A survey of Cyclotomic eigenvalue problems in algebraic - - PowerPoint PPT Presentation

A survey of Cyclotomic eigenvalue problems in algebraic combinatorics

Allen Herman University of Regina Conference on Methods of Algebraic Graph Theory Villanova University June 2 - 5, 2014

The Cyclotomic Eigenvalue Problem

Let ζn denote a primitive n-th root of unity in C for any positive integer n. We will say that λ ∈ C is cyclotomic if λ ∈ Q(ζn) for some positive integer n. The Cyclotomic Eigenvalue Problem: Let (X, S) be a finite commutative association scheme. Is it true that, for every adjacency matrix σs of a relation s ∈ S, all eigenvalues of σs are cyclotomic? [Posed by Norton at Oberwolfach 1980]

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Definition of CC and association scheme

A finite Coherent Configuration (X, S) of order n is a finite set of relations S on an n- set X whose collection of adjacency matrices is a linearly independent set B of n × n (0, 1)-matrices satisfying:

• B2 ⊂ NB, (nonnegative integer structure constants)
• ∃∆ ⊂ B (the fibres) such that

∆ = I,

• BT = B, and

B = J, (The hat notation means sum of all elements in the set.) The CC is an association scheme (aka. homogeneous CC) if ∆ = {I}, and it is a commutative association scheme if the adjacency matrices of S pairwise commute (which occurs, for example, when they are all symmetric).

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Variations of the Cyclotomic Eigenvalue Problem

• 1. Are the eigenvalues of distance-regular graphs cyclotomic?
• 2. (Cyclotomic Eigenvalue Problem) Are the eigenvalues of

commutative association schemes cyclotomic?

• 3. Do the adjacency matrices of the relations in a finite coherent

configuration have cyclotomic eigenvalues?

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Variations of the CEP for Representation Theorists

• 4. (Character Value Problem) Are the character values of finite

coherent configurations cyclotomic?

• 5. (Schur Subgroup Problem) Is the collection of division algebras

that occur in simple components of rational adjacency algebras of finite CCs the same as the collection of division algebras that occur among rational group algebras of finite groups? [asked in H.-Rahnamai-Barghi,’11].

Remark: Division algebras in QG for specific finite groups G can be now be determined using the GAP package wedderga [H. et. al., 2014]

Notes: 3 = ⇒ 4 = ⇒ 2 = ⇒ 1, and 5 = ⇒ 4.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Which regular graphs are cyclotomic?

For 1: All known distance-regular graphs are cyclotomic. Not all regular graphs are cyclotomic. These families of regular graphs are known to be cyclotomic:

• all strongly regular graphs/digraphs

[since their eigenvalues are at worst quadratic over Q],

• NEPS products or line graphs of cyclotomic regular graphs

[by their eigenvalue formuli],

• all Cayley graphs [because their eigenvalues can be written in

terms of values of characters of finite groups], and

• distance-regular graphs of diameter ≥ 34 whose association

schemes are both P-and Q-polynomial [since the eigenvalues are either rational or arise from an ordinary n-gon by (Bannai-Ito86, III.7)].

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Positive results for the Cyclotomic Eigenvalue Problem

For 4 and 5: These hold automatically for all thin schemes, and hence [by Morita equivalence], all Schurian schemes, and indeed then for all Schurian CCs (= 2-orbit CCs arising from an action of a group G on a set X); and for any configuration (X, S) for which QS is ring isomorphic to the rational adjacency algebra of a Schurian CC. For 2: All known association schemes have cyclotomic eigenvalues. These families of commutative association schemes have cyclotomic eigenvalues:

• all commutative Schurian schemes [because then 4 holds];
• fusions of commutative association schemes with cyclotomic

eigenvalues [by the Bannai-Muzychuk criterion], and

• commutative quasithin schemes [H.-Rahnamai-Barghi,08],

[Muzychuk-Ponomarenko,12].

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Remaining cases for the Cyclotomic Eigenvalue Problem

Case 1: (Schur rings of nonabelian groups) (X, S) is a fusion of a noncommutative Schurian scheme with transitive automorphism group.

Remark: Sourav Sikdar and I have used GAP to generate (almost complete) data for fusions of association schemes of order up to 30. See: http://uregina.ca/~hermana/HermanSouravMITACS2011.html

Case 2: (Schemes with intransitive automorphism groups) (X, S) is a commutative association scheme of rank ≥ 4 for which the action of Aut(S) on X is not transitive.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Integral table algebras

The adjacency algebra of a commutative association scheme is a special type of integral table algebra. An integral table algebra (A, B) is a commutative semisimple unital C-algebra with involution ′ that is the C-span of a finite distinguished basis B with these properties: B2 ⊂ NB, 1 ∈ B, B′ = B, and ∀b, c ∈ B, 1 ∈ suppB(bc) ⇐ ⇒ c = b′; PLUS it has a unique 1-dimensional representation that takes nonnegative integer values on elements of B. The splitting field L of (A, B) is the minimal extension of Q that contains the eigenvalues of elements of B when these are viewed as matrices in the regular representation of (A, B).

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

A table algebra with a noncyclotomic splitting field

Elements of integral table algebra bases need not have cyclotomic eigenvalues! Example 1: Not all members of the family H(2n, m) appearing in Blau’s classification of homogeneous commutative integral table algebras of degree 2 with a faithful element [Blau,’97] have cyclotomic eigenvalues. The first such example in this family is H(6, 2), which has rank 14. The splitting field of the characteristic polynomial of its basis element c1 has a nonabelian dihedral Galois group of order 8, so this basis element has noncyclotomic eigenvalues.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

A Krein field for commutative semisimple Q-algebras

Let QB be a commutative semisimple Q-algebra with basis B and splitting field L (= smallest field extension of Q containing eigenvalues of elements of B in the regular representation). Let G = Gal(L/Q). We know that the CEP holds for QB ⇐ ⇒ L is cyclotomic ⇐ ⇒ G is abelian. Let X be the character table of LB whose columns are indexed by elements of B and whose rows are indexed by the irreducible characters of LB . Every element of G is a row permutation of X. Let H be the set of elements G that can be realized by a column permutation of X. Let K be the subfield of L fixed by H. Theorem (Extension of Munemasa’s theorem to table algebras) H = Gal(L/K) ⊆ Z(Gal(L/Q)) = Z(G). In particular, L is cyclotomic whenever G/H = Gal(K/Q) is cyclic.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

A noncyclotomic Krein field

Example 2: Character table of Q(

3

√ 2) = Q[1, x, x2 : x3 = 2] 1 x x2 χ1 1

3

√ 2

3

√ 4 χ2 1

3

√ 2ζ3

3

√ 4ζ2

3

χ3 1

3

√ 2ζ2

3

3

√ 4ζ3 We have H = 1, L = K = Q(

3

√ 2, ζ3) is not cyclotomic, G = Gal(L/Q) ≃ S3, and the character values are not all cyclotomic.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Schemes with big Krein fields

Example 3: The only distance-regular graphs I know of for which Gal(K/Q) is not cyclic are generalized hexagons GH(q, q) when q and 3q are non-squares. The smallest of these is the incidence graph of GH(2, 2), aka. Tutte’s 12-cage, whose incidence array is {3, 2, 2, 2, 2, 2; 1, 1, 1, 1, 1, 3}. The character table of its P-polynomial scheme is:

I σ σ2 σ3 σ4 σ5 σ6 χ1 1 3 6 12 24 48 32 χ2 1 −3 6 −4 χ3 1 −3 6 −12 24 48 32 χ4 1 √ 2 −1 −3 √ 2 −4 2 √ 2 4 χ5 1 − √ 2 −1 3 √ 2 −4 −2 √ 2 4 χ6 1 √ 6 3 √ 6 −2 √ 6 −4 χ7 1 − √ 6 3 − √ 6 2 √ 6 −4

We have that L = Q( √ 2, √ 6) is cyclotomic, Gal(L/Q) ≃ C2 × C2, H = 1, and K = L.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Schemes with big Krein fields, II

Example 4: The Biggs-Smith graph on 102 vertices is the distance regular graph with incidence array {3,2,2,2,1,1,1;1,1,1,1,1,1,3}. The character table of the corresponding association scheme is:

I σ σ2 σ3 σ4 σ5 σ6 σ7 χ1 1 3 6 12 24 24 24 8 χ2 1 2 1 −2 −6 −2 4 2 χ3 1 −3 6 −6 2 χ4 1 (1 + √ 17)/2 (3 + √ 17)/2 4 −1 + √ 17 1 − √ 17 −8 1 − √ 17 χ5 1 (1 − √ 17)/2 (3 − √ 17)/2 4 −1 − √ 17 1 + √ 17 −8 1 + √ 17 χ6 1 −1 − α1 α1 − α4 α1 − α4 α2 − α4 2 − 2α1 + α2 1 + 2α2 −2 − α4 χ7 1 −1 − α4 α4 − α2 α4 − α2 α1 − α2 2 − 2α4 + α1 1 + 2α1 −2 − α2 χ8 1 −1 − α2 α2 − α1 α2 − α1 α4 − α1 2 − 2α2 + α4 1 + 2α4 −2 − α1

Here α1 = ζ9 + ζ8

9, α2 = ζ2 9 + ζ7 9, and α4 = ζ4 9 + ζ5 9.

Q(α1) = Q(ζ9)+ has degree 3 over Q. We have that L = Q( √ 17, α1) is cyclotomic, with Gal(L/Q) ≃ C6. Here H = 1, so K = L. This is the only distance-regular graph I know of that has a nonquadratic eigenvalue!

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

A small noncyclotomic regular graph

Q11 in Read and Wilson’s An Atlas of Graphs is a 4-regular graph

• n 9 vertices whose characteristic polynomial factors as

(x − 4)(x + 2)(x3 − 3x − 1)(x4 + 2x3 − 3x + 2). Q11: The roots of (x3 − 3x − 1) lie in Q(ζ9)+. For (x4 + 2x3 − 3x + 2), the Galois group of its splitting field (computed using GAP’s GaloisType command) is S4. Since this is nonabelian, the splitting field L of this graph is not cyclotomic.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Noncyclotomic regular graphs on 9 vertices

5 of the 16 connected 4-regular graphs on 9 vertices are noncyclotomic: Q11, Q13, Q14, and Q15 have Galois group ⊇ S4, and Q12 has nonabelian Galois group ≃ D4. Q12: Observation: These are the smallest noncyclotomic regular graphs. All regular graphs with 8 vertices or less have cyclotomic eigenvalues.

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

A small noncyclotomic regular graph on 10 vertices

C9 in Read and Wilson’s An Atlas of Graphs is a 3-regular graph

• n 10 vertices with characteristic polynomial

(x − 3)(x − 1)x(x + 1)(x + 2)(x5 + x4 − 7x3 − 5x2 + 10x + 4). The Galois group of the splitting field of (x5 + x4 − 7x3 − 5x2 + 10x + 4) is isomorphic to S5. C9:

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato

Noncyclotomic regular graphs on 10 vertices

9 of the 19 connected 3-regular graphs on 10 vertices are noncyclotomic: C9, C13, and C14 have Galois group ⊇ S5, C10 has Galois group ⊇ S4, and C12, C16, C18, C22, and C25 have Galois group ⊇ S3. C12:

Allen Herman University of Regina A survey of Cyclotomic eigenvalue problems in algebraic combinato