a Smith normal form H 0 , Y = H = diag ( s 1 , s 2 , . . . , s r ) , s 1 | s 2 | · · · | s r . 0 0
Smith normal form Two integer matrices X and Y are equivalent iff there exist unimodular integer matrices P and Q such that PXQ = Y Each equivalence class contains a Smith normal form H 0 , Y = H = diag ( s 1 , s 2 , . . . , s r ) , s 1 | s 2 | · · · | s r . 0 0 Similarly for PIDs.
Smith normal form Two integer matrices X and Y are equivalent iff there exist unimodular integer matrices P and Q such that PXQ = Y Each equivalence class contains a Smith normal form H 0 , Y = H = diag ( s 1 , s 2 , . . . , s r ) , s 1 | s 2 | · · · | s r . 0 0 Similarly for PIDs. The SNF of the Laplacian (resp. adjacency matrix) gives the structure of the critical group (resp. Smith group).
Smith normal form Two integer matrices X and Y are equivalent iff there exist unimodular integer matrices P and Q such that PXQ = Y Each equivalence class contains a Smith normal form H 0 , Y = H = diag ( s 1 , s 2 , . . . , s r ) , s 1 | s 2 | · · · | s r . 0 0 Similarly for PIDs. The SNF of the Laplacian (resp. adjacency matrix) gives the structure of the critical group (resp. Smith group). A recent survey of Smith normal forms in combinatorics was written by Richard Stanley (just published in the Special Issue on 50 years of JCTA).
Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Smith group of Paley graphs Critical group of Paley graphs
◮ Trees, K (Γ) = { 0 } .
◮ Trees, K (Γ) = { 0 } . ◮ Complete graphs, K ( K n ) ∼ = ( Z / n Z ) n − 2 . (A refinement of Cayley’s theorem saying that the number of spanning trees of K n is n n − 2 .)
◮ Trees, K (Γ) = { 0 } . ◮ Complete graphs, K ( K n ) ∼ = ( Z / n Z ) n − 2 . (A refinement of Cayley’s theorem saying that the number of spanning trees of K n is n n − 2 .) ◮ Wheel graphs W n , K (Γ) ∼ = ( Z /ℓ n ) 2 , if n is odd (Biggs). Here ℓ n is a Lucas number.
◮ Trees, K (Γ) = { 0 } . ◮ Complete graphs, K ( K n ) ∼ = ( Z / n Z ) n − 2 . (A refinement of Cayley’s theorem saying that the number of spanning trees of K n is n n − 2 .) ◮ Wheel graphs W n , K (Γ) ∼ = ( Z /ℓ n ) 2 , if n is odd (Biggs). Here ℓ n is a Lucas number. ◮ Complete multipartite graphs (Jacobson, Niedermaier, Reiner).
◮ Trees, K (Γ) = { 0 } . ◮ Complete graphs, K ( K n ) ∼ = ( Z / n Z ) n − 2 . (A refinement of Cayley’s theorem saying that the number of spanning trees of K n is n n − 2 .) ◮ Wheel graphs W n , K (Γ) ∼ = ( Z /ℓ n ) 2 , if n is odd (Biggs). Here ℓ n is a Lucas number. ◮ Complete multipartite graphs (Jacobson, Niedermaier, Reiner). ◮ Conference graphs on a square-free number of vertices (Lorenzini).
Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Smith group of Paley graphs Critical group of Paley graphs
Paley graphs P ( q ) ◮ Vertex set is F q , q = p t ≡ 1 ( mod 4 )
Paley graphs P ( q ) ◮ Vertex set is F q , q = p t ≡ 1 ( mod 4 ) ◮ S = set of nonzero squares in F q
Paley graphs P ( q ) ◮ Vertex set is F q , q = p t ≡ 1 ( mod 4 ) ◮ S = set of nonzero squares in F q ◮ two vertices x and y are joined by an edge iff x − y ∈ S .
Paley graphs are Cayley graphs We can view P ( q ) as a Cayley graph on ( F q , +) with connecting set S
Paley graphs are strongly regular graphs It is well known and easily checked that P ( q ) is a strongly 2 , r = − 1 + √ q regular graph and that its eigenvalues are k = q − 1 2 and s = − 1 −√ q , with multiplicities 1, q − 1 and q − 1 2 , respectively. 2 2
Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Smith group of Paley graphs Critical group of Paley graphs
◮ | S ( P ( q )) | = k ( k / 2 ) k , so gcd ( | S ( P ( q )) | , q ) = 1 .
◮ | S ( P ( q )) | = k ( k / 2 ) k , so gcd ( | S ( P ( q )) | , q ) = 1 . ◮ X , complex character table of ( F q , +)
◮ | S ( P ( q )) | = k ( k / 2 ) k , so gcd ( | S ( P ( q )) | , q ) = 1 . ◮ X , complex character table of ( F q , +) ◮ X is a matrix over Z [ ζ ] , ζ a complex primitive p -th root of unity.
◮ | S ( P ( q )) | = k ( k / 2 ) k , so gcd ( | S ( P ( q )) | , q ) = 1 . ◮ X , complex character table of ( F q , +) ◮ X is a matrix over Z [ ζ ] , ζ a complex primitive p -th root of unity. t = I . 1 q XX ◮
◮ | S ( P ( q )) | = k ( k / 2 ) k , so gcd ( | S ( P ( q )) | , q ) = 1 . ◮ X , complex character table of ( F q , +) ◮ X is a matrix over Z [ ζ ] , ζ a complex primitive p -th root of unity. t = I . 1 q XX ◮ ◮ 1 t = diag ( ψ ( S )) ψ , q XAX (1) where ψ runs through the additive characters of F q .
Theorem S ( P ( q )) ∼ = Z / 2 µ Z ⊕ ( Z /µ Z ) 2 µ , where µ = q − 1 4 . Remark This theorem was conjectured by Joe Rushanan in his Caltech PhD thesis (1988).
Critical groups of graphs Outline Laplacian matrix of a graph Chip-firing game Smith normal form Some families of graphs with known critical groups Paley graphs Smith group of Paley graphs Critical group of Paley graphs
Symmetries ◮ � q + √ q � k � q − √ q � k | K ( P ( q )) | = 1 q − 3 2 µ k , = q q 2 2 where µ = q − 1 4 .
Symmetries ◮ � q + √ q � k � q − √ q � k | K ( P ( q )) | = 1 q − 3 2 µ k , = q q 2 2 where µ = q − 1 4 . ◮ Aut ( P ( q )) ≥ F q ⋊ S .
Symmetries ◮ � q + √ q � k � q − √ q � k | K ( P ( q )) | = 1 q − 3 2 µ k , = q q 2 2 where µ = q − 1 4 . ◮ Aut ( P ( q )) ≥ F q ⋊ S . ◮ K ( P ( q )) = K ( P ( q )) p ⊕ K ( P ( q )) p ′
Symmetries ◮ � q + √ q � k � q − √ q � k | K ( P ( q )) | = 1 q − 3 2 µ k , = q q 2 2 where µ = q − 1 4 . ◮ Aut ( P ( q )) ≥ F q ⋊ S . ◮ K ( P ( q )) = K ( P ( q )) p ⊕ K ( P ( q )) p ′ ◮ Use F q -action to help compute p ′ -part.
Symmetries ◮ � q + √ q � k � q − √ q � k | K ( P ( q )) | = 1 q − 3 2 µ k , = q q 2 2 where µ = q − 1 4 . ◮ Aut ( P ( q )) ≥ F q ⋊ S . ◮ K ( P ( q )) = K ( P ( q )) p ⊕ K ( P ( q )) p ′ ◮ Use F q -action to help compute p ′ -part. ◮ Use S -action to help compute p -part.
p ′ -part ◮ X , complex character table of ( F q , +)
p ′ -part ◮ X , complex character table of ( F q , +) ◮ X is a matrix over Z [ ζ ] , ζ a complex primitive p -th root of unity.
p ′ -part ◮ X , complex character table of ( F q , +) ◮ X is a matrix over Z [ ζ ] , ζ a complex primitive p -th root of unity. t = I . 1 q XX ◮
p ′ -part ◮ X , complex character table of ( F q , +) ◮ X is a matrix over Z [ ζ ] , ζ a complex primitive p -th root of unity. t = I . 1 q XX ◮ ◮ 1 t = diag ( k − ψ ( S )) ψ , q XLX (2)
Theorem K ( P ( q )) p ′ ∼ = ( Z /µ Z ) 2 µ , where µ = q − 1 4 .
The p -part Let L = kI − A . There exist invertible matrices P and Q over the ring of p -adic integers such that Y 0 Y = diag ( 1 , 1 , . . . 1 , p , p , . . . p , p 2 , p 2 . . . , p 2 , . . . ) . , PLQ = 0 0 The number of 1’s on the diagonal of Y is the p -rank of L , and it 2 ) t (a result of Brouwer and Van Eijl, 1992). is equal to ( p + 1
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q .
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q . ◮ T : F × q → R × , the Teichmüller character.
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q . ◮ T : F × q → R × , the Teichmüller character. ◮ T generates the cyclic group Hom ( F × q , R × ) .
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q . ◮ T : F × q → R × , the Teichmüller character. ◮ T generates the cyclic group Hom ( F × q , R × ) . ◮ Let R F q be the free R -module with basis indexed by the elements of F q ; write the basis element corresponding to x ∈ F q as [ x ] .
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q . ◮ T : F × q → R × , the Teichmüller character. ◮ T generates the cyclic group Hom ( F × q , R × ) . ◮ Let R F q be the free R -module with basis indexed by the elements of F q ; write the basis element corresponding to x ∈ F q as [ x ] . ◮ F × q acts on R F q , permuting the basis by field multiplication,
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q . ◮ T : F × q → R × , the Teichmüller character. ◮ T generates the cyclic group Hom ( F × q , R × ) . ◮ Let R F q be the free R -module with basis indexed by the elements of F q ; write the basis element corresponding to x ∈ F q as [ x ] . ◮ F × q acts on R F q , permuting the basis by field multiplication, ◮ R F q decomposes as the direct sum R [ 0 ] ⊕ R F × q of a trivial module with the regular module for F × q .
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q . ◮ T : F × q → R × , the Teichmüller character. ◮ T generates the cyclic group Hom ( F × q , R × ) . ◮ Let R F q be the free R -module with basis indexed by the elements of F q ; write the basis element corresponding to x ∈ F q as [ x ] . ◮ F × q acts on R F q , permuting the basis by field multiplication, ◮ R F q decomposes as the direct sum R [ 0 ] ⊕ R F × q of a trivial module with the regular module for F × q . ◮ R F × q = ⊕ q − 2 i = 0 E i , E i affording T i .
F × q -action ◮ R = Z p [ ξ q − 1 ] , pR maximal ideal of R , R / pR ∼ = F q . ◮ T : F × q → R × , the Teichmüller character. ◮ T generates the cyclic group Hom ( F × q , R × ) . ◮ Let R F q be the free R -module with basis indexed by the elements of F q ; write the basis element corresponding to x ∈ F q as [ x ] . ◮ F × q acts on R F q , permuting the basis by field multiplication, ◮ R F q decomposes as the direct sum R [ 0 ] ⊕ R F × q of a trivial module with the regular module for F × q . ◮ R F × q = ⊕ q − 2 i = 0 E i , E i affording T i . ◮ A basis element for E i is � T i ( x − 1 )[ x ] . e i = x ∈ F × q
S -action q . Note that T i = T i + k on S . ◮ Consider the action of S on R F ×
S -action q . Note that T i = T i + k on S . ◮ Consider the action of S on R F × ◮ S -isotypic components on R F × q are each 2-dimensional.
S -action q . Note that T i = T i + k on S . ◮ Consider the action of S on R F × ◮ S -isotypic components on R F × q are each 2-dimensional. ◮ { e i , e i + k } is basis of M i = E i + E i + k
S -action q . Note that T i = T i + k on S . ◮ Consider the action of S on R F × ◮ S -isotypic components on R F × q are each 2-dimensional. ◮ { e i , e i + k } is basis of M i = E i + E i + k ◮ The S -fixed subspace M 0 has basis { 1 , [ 0 ] , e k } .
S -action q . Note that T i = T i + k on S . ◮ Consider the action of S on R F × ◮ S -isotypic components on R F × q are each 2-dimensional. ◮ { e i , e i + k } is basis of M i = E i + E i + k ◮ The S -fixed subspace M 0 has basis { 1 , [ 0 ] , e k } . ◮ L is S -equivariant endomorphisms of R F q , � L ([ x ]) = k [ x ] − [ x + s ] , x ∈ F q . s ∈ S
S -action q . Note that T i = T i + k on S . ◮ Consider the action of S on R F × ◮ S -isotypic components on R F × q are each 2-dimensional. ◮ { e i , e i + k } is basis of M i = E i + E i + k ◮ The S -fixed subspace M 0 has basis { 1 , [ 0 ] , e k } . ◮ L is S -equivariant endomorphisms of R F q , � L ([ x ]) = k [ x ] − [ x + s ] , x ∈ F q . s ∈ S ◮ L maps each M i to itself.
Jacobi Sums The Jacobi sum of two nontrivial characters T a and T b is � J ( T a , T b ) = T a ( x ) T b ( 1 − x ) . x ∈ F q
Jacobi Sums The Jacobi sum of two nontrivial characters T a and T b is � J ( T a , T b ) = T a ( x ) T b ( 1 − x ) . x ∈ F q Lemma Suppose 0 ≤ i ≤ q − 2 and i � = 0 , k. Then L ( e i ) = 1 2 ( qe i − J ( T − i , T k ) e i + k )
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