Tatra Schemes and Their Mergings Sven Reichard TU Dresden Plze n, - - PowerPoint PPT Presentation

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

1

Motivation

2

Preliminaries

3

Tatra schemes

4

Non-commutative schemes of rank 6

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Contents

1

Motivation

2

Preliminaries

3

Tatra schemes

4

Non-commutative schemes of rank 6

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

• f 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

1

Motivation

2

Preliminaries

3

Tatra schemes

4

Non-commutative schemes of rank 6

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 ≤ Cn×n with a special basis A1, · · · , Ad, satisfying the following axioms: Each Ai is a (0, 1)-matrix.

• i Ai = Jn

In ∈ W W is closed under transposition. It follows, that for each i there is an i∗ with AT

i = Ai∗

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 R1, · · · , Rd be binary relations on a set Ω, with the following properties: The relations form a partition of Ω2. Each relation is either reflexive or anti-reflexive. For each i there is an i∗ with R−1

i

= Ri∗. There are numbers pk

ij such for any (x, y) ∈ Rk,

|Ri(x) ∪ R−1

j

(y)| = pk

ij.

Then the relations Ri 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 idΩ 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

1

Motivation

2

Preliminaries

3

Tatra schemes

4

Non-commutative schemes of rank 6

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: f (u, v) =

• u1

v1 u2 v2

• = u1v2 − u2v1.

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) ∈ Rx iff v = xu. For y ∈ F ∗, (u, v) ∈ Sy iff f (u, v) = y.

Then each pair (u, v) not containing the origin is in exactly

• ne of the relations Rx or Sy.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

In terms of matrices, let Ai and Bj be the adjacency matrices of Si and Rj, respectively. Then we have for any i, j ∈ Zr: AiAj = Aij AiBj = Bj/i BiAj = Bij BiBj = qAj/i +

j Bj.

Moreover, AT

i = Ai−1

BT

i

= B−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 Si and Ri 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 Bx) and the “scaling factor” (for Ax) now lie in the quotient group F ∗/K. We can check that everything is well-defined. Again we look at the relations Sx with matrices Ax, and Rx with matrices Bx.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

In this case, we get the following: AxAy = Axy AxBy = By/x BxAy = Bxy BxBy = qAy/x + m

j Bj.

We also have AT

x = Ax−1.

If furthermore −1 ∈ K, then Bx 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 2r and order (q2 − 1)/m = (q + 1)r. The relations given by Ax are thin. They form a closed subset, in fact the only non-trivial one. The relations given by Bx have valency q. They are distance regular covers of Kq+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 Ax, 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

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

More History

The distance-regular graphs were first described by

• R. Mathon.

The group related to M(q, r) appeared in work by Nevo and Thurston. Klin observed that it had a lot of non-schurian mergings. Kalmanovich did a first investigation of those mergings. One particular non-schurian merging (of rank 4) was explained during a bus ride from Bansk´ a Bystrica to Nov´ y Smokovec (through the Low Tatras, to the High Tatras). This led to the suggestion of the name “Tatra schemes”.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Conjecture The automorphism group of Mathon’s distance-regular graph is equal to Aut(M(q, r). In particular, the graph is non-schurian.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Mergings from S-rings

Now we look at the group F ∗/K ∼ = Zr. Let A = T0, . . . , Ts be an S-ring over Zr. For k = 0, . . . , s, set Ck =

• x∈Tk

Ax Dk =

• x∈Tk

Bx

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Lemma Let A = T0, . . . , Td−1 be an S-ring over H with structure constants pk

ij.

Let Ci and Di, 0 ≤ i < d be defined as above. Then for any i, j: Ci · Cj =

d−1

• k=0

pk

ijCk

Di · Cj =

d−1

• k=0

pk

ijDk

Ci · Dj =

d−1

• k=0

pk

ji∗Dk

Di · Dj =

d−1

• k=0

pk

i∗j

• Ck + m|Tk|

d−1

• h=0

Dh

• Sven Reichard

Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

The idea of the proof: x → Ax is a group monomorphism. Extend to a ring homomorphism C[H] → Cn×n. Apply this to a product Ti · Tj in A to get Ci · Cj. Use the fact that Di can be written as a product Ci · B1.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Lemma C T

i

= Ci∗. DT

i

= Di. Corollary The matrices Ci and Dj generate a homogeneous coherent algebra

• f rank 2d.

It is commutative iff the S-ring A is symmetric.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

This construction leads to a lot association schemes with various parameter sets. If the conjecture about Aut(drg) is true, they are all non-schurian. However, not all mergings of Tatra schemes arise in this way.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Contents

1

Motivation

2

Preliminaries

3

Tatra schemes

4

Non-commutative schemes of rank 6

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Paley tournament

Let p ≡ 3 mod 4 be a prime. Consider the group Zp. Define the following sets: T0 = {0} T1 = {x2|x = 0} T2 = −T1. Then T0, T1, T2 form an antisymmetric S-ring of rank 3. The directed graphs corresponding to T1 and T2 are doubly regular tournaments, named after Paley.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Construction

We now have a construction for non-commutative rank 6 schemes: Let r ≡ 3 mod 4 be a prime. Let A be the S-ring corresponding to a Paley tournament over Zr. Let q = mr + 1 be a prime power. Take the scheme M(q, r), and let W be its merging defined by A. Then W is rank 6, non-commutative, of order r(q + 1). If q is a power of two, and m = 1, then r = q − 1 is a Mersenne prime, and we reproduce the construction of Drabkin-French.

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Outlook

Prove the conjecture. Characterize all mergings of M(q, r) in terms of the group ring C[K]. Can the scheme M(q, r) be replaced by other schemes?

Sven Reichard Tatra Schemes and Their Mergings

Motivation Preliminaries Tatra schemes Non-commutative schemes of rank 6

Dˇ ekuji v´ am za pozornost.

Sven Reichard Tatra Schemes and Their Mergings