Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Tatra Schemes and Their Mergings Sven Reichard TU Dresden Plzeˇ n, 2016-10-05 Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Contents Motivation 1 Preliminaries 2 Tatra schemes 3 Non-commutative schemes of rank 6 4 Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Contents Motivation 1 Preliminaries 2 Tatra schemes 3 Non-commutative schemes of rank 6 4 Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Acknowledgments Misha Klin Eran Nevo, Danny Kalmanovich Roman Nedela Misha Muzychuk, Ilya Ponamarenko, Paul-Hermann Zieschang Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Motivation Theorem All groups of order n ≤ 5 are abelian. Theorem Up to isomorphism, there are two groups of order 6, one of them is not abelian. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Association schemes are generalizations of groups. Can we generalize the previous results? Theorem All association schemes of rank d ≤ 5 are commutative. Theorem There are infinitely many non-isomorphic association schemes of rank 6. Some of them are non-commutative. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Can we understand the structure of non-commutative schemes of rank 6? Symmetric schemes are commutative, so our scheme should contain at least one antisymmetric pair of relations. We can distinguish several distinct cases: primitive schemes; imprimitive schemes with symmetric closed set; imprimitive schemes with non-symmetric closed set. We will concentrate on the last case. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 History Hanaki and Zieschang started a theoretical analysis of such schemes. Three classes of examples were described which are well understood. (Coxeter schemes, semidirect products, schemes with thin normal closed sets) Drabkin and French gave a construction of such schemes on n = p ( p + 2) points, where p > 3 is a Mersenne prime. We give a generalization of their construction to a wider class of parameters ( n = r ( q + 1), q prime power, r | ( q − 1) odd prime). In the process we obtain a lot of non-isomorphic non-schurian imprimitive schemes of arbitrary even rank. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Contents Motivation 1 Preliminaries 2 Tatra schemes 3 Non-commutative schemes of rank 6 4 Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Coherent algebras A coherent algebra is a matrix algebra W ≤ C n × n with a special basis A 1 , · · · , A d , satisfying the following axioms: Each A i is a (0 , 1)-matrix. � i A i = J n I n ∈ W W is closed under transposition. It follows, that for each i there is an i ∗ with A T i = A i ∗ Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Coherent configurations We can consider each (0 , 1)-matrix as the incidence matrix of a binary relation on an n -element set. Hence we can give an equivalent definition in terms of relations. Let R 1 , · · · , R d be binary relations on a set Ω, with the following properties: The relations form a partition of Ω 2 . Each relation is either reflexive or anti-reflexive. For each i there is an i ∗ with R − 1 = R i ∗ . i There are numbers p k ij such for any ( x , y ) ∈ R k , | R i ( x ) ∪ R − 1 ( y ) | = p k ij . j Then the relations R i form a coherent configurations on Ω. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 A set of relations forms a coherent configuration iff the incidence matrices are the standard basis of a coherent algebra. Hence we may freely mix both languages. A coherent algebra is homogeneous if the identity matrix is in the standard basis. A coherent configuration is homogeneous if id Ω is one of the relations. Homogeneous configurations are also called association schemes. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 S-rings Assume that we have a Cayley association scheme: invariant under a regular group H . We can identify points with group elements, and relations with subsets of H corresponding to the neighbors of the identity. This leads to the notion of an S-ring, which can be considered as a particular subring of the group ring C [ H ]. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Contents Motivation 1 Preliminaries 2 Tatra schemes 3 Non-commutative schemes of rank 6 4 Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Basic construction Let q be a prime power. Let F = GF ( q ) be a field with q elements. Let α be a primitive element of F . Let V = F 2 be a 2-dimensional vector space over F . Let f : V × V → F be given by the determinant: � � u 1 v 1 � � f ( u , v ) = � � u 2 v 2 � � = u 1 v 2 − u 2 v 1 . Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 In what follows let u , v ∈ V \ { 0 } . If f ( u , v ) = 0, then the vectors are linearly dependent, hence v = xu for some x ∈ F ∗ . This allows us to define the following binary relations on V : For x ∈ F ∗ , ( u , v ) ∈ R x iff v = xu . For y ∈ F ∗ , ( u , v ) ∈ S y iff f ( u , v ) = y . Then each pair ( u , v ) not containing the origin is in exactly one of the relations R x or S y . Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 In terms of matrices, let A i and B j be the adjacency matrices of S i and R j , respectively. Then we have for any i , j ∈ Z r : A i A j = A ij A i B j = B j / i B i A j = B ij B i B j = qA j / i + � j B j . Moreover, A T i = A i − 1 B T = B − i . i Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 This gives us the following: Theorem The relations S i and R i form an association scheme of rank 2( q − i ) on the set V \ { 0 } . This scheme is schurian. Its automorphism group is SL (2 , q ). Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Now we will construct a quotient structure. Let K be a subgroup of the multiplicative group of F . Say, | K | = m , where q = mr + 1. Consider vectors “modulo K ”: Ω = ( V \ { 0 } ) / K . If K = 1 we don’t get anything new. If K = F ∗ we get the projective line. Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 The relations considered before can be transferred to these “quasi-projective points”. The values of the determinant (for B x ) and the “scaling factor” (for A x ) now lie in the quotient group F ∗ / K . We can check that everything is well-defined. Again we look at the relations S x with matrices A x , and R x with matrices B x . Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 In this case, we get the following: A x A y = A xy A x B y = B y / x B x A y = B xy B x B y = qA y / x + m � j B j . We also have A T x = A x − 1 . If furthermore − 1 ∈ K , then B x is symmetrical, and we get an association scheme. Definition The association scheme described above will be denoted by M ( q , r ). Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 Some properties M ( q , r ) is a schurian association scheme of rank 2 r and order ( q 2 − 1) / m = ( q + 1) r . The relations given by A x are thin. They form a closed subset, in fact the only non-trivial one. The relations given by B x have valency q . They are distance regular covers of K q +1 . Sven Reichard Tatra Schemes and Their Mergings
Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6 We have a closed set formed by the A x , in other words a spread. The other relations are antipodal distance regular graphs of diameter 3. This looks similar to a Siamese association scheme. However, here, joining a drg and the spread does not give us a strongly regular graph. Sven Reichard Tatra Schemes and Their Mergings
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