Quotients of Coxeter groups associated to signed line graphs. Robert Shwartz Ariel University ISRAEL
Let G be a connected undirected graph without loops, with n vertices and k edges. Then the line graph of G is the undirected graph L ( G ), where the following holds: • Each vertex of L ( G ) corresponds to a cer- tain edge of G ; • Two vertices of L ( G ) are connected by an edge if the corresponding edges in in G have a common endpoint. 1
For example, if the graph G is: 1 6 2 3 4 5 Then the graph L ( G ) is: 13 45 12 23 34 46 A vertex ij in L ( G ) corresponds to the edge of G which connects the vertices i and j of G . Notice that the triangle with vertices 12, 23, 13 in L ( G ) is induced from the triangle with vertices 1, 2, 3 in G , while all other cycles of L ( G ) are not induced by a cycle of G . 2
Signed line graph Let f be a function from the edges of L ( G ) to the set {− 1 , +1 } . L ( G ) f is a signed line graph for the graph G , where The edge e of L ( G ) f is signed by f ( e ). A cycle in a signed graph is called balanced if the product of the values of f along this cycle is equal to +1. A cycle in a signed graph is called non-balanced if the product of the values of f along this cycle is equal to − 1. 3
Signed Coxeter groups The canonical construction of the standard ge- ometric representation of a simply laced Cox- eter group W assoiated to the Coxeter graph Γ can be generalized for a signed Coxeter graph Γ f in the following way. Let ( W, S ) be a simply laced Coxeter system, where S = { s 1 , s 2 , ..., s n } , let Γ be its Coxeter graph, and let f be a function on edges of Γ �� �� with values ± 1, i.e., f ∈ { 1 , − 1 } when s i , s j m ij = 3. 4
Let us construct the mapping: • The generator s i is mapped to the n × n matrix ω i which differs from the identity matrix only by the i -th row; • The i -th row of the matrix ω i has − 1 at the position ( i, i ); �� �� • It has f in the position ( i, j ) when s i , s j the node s j is connected to the node s i , i.e., when m ij = 3, and it has 0 in the position ( i, j ) when the nodes s j and s i are not connected by an edge, i.e., when s j and s i commute. Thus, we defined the mapping R Γ ,f : S → GL n ( C ), R Γ ,f ( s i ) = ω i . 5
Example: Consider for example a signed Coxeter graph of − 1 1 the symmetric group S 4 : s 1 s 2 s 3 − 1 1 0 1 0 0 s 1 �→ ω 1 = 0 1 0 , s 2 �→ ω 2 = 1 − 1 − 1 0 0 1 0 0 1 1 0 0 s 3 �→ ω 3 = 0 1 0 . 0 − 1 − 1 6
The following propositions holds: The matrices ω 1 , ω 2 ,..., ω n satisfy the Coxeter relations of the group W . The mapping R Γ ,f ( s i ) = ω i can be extended to a group homomorphism W → GL n ( C ). In other words, the matrix group Ω Γ ,f = � ω 1 , ω 2 , ..., ω n � is isomorphic to some quotient, may be proper, of the simply laced Coxeter group W . The standard geometric representation is a par- ticular case of the representation R Γ ,f when the function f maps every edge to 1. 7
It is natural to inquire how many different (non- isomorphic) matrix groups Ω can we get this way from a given Coxeter graph Γ. More pre- cisely: Given an undirected graph Γ = Problem. To each of 2 | E | functions from E to ( V, E ). { 1 , − 1 } we associate the matrix group Ω Γ ,f as it is described above. How many different groups do we get this way and what can be said about the structure of these groups? 8
A partial answer to this question was given in the paper: V. Bugaenko, Y. Cherniavsky, T. Nagnibeda, R. Shwartz,”Weighted Coxeter graphs and gen- eralized geometric representations of Coxeter groups”, Discrete Applied Mathematics 192 (2015) Let Γ f = ( V, E, f ) be a signed Coxeter graph. Then the representation R Γ ,f is faithful if and only if Γ f is balanced, i.e., if and only if every cycle in the graph has an even number of − 1’s. Thus, for all functions f : E → { 1 , − 1 } such that the signed graph Γ f is balanced, the group Ω Γ ,f is isomorphic to the simply laced Coxeter group associated to the graph Γ. 9
It seems to be a difficult problem to distinguish the cases of non-faithful representations R Γ ,f . There is a partial answer to the formulated above problem: We describe the group Ω Γ ,f when Γ is a line graph L ( G ) with certain restriction. 10
The main Theorem Let Γ f be a signed graph with k vertices. As- sume that Γ = L ( G ), i.e., Γ is the line graph of a certain graph G with n vertices and k edges. Assume that every cycle of Γ f , which is not induced from a cycle of G , is not balanced. 1. If every cycle of Γ f , which is induced from a cycle of G , is balanced, then the group Ω Γ ,f is isomorphic to a certain semidirect product of Z ( n − 1)( k − n +1) with the symmetric group S n . If there exists at least one non-balanced 2. cycle in Γ f , which is induced from a cycle of G , then the group Ω Γ ,f is isomorphic to a certain semidirect product of Z n ( k − n ) with the Coxeter group D n . 11
In order to prove the Theorem we construct a certain matrix α such that α · Ω Γ ,f · α − 1 = X n,k where the group X n,k ≃ Z ( n − 1)( k − n +1) ⋊ S n , or α · Ω Γ ,f · α − 1 = Y n,k where the group Y n,k ≃ Z n ( k − n ) ⋊ D n . 12
The subgroup S n of GL n − 1 ( C ) . A matrix of S n is either a certain ( n − 1) × ( n − 1) permutation matrix, or is a matrix which has the following structure: • For a certain i ∈ { 1 , 2 , ..., n − 1 } , all the elements of the i -th row equal to − 1; • There exists j ∈ { 1 , 2 , ..., n − 1 } such that all the elements of the j -th column are zeros except the element in the position ij which is − 1; • If we delete the i -th row and the j -th col- umn we obtain a certain ( n − 2) × ( n − 2) permutation matrix. Then S n is a subgroup of GL n − 1 ( C ), and S n is isomorphic to the symmetric group S n . 13
Example: The matrices 0 1 0 1 0 0 1 0 0 0 0 1 , 0 0 1 0 1 0 1 0 0 0 1 0 − 1 − 1 − 1 generate the subgroup S 4 of GL 3 ( C ) which is isomorphic to S 4 . 14
The subgroup D n of GL n − 1 ( C ) . Let D n be a subset of GL n ( C ), which consists of all matrices having the following structure: • A matrix of D n has the unique non-zero entry in each row and each column, which is 1 or − 1; • The number of − 1’s is even. Then D n is a subgroup of GL n ( C ), and D n is isomorphic to the Coxeter group D n . 15
The subgroup X n,k of GL k ( C ) . Let k and n be natural numbers such that k � n − 1. Let X n,k be the following subset of GL k ( C ): �� �� 0 ( n − 1) × ( k − n +1) P X n,k = Q I k − n +1 such that: P ∈ S n , Q ∈ Z ( k − n +1) × ( n − 1) Then: • X n,k is a subgroup of GL k ( C ); • X n,k is isomorphic to a semidirect product of Z ( n − 1)( k − n +1) with the symmetric group S n . 16
The subgroup Y n,k of GL k ( C ) . Let k and n be natural numbers such that k � n . Let Y n,k be the following subset of GL k ( C ): �� �� 0 n × ( k − n ) P Y n,k = Q I k − n such that: P ∈ D n , Q ∈ Z ( k − n ) × n Then: • Y n,k is a subgroup of GL k ( C ); • Y n,k is isomorphic to a semidirect product of Z n ( k − n ) with the Coxeter group D n . 17
The structure of the conjugating matrix α α = A (Γ f ) · D (Γ f ), where The matrices A (Γ f ) and D (Γ f ) depends on the graph Γ f , which is the signed line graph of G . Now, we describe the structures of these ma- trices 18
Let T ( G ) be a spanning tree of the graph G . Let C 1 , C 2 , ... , C k − n +1 be a certain basis of the binary cycle space of G . Let C ′ i (Γ) be the cycle of Γ f which is induced from the cycle C i ( G ). The vertices of C ′ i (Γ) correspond to the edges of C i ( G ) in G . Consider two cases: • Case 1 - Every cycle C ′ i (Γ) is a balanced cycle in Γ f ; • Case 2 - There exists at least one non- balanced cycle C ′ i (Γ) in Γ f . In this case, without loss of generality, assume that C ′ 1 (Γ) is a non-balanced cycle in Γ f . 19
Case 1: . For 1 ≤ i ≤ n , denote by v i the vertices of G . Assign the numbers 1, 2, ... , n − 1 to the n − 1 edges of T ( G ) in such a way that a vertex v i is an endpoint of the edge e i . Notice that such an indexing of edges of T ( G ) is unique for a fixed indexing of vertices of G . Let us index the remained k − n + 1 edges of G in the following way: e n should belong to the cycle C 1 ( G ), e n +1 should belong to C 2 ( G ), ... , e k should belong to C k − n +1 ( G ). Let ℓ i be the vertex of Γ f which corresponds to the edge e i of G . 20
The matrix A (Γ f ) Let A (Γ f ) be a k × k matrix defined as follows: • A (Γ f ) i,i = 1 for every 1 ≤ i ≤ k ; � � • A (Γ f ) i,j = − f when the edges e i and ℓ i , ℓ j e j in G have a common endpoint v i , and 1 ≤ i ≤ n − 1; • A (Γ f ) i,j = 0 otherwise. The matrix A (Γ f ) is an invertible matrix with determinant 1. 21
The matrix D (Γ f ) Let D (Γ f ) be a k × k diagonal matrix defined as follows: • For n ≤ i ≤ k , D (Γ f ) i,i = 1; • For 1 ≤ i ≤ n − 1, D (Γ f ) i,i = ( − 1) d i , where ( d 1 , d 2 , . . . , d n − 1 ) is a solution for the fol- lowing system of the linear equations over F 2 : 22
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