Automorphism groups of Schur rings over cyclic groups of prime-power - - PowerPoint PPT Presentation

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Automorphism groups of Schur rings over cyclic groups of prime-power order

Reinhard Pöschel joint work with Mikhail Klin

Institut für Algebra Technische Universität Dresden Germany Ben-Gurion University of the Negev Israel

Conference Modern Trends in Algebraic Graph Theory Villanova University June 2-5, 2014

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (1/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Some remarks about and for My coauthor Misha Klin

(unknown to him up to now)

Reinhard Pöschel

Institut für Algebra Technische Universität Dresden Germany

Conference Modern Trends in Algebraic Graph Theory Villanova University June 4, 2014

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (2/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Our roots: Issai Schur and Lev Arkad’evič Kalužnin

Kiev 1971/72: Mikhail Klin — Lev Arkad’evič Kalužnin ▲❡✈ ❆r❦❛❞⑦❡✈✐q ❑❛❧✉✙♥✐♥ (31.1.1914 – 6.12.1990)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Our roots: Issai Schur and Lev Arkad’evič Kalužnin

Issai Schur (10.1.1875 – 10.1.1941)

• n the tombstone:

Yeshiyahu Schur Professor of Mathematics

4 Shevet 5635 (10.1.1875) 12 Tevet 5701 (11.1.1941 [starting 10.1., 6 p.m.]) Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Our roots: Issai Schur and Lev Arkad’evič Kalužnin

Mikhail Klin – Issai Schur – Andy Woldar

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Our roots: Issai Schur and Lev Arkad’evič Kalužnin

Kiev 1971/72: Mikhail Klin — Lev Arkad’evič Kalužnin — Reinhard Pöschel ▲❡✈ ❆r❦❛❞⑦❡✈✐q ❑❛❧✉✙♥✐♥ (31.1.1914 – 6.12.1990) 1978

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (3/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Outline

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (4/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Outline

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (5/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings

S-ring (S=Schur): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup.

• I. Schur used S-rings (in form of transitivity modules

) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by

• W. Burnside (in 1900) for a special case).
• I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen.

S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group)

• H. Wielandt 1964, Finite permutation groups.

and in the papers

• R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring

von p–Gruppen.

• M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for

cyclic graphs and the method of Schur rings.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings

S-ring (S=Schur): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup.

• I. Schur used S-rings (in form of transitivity modules

) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by

• W. Burnside (in 1900) for a special case).
• I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen.

S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group)

• H. Wielandt 1964, Finite permutation groups.

and in the papers

• R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring

von p–Gruppen.

• M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for

cyclic graphs and the method of Schur rings.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings

S-ring (S=Schur): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup.

• I. Schur used S-rings (in form of transitivity modules

) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by

• W. Burnside (in 1900) for a special case).
• I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen.

S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group)

• H. Wielandt 1964, Finite permutation groups.

and in the papers

• R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring

von p–Gruppen.

• M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for

cyclic graphs and the method of Schur rings.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings

S-ring (S=Schur): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup.

• I. Schur used S-rings (in form of transitivity modules

) to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by

• W. Burnside (in 1900) for a special case).
• I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen.

S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group)

• H. Wielandt 1964, Finite permutation groups.

and in the papers

• R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring

von p–Gruppen.

• M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for

cyclic graphs and the method of Schur rings.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings

S-ring (S=Schur): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup.

• I. Schur used S-rings (in form of transitivity modules (“Schurian S-rings”))

to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by

• W. Burnside (in 1900) for a special case).
• I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen.

S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group)

• H. Wielandt 1964, Finite permutation groups.

and in the papers

• R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring

von p–Gruppen.

• M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for

cyclic graphs and the method of Schur rings.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings

S-ring (S=Schur): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup.

• I. Schur used S-rings (in form of transitivity modules (“Schurian S-rings”))

to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by

• W. Burnside (in 1900) for a special case).
• I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen.

S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group)

• H. Wielandt 1964, Finite permutation groups.

and in the papers

• R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring

von p–Gruppen.

• M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for

cyclic graphs and the method of Schur rings.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings

S-ring (S=Schur): combinatorial approach to the structure of the 2-orbits of a transitive permutation group containing a regular subgroup.

• I. Schur used S-rings (in form of transitivity modules (“Schurian S-rings”))

to prove that each overgroup of a regular cyclic group of composite order is either primitive or 2-transitive (without character theory used by

• W. Burnside (in 1900) for a special case).
• I. Schur 1933, Zur Theorie der einfach transitiven Permutationsgruppen.

S-ring theory was further developed by H. Wielandt (e.g., disproving an assumption of Schur, that each S-ring is the transitivity module of a permutation group)

• H. Wielandt 1964, Finite permutation groups.

and in the papers

• R. Pöschel 1974, Untersuchungen von S-Ringen, insbesondere im Gruppenring

von p–Gruppen.

• M. H. Klin, R. Pöschel 1981, The König problem, the isomorphism problem for

cyclic graphs and the method of Schur rings.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (6/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Definition)

S-ring S over a group H (here H = Zpm): subring of the group ring Z(H); +, ∗ (formal sums

h∈H αhh, αh ∈ Z) with the following

properties: Example: H = Zn = {0, 1, 2, 3, 4, 5, 6, 7, 8}, n = pm = 32 = 9: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z Z-module basis of simple quantities T :=

h∈T h for T ⊆ Zn,

which form a partition(!) of Zn. Let T(x) denote the subset in the basis which contains x ∈ Zn. Then it is required: T(0) = {0} and kT(pi) = T(kpi) for k ∈ Z∗

n

(i ∈ {0, 1, . . . , m − 1}), e.g. 2T(1) = 2{1, 8} = {2, 7} = T(2). subring property ensures the existence of structure constants pk

ij for

S-ring S = T0, T1, . . . , Tr−1 such that Ti ∗ Tj = r−1

k=0 pk ijTk,

e.g., T(1) ∗ T(2) = 1, 8 ∗ 2, 7 = 3, 8, 1, 6 = 1, 8 + 3, 6 = T(1) + T(3).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (7/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Definition)

S-ring S over a group H (here H = Zpm): subring of the group ring Z(H); +, ∗ (formal sums

h∈H αhh, αh ∈ Z) with the following

properties: Example: H = Zn = {0, 1, 2, 3, 4, 5, 6, 7, 8}, n = pm = 32 = 9: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z Z-module basis of simple quantities T :=

h∈T h for T ⊆ Zn,

which form a partition(!) of Zn. Let T(x) denote the subset in the basis which contains x ∈ Zn. Then it is required: T(0) = {0} and kT(pi) = T(kpi) for k ∈ Z∗

n

(i ∈ {0, 1, . . . , m − 1}), e.g. 2T(1) = 2{1, 8} = {2, 7} = T(2). subring property ensures the existence of structure constants pk

ij for

S-ring S = T0, T1, . . . , Tr−1 such that Ti ∗ Tj = r−1

k=0 pk ijTk,

e.g., T(1) ∗ T(2) = 1, 8 ∗ 2, 7 = 3, 8, 1, 6 = 1, 8 + 3, 6 = T(1) + T(3).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (7/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Definition)

S-ring S over a group H (here H = Zpm): subring of the group ring Z(H); +, ∗ (formal sums

h∈H αhh, αh ∈ Z) with the following

properties: Example: H = Zn = {0, 1, 2, 3, 4, 5, 6, 7, 8}, n = pm = 32 = 9: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z Z-module basis of simple quantities T :=

h∈T h for T ⊆ Zn,

which form a partition(!) of Zn. Let T(x) denote the subset in the basis which contains x ∈ Zn. Then it is required: T(0) = {0} and kT(pi) = T(kpi) for k ∈ Z∗

n

(i ∈ {0, 1, . . . , m − 1}), e.g. 2T(1) = 2{1, 8} = {2, 7} = T(2). subring property ensures the existence of structure constants pk

ij for

S-ring S = T0, T1, . . . , Tr−1 such that Ti ∗ Tj = r−1

k=0 pk ijTk,

e.g., T(1) ∗ T(2) = 1, 8 ∗ 2, 7 = 3, 8, 1, 6 = 1, 8 + 3, 6 = T(1) + T(3).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (7/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Definition)

S-ring S over a group H (here H = Zpm): subring of the group ring Z(H); +, ∗ (formal sums

h∈H αhh, αh ∈ Z) with the following

properties: Example: H = Zn = {0, 1, 2, 3, 4, 5, 6, 7, 8}, n = pm = 32 = 9: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z Z-module basis of simple quantities T :=

h∈T h for T ⊆ Zn,

which form a partition(!) of Zn. Let T(x) denote the subset in the basis which contains x ∈ Zn. Then it is required: T(0) = {0} and kT(pi) = T(kpi) for k ∈ Z∗

n

(i ∈ {0, 1, . . . , m − 1}), e.g. 2T(1) = 2{1, 8} = {2, 7} = T(2). subring property ensures the existence of structure constants pk

ij for

S-ring S = T0, T1, . . . , Tr−1 such that Ti ∗ Tj = r−1

k=0 pk ijTk,

e.g., T(1) ∗ T(2) = 1, 8 ∗ 2, 7 = 3, 8, 1, 6 = 1, 8 + 3, 6 = T(1) + T(3).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (7/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Definition)

S-ring S over a group H (here H = Zpm): subring of the group ring Z(H); +, ∗ (formal sums

h∈H αhh, αh ∈ Z) with the following

properties: Example: H = Zn = {0, 1, 2, 3, 4, 5, 6, 7, 8}, n = pm = 32 = 9: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z Z-module basis of simple quantities T :=

h∈T h for T ⊆ Zn,

which form a partition(!) of Zn. Let T(x) denote the subset in the basis which contains x ∈ Zn. Then it is required: T(0) = {0} and kT(pi) = T(kpi) for k ∈ Z∗

n

(i ∈ {0, 1, . . . , m − 1}), e.g. 2T(1) = 2{1, 8} = {2, 7} = T(2). subring property ensures the existence of structure constants pk

ij for

S-ring S = T0, T1, . . . , Tr−1 such that Ti ∗ Tj = r−1

k=0 pk ijTk,

e.g., T(1) ∗ T(2) = 1, 8 ∗ 2, 7 = 3, 8, 1, 6 = 1, 8 + 3, 6 = T(1) + T(3).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (7/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Definition)

S-ring S over a group H (here H = Zpm): subring of the group ring Z(H); +, ∗ (formal sums

h∈H αhh, αh ∈ Z) with the following

properties: Example: H = Zn = {0, 1, 2, 3, 4, 5, 6, 7, 8}, n = pm = 32 = 9: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z Z-module basis of simple quantities T :=

h∈T h for T ⊆ Zn,

which form a partition(!) of Zn. Let T(x) denote the subset in the basis which contains x ∈ Zn. Then it is required: T(0) = {0} and kT(pi) = T(kpi) for k ∈ Z∗

n

(i ∈ {0, 1, . . . , m − 1}), e.g. 2T(1) = 2{1, 8} = {2, 7} = T(2). subring property ensures the existence of structure constants pk

ij for

S-ring S = T0, T1, . . . , Tr−1 such that Ti ∗ Tj = r−1

k=0 pk ijTk,

e.g., T(1) ∗ T(2) = 1, 8 ∗ 2, 7 = 3, 8, 1, 6 = 1, 8 + 3, 6 = T(1) + T(3).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (7/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T Example

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T Example

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T Example

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T Example

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T

Γ(Z9, {1, 8}) 1 2 4 3 5 6 7 8 Γ(Z9, {2, 7}) Γ(Z9, {4, 5}) Γ(Z9, {3, 6})

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T

Γ(Z9, {1, 8}) 1 2 4 3 5 6 7 8 Γ(Z9, {2, 7}) Γ(Z9, {4, 5}) Γ(Z9, {3, 6})

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T

Γ(Z9, {1, 8}) 1 2 4 3 5 6 7 8 Γ(Z9, {2, 7}) Γ(Z9, {4, 5}) Γ(Z9, {3, 6})

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T

Γ(Z9, {1, 8}) 1 2 4 3 5 6 7 8 Γ(Z9, {2, 7}) Γ(Z9, {4, 5}) Γ(Z9, {3, 6})

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Connection to algebraic graph theory

important observation: S-rings colored circulant graphs

(Cayley graphs over Zn, full cycle (0 1 . . . n − 1) ∈ Sym(n) is an automorphism)

as well as to coherent configurations, centralizer algebras (V-rings) and other notions ... Namely: To basis set T ∈ S corresponds the (in general directed) graph Γ(Zn, T) := (Zn, E) with (x, y) ∈ E : ⇐ ⇒ y − x ∈ T

Γ(Z9, {1, 8}) 1 2 4 3 5 6 7 8 Γ(Z9, {2, 7}) Γ(Z9, {4, 5}) Γ(Z9, {3, 6})

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (8/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Examples)

H = Zn, consider G ≤ Sym(Zn) with full cycle (0 1 2 . . . n − 1) ∈ G

(i.e., G contains Zn as regular subgroup)

Submodule of Z(Zn) generated by the 1-orbits of the stabilizer G0 ({T0, . . . , Tr−1} := 1-Orb(G0)) S(G, Zn) := T0, . . . , Tr−1Z is called the transitivity module of (G, Zn). Theorem (I. Schur 1933): S(G, Zn) is an S-ring. (schurian S-ring)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (9/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Examples)

H = Zn, consider G ≤ Sym(Zn) with full cycle (0 1 2 . . . n − 1) ∈ G

(i.e., G contains Zn as regular subgroup)

Submodule of Z(Zn) generated by the 1-orbits of the stabilizer G0 ({T0, . . . , Tr−1} := 1-Orb(G0)) S(G, Zn) := T0, . . . , Tr−1Z is called the transitivity module of (G, Zn). Theorem (I. Schur 1933): S(G, Zn) is an S-ring. (schurian S-ring)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (9/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Examples)

H = Zn, consider G ≤ Sym(Zn) with full cycle (0 1 2 . . . n − 1) ∈ G

(i.e., G contains Zn as regular subgroup)

Submodule of Z(Zn) generated by the 1-orbits of the stabilizer G0 ({T0, . . . , Tr−1} := 1-Orb(G0)) S(G, Zn) := T0, . . . , Tr−1Z is called the transitivity module of (G, Zn). Theorem (I. Schur 1933): S(G, Zn) is an S-ring. (schurian S-ring)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (9/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (Examples)

H = Zn, consider G ≤ Sym(Zn) with full cycle (0 1 2 . . . n − 1) ∈ G

(i.e., G contains Zn as regular subgroup)

Submodule of Z(Zn) generated by the 1-orbits of the stabilizer G0 ({T0, . . . , Tr−1} := 1-Orb(G0)) S(G, Zn) := T0, . . . , Tr−1Z is called the transitivity module of (G, Zn). Theorem (I. Schur 1933): S(G, Zn) is an S-ring. (schurian S-ring)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (9/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (special examples)

n := 9 = 32 Let (G, Zn) be the subgroup of the affine group consisting of all permutations of the form µa,b : x → ax + b (all counting modulo n), (a ∈ Z∗

n, x, b ∈ Zn)

with a ∈ {1, 8} ≤ Z∗

n (note 82 = 1)

The transitivity module S(G, Z9) has the simple quantities of the previous example: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z n := 27 = 33 The following simple quantities are a basis of an S-ring: M2

3(1, 0) = 0 , 1, 8, 10, 17, 19, 26 , 2, 7, 11, 16, 20, 25 ,

4, 5, 13, 14, 22, 23 , 3, 24 , 6, 21 , 12, 15 , 9, 18 M2

3(1, 0) is also the transitivity module of an affine subgroup (here

a ∈ {1, −1} + p2 = {1, 8, 10, 17, 19, 26} ≤ Z∗

p3)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (10/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (special examples)

n := 9 = 32 Let (G, Zn) be the subgroup of the affine group consisting of all permutations of the form µa,b : x → ax + b (all counting modulo n), (a ∈ Z∗

n, x, b ∈ Zn)

with a ∈ {1, 8} ≤ Z∗

n (note 82 = 1)

The transitivity module S(G, Z9) has the simple quantities of the previous example: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z n := 27 = 33 The following simple quantities are a basis of an S-ring: M2

3(1, 0) = 0 , 1, 8, 10, 17, 19, 26 , 2, 7, 11, 16, 20, 25 ,

4, 5, 13, 14, 22, 23 , 3, 24 , 6, 21 , 12, 15 , 9, 18 M2

3(1, 0) is also the transitivity module of an affine subgroup (here

a ∈ {1, −1} + p2 = {1, 8, 10, 17, 19, 26} ≤ Z∗

p3)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (10/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (special examples)

n := 9 = 32 Let (G, Zn) be the subgroup of the affine group consisting of all permutations of the form µa,b : x → ax + b (all counting modulo n), (a ∈ Z∗

n, x, b ∈ Zn)

with a ∈ {1, 8} ≤ Z∗

n (note 82 = 1)

The transitivity module S(G, Z9) has the simple quantities of the previous example: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z n := 27 = 33 The following simple quantities are a basis of an S-ring: M2

3(1, 0) = 0 , 1, 8, 10, 17, 19, 26 , 2, 7, 11, 16, 20, 25 ,

4, 5, 13, 14, 22, 23 , 3, 24 , 6, 21 , 12, 15 , 9, 18 M2

3(1, 0) is also the transitivity module of an affine subgroup (here

a ∈ {1, −1} + p2 = {1, 8, 10, 17, 19, 26} ≤ Z∗

p3)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (10/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

S-rings (special examples)

n := 9 = 32 Let (G, Zn) be the subgroup of the affine group consisting of all permutations of the form µa,b : x → ax + b (all counting modulo n), (a ∈ Z∗

n, x, b ∈ Zn)

with a ∈ {1, 8} ≤ Z∗

n (note 82 = 1)

The transitivity module S(G, Z9) has the simple quantities of the previous example: S = 0, 1, 8, 2, 7, 4, 5, 3, 6Z n := 27 = 33 The following simple quantities are a basis of an S-ring: M2

3(1, 0) = 0 , 1, 8, 10, 17, 19, 26 , 2, 7, 11, 16, 20, 25 ,

4, 5, 13, 14, 22, 23 , 3, 24 , 6, 21 , 12, 15 , 9, 18 M2

3(1, 0) is also the transitivity module of an affine subgroup (here

a ∈ {1, −1} + p2 = {1, 8, 10, 17, 19, 26} ≤ Z∗

p3)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (10/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The structure of S-rings over Zpm

case n = pm, p odd prime: R. Pöschel 1974, Leung/Man, Muzychuk case n = 2m: Klin, Golfand, Najmark, Pöschel, Muzychuk, Leung/Man

(1978 – 1993)

Decomposition of S-rings (a “wreath product construction”) leads to decomposable and indecomposable S-rings. Structure Theorem [Klin/Pö ] There is a 1-1-correspondence between indecomposable S-rings S over Zpm and pairs (λ, f ), where λ is an atomic sequence of exponent m and f is an divisor of p−1. The S-ring corresponding to (λ, f ) is the so-called atomic Z-submodule Mf

m(λ).

What this is good for? From (λ, f ) one easily and directly can compute the automorphism group of the corresponding S-ring. It provides good isomorphism criteria for circulant graphs (and algorithms for isomorphism testing).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (11/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The structure of S-rings over Zpm

case n = pm, p odd prime: R. Pöschel 1974, Leung/Man, Muzychuk case n = 2m: Klin, Golfand, Najmark, Pöschel, Muzychuk, Leung/Man

(1978 – 1993)

Decomposition of S-rings (a “wreath product construction”) leads to decomposable and indecomposable S-rings. Structure Theorem [Klin/Pö ] There is a 1-1-correspondence between indecomposable S-rings S over Zpm and pairs (λ, f ), where λ is an atomic sequence of exponent m and f is an divisor of p−1. The S-ring corresponding to (λ, f ) is the so-called atomic Z-submodule Mf

m(λ).

What this is good for? From (λ, f ) one easily and directly can compute the automorphism group of the corresponding S-ring. It provides good isomorphism criteria for circulant graphs (and algorithms for isomorphism testing).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (11/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The structure of S-rings over Zpm

case n = pm, p odd prime: R. Pöschel 1974, Leung/Man, Muzychuk case n = 2m: Klin, Golfand, Najmark, Pöschel, Muzychuk, Leung/Man

(1978 – 1993)

Decomposition of S-rings (a “wreath product construction”) leads to decomposable and indecomposable S-rings. Structure Theorem [Klin/Pö ] There is a 1-1-correspondence between indecomposable S-rings S over Zpm and pairs (λ, f ), where λ is an atomic sequence of exponent m and f is an divisor of p−1. The S-ring corresponding to (λ, f ) is the so-called atomic Z-submodule Mf

m(λ).

What this is good for? From (λ, f ) one easily and directly can compute the automorphism group of the corresponding S-ring. It provides good isomorphism criteria for circulant graphs (and algorithms for isomorphism testing).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (11/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The structure of S-rings over Zpm

case n = pm, p odd prime: R. Pöschel 1974, Leung/Man, Muzychuk case n = 2m: Klin, Golfand, Najmark, Pöschel, Muzychuk, Leung/Man

(1978 – 1993)

Decomposition of S-rings (a “wreath product construction”) leads to decomposable and indecomposable S-rings. Structure Theorem [Klin/Pö ] There is a 1-1-correspondence between indecomposable S-rings S over Zpm and pairs (λ, f ), where λ is an atomic sequence of exponent m and f is an divisor of p−1. The S-ring corresponding to (λ, f ) is the so-called atomic Z-submodule Mf

m(λ).

What this is good for? From (λ, f ) one easily and directly can compute the automorphism group of the corresponding S-ring. It provides good isomorphism criteria for circulant graphs (and algorithms for isomorphism testing).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (11/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The structure of S-rings over Zpm

case n = pm, p odd prime: R. Pöschel 1974, Leung/Man, Muzychuk case n = 2m: Klin, Golfand, Najmark, Pöschel, Muzychuk, Leung/Man

(1978 – 1993)

Decomposition of S-rings (a “wreath product construction”) leads to decomposable and indecomposable S-rings. Structure Theorem [Klin/Pö ] There is a 1-1-correspondence between indecomposable S-rings S over Zpm and pairs (λ, f ), where λ is an atomic sequence of exponent m and f is an divisor of p−1. The S-ring corresponding to (λ, f ) is the so-called atomic Z-submodule Mf

m(λ).

What this is good for? From (λ, f ) one easily and directly can compute the automorphism group of the corresponding S-ring. It provides good isomorphism criteria for circulant graphs (and algorithms for isomorphism testing).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (11/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The structure of S-rings over Zpm

case n = pm, p odd prime: R. Pöschel 1974, Leung/Man, Muzychuk case n = 2m: Klin, Golfand, Najmark, Pöschel, Muzychuk, Leung/Man

(1978 – 1993)

Decomposition of S-rings (a “wreath product construction”) leads to decomposable and indecomposable S-rings. Structure Theorem [Klin/Pö ] There is a 1-1-correspondence between indecomposable S-rings S over Zpm and pairs (λ, f ), where λ is an atomic sequence of exponent m and f is an divisor of p−1. The S-ring corresponding to (λ, f ) is the so-called atomic Z-submodule Mf

m(λ).

What this is good for? From (λ, f ) one easily and directly can compute the automorphism group of the corresponding S-ring. It provides good isomorphism criteria for circulant graphs (and algorithms for isomorphism testing).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (11/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The structure of S-rings over Zpm

case n = pm, p odd prime: R. Pöschel 1974, Leung/Man, Muzychuk case n = 2m: Klin, Golfand, Najmark, Pöschel, Muzychuk, Leung/Man

(1978 – 1993)

Decomposition of S-rings (a “wreath product construction”) leads to decomposable and indecomposable S-rings. Structure Theorem [Klin/Pö ] There is a 1-1-correspondence between indecomposable S-rings S over Zpm and pairs (λ, f ), where λ is an atomic sequence of exponent m and f is an divisor of p−1. The S-ring corresponding to (λ, f ) is the so-called atomic Z-submodule Mf

m(λ).

What this is good for? From (λ, f ) one easily and directly can compute the automorphism group of the corresponding S-ring. It provides good isomorphism criteria for circulant graphs (and algorithms for isomorphism testing).

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (11/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Example

Example for n = pm = 33, f = 2, λ = (1, 0) : Wm,f = W3,2 = {1, 26} ≤ Z∗

27

T(1) = T(p0) = {a + kp3−1 | a ∈ Wm,f , k ∈ Zp1} = {1, 10, 19, 26, 17, 8} T(3) = T(p1) = {ap + kp3−0 | a ∈ Wm,f , k ∈ Zp0} = {3, 24} T(9) = T(p2) = {ap2 | a ∈ Wm,f } = {9, 18}

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (12/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Example

Example for n = pm = 33, f = 2, λ = (1, 0) : Wm,f = W3,2 = {1, 26} ≤ Z∗

27

T(1) = T(p0) = {a + kp3−1 | a ∈ Wm,f , k ∈ Zp1} = {1, 10, 19, 26, 17, 8} T(3) = T(p1) = {ap + kp3−0 | a ∈ Wm,f , k ∈ Zp0} = {3, 24} T(9) = T(p2) = {ap2 | a ∈ Wm,f } = {9, 18}

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (12/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Example

Example for n = pm = 33, f = 2, λ = (1, 0) : Let Wm,f be the unique subgroup of Z∗

pm of order f

Wm,f = W3,2 = {1, 26} ≤ Z∗

27

(f = 2) T(1) = T(p0) = {a + kp3−1 | a ∈ Wm,f , k ∈ Zp1} = {1, 10, 19, 26, 17, 8} T(3) = T(p1) = {ap + kp3−0 | a ∈ Wm,f , k ∈ Zp0} = {3, 24} T(9) = T(p2) = {ap2 | a ∈ Wm,f } = {9, 18}

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (12/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Example

Example for n = pm = 33, f = 2, λ = (1, 0) : Let Wm,f be the unique subgroup of Z∗

pm of order f

Wm,f = W3,2 = {1, 26} ≤ Z∗

27

(f = 2) T(1) = T(p0) = {a + kp3−1 | a ∈ Wm,f , k ∈ Zp1} = {1, 10, 19, 26, 17, 8} T(3) = T(p1) = {ap + kp3−0 | a ∈ Wm,f , k ∈ Zp0} = {3, 24} T(9) = T(p2) = {ap2 | a ∈ Wm,f } = {9, 18}

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (12/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Example

Example for n = pm = 33, f = 2, λ = (1, 0) : Let Wm,f be the unique subgroup of Z∗

pm of order f

Wm,f = W3,2 = {1, 26} ≤ Z∗

27

(f = 2) T(1) = T(p0) = {a + kp3−1 | a ∈ Wm,f , k ∈ Zp1} = {1, 10, 19, 26, 17, 8} T(3) = T(p1) = {ap + kp3−0 | a ∈ Wm,f , k ∈ Zp0} = {3, 24} T(9) = T(p2) = {ap2 | a ∈ Wm,f } = {9, 18}

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (12/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Example

Example for n = pm = 33, f = 2, λ = (1, 0) : Let Wm,f be the unique subgroup of Z∗

pm of order f

Wm,f = W3,2 = {1, 26} ≤ Z∗

27

(f = 2) T(1) = T(p0) = {a + kp3−1 | a ∈ Wm,f , k ∈ Zp1} = {1, 10, 19, 26, 17, 8} T(3) = T(p1) = {ap + kp3−0 | a ∈ Wm,f , k ∈ Zp0} = {3, 24} T(9) = T(p2) = {ap2 | a ∈ Wm,f } = {9, 18}

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (12/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Example

Example for n = pm = 33, f = 2, λ = (1, 0)= (u0, u1) : Let Wm,f be the unique subgroup of Z∗

pm of order f

Wm,f = W3,2 = {1, 26} ≤ Z∗

27

(f = 2) The basis of Mf

m(λ) = Mf m(u0, u1, . . . , um−2) is given by

T(pi) := {api + kpm−ui | a ∈ Wm,f , k ∈ Zpui } for i ∈ {0, 1, . . . , m − 2} and T(pm−1) := Wm,f pm−1 (um−1 := 0) T(1) = T(p0) = {a + kp3−1 | a ∈ Wm,f , k ∈ Zp1} = {1, 10, 19, 26, 17, 8} T(3) = T(p1) = {ap + kp3−0 | a ∈ Wm,f , k ∈ Zp0} = {3, 24} T(9) = T(p2) = {ap2 | a ∈ Wm,f } = {9, 18}

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (12/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Outline

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (13/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Automorphisms of S-rings

Here: Definition via the circulant graphs Γ(Zn, Ti) corresponding to the simple quantities of an S-ring basis, S = T0, T1, . . . , Tr−1: A permutation f ∈ Sym(Zn) is an automorphism of S if it is an automorphism of each graph Γ(Zn, Ti), i.e., Aut S :=

r−1

• i=1

Γ(Zn, Ti) (automorphism group of S)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (14/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Automorphisms of S-rings

Here: Definition via the circulant graphs Γ(Zn, Ti) corresponding to the simple quantities of an S-ring basis, S = T0, T1, . . . , Tr−1: A permutation f ∈ Sym(Zn) is an automorphism of S if it is an automorphism of each graph Γ(Zn, Ti), i.e., Aut S :=

r−1

• i=1

Γ(Zn, Ti) (automorphism group of S)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (14/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Some history

Oberwolfach 1993 (Klin/Pö)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (15/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Some history

Oberwolfach 1991 (with K. Rosenbaum)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (15/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Some history

Oberwolfach 1993 (Klin/Pö) I.N. Ponomarenko 2005, (automorphism groups of circulant association schemes)

• J. Morris 2007, (automorphism groups of circulant graphs)
• M. Muzychuk 2009, (Wedge product of association schemes, Schur

rings) S.A. Evdokimov, I.N. Ponomarenko 2012, (Schurity of S-rings, generalized wreath product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (15/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Some history

Dresden (2012) (with Valery Liskovets and Misha Muzychuk) (and Ilya Ponomarenko (2010))

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (15/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

A Galois connection

The automorphism property (concerning permutations and binary relations (graphs)) induces a Galois connection:

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (16/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

A Galois connection

The automorphism property (concerning permutations and binary relations (graphs)) induces a Galois connection: S-rings S groups G ≥ Zn S − → Aut S transitivity module S(G, Zn) ← − (G, Zn)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (16/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

A Galois connection

The automorphism property (concerning permutations and binary relations (graphs)) induces a Galois connection: S-rings S groups G ≥ Zn S − → Aut S transitivity module S(G, Zn) ← − (G, Zn) Galois closures Schurian S-rings 2-closed permutation groups S(Aut S, Zn) G (2) := Aut(Γ(G, Zn)) = Aut Inv(2) G = Aut 2-OrbG

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (16/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

A Galois connection

The automorphism property (concerning permutations and binary relations (graphs)) induces a Galois connection: S-rings S groups G ≥ Zn S − → Aut S transitivity module S(G, Zn) ← − (G, Zn) Galois closures Schurian S-rings 2-closed permutation groups S(Aut S, Zn) G (2) := Aut(Γ(G, Zn)) = Aut Inv(2) G = Aut 2-OrbG Zpm is schurian, i.e. each S-ring over Zpm is a Schurian S-ring.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (16/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism groups Aut S

Let S-ring S over Zpm be given, how to find Aut S? Case 1: S decomposable: S = S1[S2] S1 S-ring over Zps, S2 S-ring over Zpt, s + t = m Then Aut S = Aut S1 ≀ Aut S2 (wreath product) and | Aut S| = | Aut S1| · | Aut S2|ps Case 2: S indecomposable, i.e., S = Mf

m(λ) (by Theorem):

• Compute the “group formula” A(λ) of (λ, f ) for each atomic

sequence λ (the group structure of Aut S and its cardinality easily and directly follow from f and the formula A(λ))

• Interpret A(λ) as subgroup of Sym(Zpm) (subwreath product)
• Then Aut S = A(λ)•Wm,f (semidirect product)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (17/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

From atomic sequence λ to group formula A(λ)

Reduction process (inductive definition of A(λ)): For λ = (u0, u1, . . . , um−2) (atomic - in particular decreasing - sequence of exponent m) compute “derivation” sequence λ′ = (v0, v1, . . . , vm′−2) of exponent m′ = m − ur−1 (where r := min{i | ui = 0}) vi := ui − ur−1 for i ∈ {0, 1, . . . , r − 1}, vj := 0 for j ∈ {r, . . . , m′ − 2}, and put A(λ) :=

• Zpm

if r = 0, (Zpur−1)pr

• A(λ′)

if r ≥ 1.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (18/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

From atomic sequence λ to group formula A(λ)

Reduction process (inductive definition of A(λ)): For λ = (u0, u1, . . . , um−2) (atomic - in particular decreasing - sequence of exponent m) compute “derivation” sequence λ′ = (v0, v1, . . . , vm′−2) of exponent m′ = m − ur−1 (where r := min{i | ui = 0}) vi := ui − ur−1 for i ∈ {0, 1, . . . , r − 1}, vj := 0 for j ∈ {r, . . . , m′ − 2}, and put A(λ) :=

• Zpm

if r = 0, (Zpur−1)pr

• A(λ′)

if r ≥ 1.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (18/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Examples: From λ to group formula A(λ)

m λ λ′ A(λ) [= (Zpur−1 )pr

• A(λ′)]
• (A(λ))

m = 2 (0) (0) Z32 32 m = 3 (0, 0) (0, 0) Z33 33 (1, 0) (0) (Z3)3•A(λ′) = (Z3)3•Z32 35 m = 4 (0, 0, 0) (0, 0, 0) Z34, 34 (1, 0, 0) (0, 0) (Z3)3•A(λ′) = (Z3)3•Z33 36 (1, 1, 0) (0, 0) (Z3)9•A(λ′) = (Z3)32

• Z33

312 (2, 0, 0) (0) (Z32)3•A(λ′) = (Z32)3•Z32 38 (2, 1, 0) (1, 0) (Z3)9•A(λ′) = (Z3)32

• ((Z3)3•Z32)

314 m = 5 (0, 0, 0, 0) (0, 0, 0, 0) Z35 35 (1, 0, 0, 0) (0, 0, 0) (Z3)3•Z34 37 (1, 1, 0, 0) (0, 0, 0) (Z3)32

• Z34

313 (1, 1, 1, 0) (0, 0, 0) (Z3)33

• Z34

331 (2, 0, 0, 0) (0, 0) (Z32)3•Z33 39 (2, 1, 0, 0) (1, 0, 0) (Z3)32

• ((Z3)3•Z33)

315 . . . . . . . . . . . . (3, 2, 1, 0) (2, 1, 0) (Z3)33

• ((Z3)32
• ((Z3)3•Z32))

341

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (19/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ) (Example)

m = 3, λ = (1, 0), group formula A(λ) = (Z3)3•Z32 The corresponding group A(λ) ≤ Sym(Zpm) is consists of all permutations of the form g : Zpm → Zpm : x → x + y(x) where y : Zpm → Zpm is a mapping of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 for x = x0 + x1p + x2p2 with arbitrary mappings y1 : Z1

p → Zp, y2 : Z2 p → Zp

but y1 must not depend on x0, and y2 must not depend on x1, thus, in fact, we have y(x) = y0 + y′

1p + y′ 2(x0)p2 for arbitrary

y0, y′

1 ∈ Zp and function y′ 2 : Zp → Zp.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (20/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ) (Example)

m = 3, λ = (1, 0), group formula A(λ) = (Z3)3•Z32 The corresponding group A(λ) ≤ Sym(Zpm) is consists of all permutations of the form g : Zpm → Zpm : x → x + y(x) where y : Zpm → Zpm is a mapping of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 for x = x0 + x1p + x2p2 with arbitrary mappings y1 : Z1

p → Zp, y2 : Z2 p → Zp

but y1 must not depend on x0, and y2 must not depend on x1, thus, in fact, we have y(x) = y0 + y′

1p + y′ 2(x0)p2 for arbitrary

y0, y′

1 ∈ Zp and function y′ 2 : Zp → Zp.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (20/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ) (Example)

m = 3, λ = (1, 0), group formula A(λ) = (Z3)3•Z32 The corresponding group A(λ) ≤ Sym(Zpm) is consists of all permutations of the form (M-representation (Muzychuk)) g : Zpm → Zpm : x → x + y(x) where y : Zpm → Zpm is a mapping of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 for x = x0 + x1p + x2p2 with arbitrary mappings y1 : Z1

p → Zp, y2 : Z2 p → Zp

but y1 must not depend on x0, and y2 must not depend on x1, thus, in fact, we have y(x) = y0 + y′

1p + y′ 2(x0)p2 for arbitrary

y0, y′

1 ∈ Zp and function y′ 2 : Zp → Zp.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (20/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ) (Example)

m = 3, λ = (1, 0), group formula A(λ) = (Z3)3•Z32 The corresponding group A(λ) ≤ Sym(Zpm) is consists of all permutations of the form (M-representation (Muzychuk)) g : Zpm → Zpm : x → x + y(x) where y : Zpm → Zpm is a mapping of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 for x = x0 + x1p + x2p2 with arbitrary mappings y1 : Z1

p → Zp, y2 : Z2 p → Zp

but y1 must not depend on x0, and y2 must not depend on x1, thus, in fact, we have y(x) = y0 + y′

1p + y′ 2(x0)p2 for arbitrary

y0, y′

1 ∈ Zp and function y′ 2 : Zp → Zp.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (20/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ) (Example)

m = 3, λ = (1, 0), group formula A(λ) = (Z3)3•Z32 The corresponding group A(λ) ≤ Sym(Zpm) is consists of all permutations of the form (M-representation (Muzychuk)) g : Zpm → Zpm : x → x + y(x) where y : Zpm → Zpm is a mapping of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 for x = x0 + x1p + x2p2 with arbitrary mappings y1 : Z1

p → Zp, y2 : Z2 p → Zp

but y1 must not depend on x0, and y2 must not depend on x1, thus, in fact, we have y(x) = y0 + y′

1p + y′ 2(x0)p2 for arbitrary

y0, y′

1 ∈ Zp and function y′ 2 : Zp → Zp.

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (20/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ)

We use p-adic expansion of x ∈ Zpm: x = x0 + x1p + · · · + xm−1pm−1

Theorem

Let λ = (u0, u1, . . . , um−2) be an atomic sequence. Then the automorphism group A(λ) consists of all permutations of the form x → x + y(x) (x ∈ Zpm) with y : Zpm → Zpm of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 + · · · + ym−1(x0, . . . , xm−2)pm−1 and satisfying the following conditions:

• For all i ∈ {0, 1, . . . , m−2} and all j with i < j < m−ui the function

yj : Zj

p → Zp : (x0, . . . , xj−1) → yj(x0, . . . , xj−1) does not depend on

the variables xi, . . . , xj−1.

• Villanova, June, 2014,
• M. Klin, R. Pöschel, Automorphism groups of Schur rings (21/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ)

We use p-adic expansion of x ∈ Zpm: x = x0 + x1p + · · · + xm−1pm−1

Theorem

Let λ = (u0, u1, . . . , um−2) be an atomic sequence. Then the automorphism group A(λ) consists of all permutations of the form x → x + y(x) (x ∈ Zpm) with y : Zpm → Zpm of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 + · · · + ym−1(x0, . . . , xm−2)pm−1 and satisfying the following conditions:

• For all i ∈ {0, 1, . . . , m−2} and all j with i < j < m−ui the function

yj : Zj

p → Zp : (x0, . . . , xj−1) → yj(x0, . . . , xj−1) does not depend on

the variables xi, . . . , xj−1.

• Villanova, June, 2014,
• M. Klin, R. Pöschel, Automorphism groups of Schur rings (21/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Characterization of the group A(λ)

We use p-adic expansion of x ∈ Zpm: x = x0 + x1p + · · · + xm−1pm−1

Theorem

Let λ = (u0, u1, . . . , um−2) be an atomic sequence. Then the automorphism group A(λ) consists of all permutations of the form x → x + y(x) (x ∈ Zpm) with y : Zpm → Zpm of the form y(x) = y0 + y1(x0)p + y2(x0, x1)p2 + · · · + ym−1(x0, . . . , xm−2)pm−1 and satisfying the following conditions:

• For all i ∈ {0, 1, . . . , m−2} and all j with i < j < m−ui the function

yj : Zj

p → Zp : (x0, . . . , xj−1) → yj(x0, . . . , xj−1) does not depend on

the variables xi, . . . , xj−1.

• Villanova, June, 2014,
• M. Klin, R. Pöschel, Automorphism groups of Schur rings (21/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism group Aut(Mf

m(λ))

above Example: n = 33, p = 3, m = 3: S = M2

3(1, 0), group formula (Z3)3•Z32

Automorphism group Aut S = A(λ)•Wm,f = A(1, 0)•W3,2 and we have (justifying the notation A(1, 0) as formula) Aut S ∼ = ((Z3)3•Z32)•W3,2

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (22/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism group Aut(Mf

m(λ))

above Example: n = 33, p = 3, m = 3: S = M2

3(1, 0), group formula (Z3)3•Z32

Automorphism group Aut S = A(λ)•Wm,f = A(1, 0)•W3,2 and we have (justifying the notation A(1, 0) as formula) Aut S ∼ = ((Z3)3•Z32)•W3,2

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (22/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The automorphism group Aut(Mf

m(λ))

above Example: n = 33, p = 3, m = 3: S = M2

3(1, 0), group formula (Z3)3•Z32

Automorphism group Aut S = A(λ)•Wm,f = A(1, 0)•W3,2 and we have (justifying the notation A(1, 0) as formula) Aut S ∼ = ((Z3)3•Z32)•W3,2

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (22/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

The isomorphism problem for circulant graphs

Γ = Γ(Zn, T) circulant graph (Cayley graph over Zn) with connection set T ⊆ Zn ((x, y) directed edge of Γ : ⇐

⇒ y − x ∈ T)

• T

: least S-ring S with T ∈ S

Proposition

Γ(Zn, T) ∼ = Γ(Zn, S) = ⇒ T = S

• (i.e. f ∈ Aut S for graph isomorphism f )

The knowledge of S-rings and their automorphism groups provides efficient isomorphism criteria and algorithms for isomorphism testing − → M. Muzychuk

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (23/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Outline

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (24/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Some further photos

Balaton 2006 (with daughter Hana)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (25/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Beer Sheva 2013 (with daughter Hana)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (26/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

For Misha

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (26/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

For Misha

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (27/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Final (personal) remark

❉♦r♦❣♦❅ ✐ ▼✐①❛✱ ❱♦t ✉✙❡ ❜♦❧❡❡ s♦r♦❦❛ ❧❡t ♠② ❦♦❧❧❡❣✐ ✐ ❞r✉③⑦✤✳ ■ ✤ ❤♦q✉ s❦❛③❛t⑦ t❡❜❡ s♣❛s✐❜♦ ③❛ ♥❛①❡ ✉s♣❡①♥♦❡ s♦tr✉❞♥✐q❡st✈♦✳ ❍♦q✉ ♣♦✙❡❧❛t⑦ t❡❜❡ ✈s❡❣♦ ♥❛✐❧✉q①❡❣♦ ✈ ❜✉❞✉✇❡♠✱ ✤r❦✐❤ ♠❛t❡♠❛t✐q❡s❦✐❤ ✐❞❡❅ ✐✱ ③❞♦r♦✈⑦✤ ♥❛ ❞♦❧❣✐❡ ❣♦❞② ✐ sq❛st⑦✤ ✈ ❧✐q♥♦❅ ✐ ✙✐③♥✐✳

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (28/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Final (personal) remark

❉♦r♦❣♦❅ ✐ ▼✐①❛✱ ❱♦t ✉✙❡ ❜♦❧❡❡ s♦r♦❦❛ ❧❡t ♠② ❦♦❧❧❡❣✐ ✐ ❞r✉③⑦✤✳ ■ ✤ ❤♦q✉ s❦❛③❛t⑦ t❡❜❡ s♣❛s✐❜♦ ③❛ ♥❛①❡ ✉s♣❡①♥♦❡ s♦tr✉❞♥✐q❡st✈♦✳ ❍♦q✉ ♣♦✙❡❧❛t⑦ t❡❜❡ ✈s❡❣♦ ♥❛✐❧✉q①❡❣♦ ✈ ❜✉❞✉✇❡♠✱ ✤r❦✐❤ ♠❛t❡♠❛t✐q❡s❦✐❤ ✐❞❡❅ ✐✱ ③❞♦r♦✈⑦✤ ♥❛ ❞♦❧❣✐❡ ❣♦❞② ✐ sq❛st⑦✤ ✈ ❧✐q♥♦❅ ✐ ✙✐③♥✐✳ Dear Misha, on behalf of all participants many thanks for all what you did for the development of algebraic combinatorics, for all the stimulating and encouraging discussions, good health and further success and all the best for the next ? years

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (28/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Final (personal) remark

❉♦r♦❣♦❅ ✐ ▼✐①❛✱ ❱♦t ✉✙❡ ❜♦❧❡❡ s♦r♦❦❛ ❧❡t ♠② ❦♦❧❧❡❣✐ ✐ ❞r✉③⑦✤✳ ■ ✤ ❤♦q✉ s❦❛③❛t⑦ t❡❜❡ s♣❛s✐❜♦ ③❛ ♥❛①❡ ✉s♣❡①♥♦❡ s♦tr✉❞♥✐q❡st✈♦✳ ❍♦q✉ ♣♦✙❡❧❛t⑦ t❡❜❡ ✈s❡❣♦ ♥❛✐❧✉q①❡❣♦ ✈ ❜✉❞✉✇❡♠✱ ✤r❦✐❤ ♠❛t❡♠❛t✐q❡s❦✐❤ ✐❞❡❅ ✐✱ ③❞♦r♦✈⑦✤ ♥❛ ❞♦❧❣✐❡ ❣♦❞② ✐ sq❛st⑦✤ ✈ ❧✐q♥♦❅ ✐ ✙✐③♥✐✳ Dear Misha, on behalf of all participants many thanks for all what you did for the development of algebraic combinatorics, for all the stimulating and encouraging discussions, good health and further success and all the best for the next 5 years

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (28/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Final (personal) remark

❉♦r♦❣♦❅ ✐ ▼✐①❛✱ ❱♦t ✉✙❡ ❜♦❧❡❡ s♦r♦❦❛ ❧❡t ♠② ❦♦❧❧❡❣✐ ✐ ❞r✉③⑦✤✳ ■ ✤ ❤♦q✉ s❦❛③❛t⑦ t❡❜❡ s♣❛s✐❜♦ ③❛ ♥❛①❡ ✉s♣❡①♥♦❡ s♦tr✉❞♥✐q❡st✈♦✳ ❍♦q✉ ♣♦✙❡❧❛t⑦ t❡❜❡ ✈s❡❣♦ ♥❛✐❧✉q①❡❣♦ ✈ ❜✉❞✉✇❡♠✱ ✤r❦✐❤ ♠❛t❡♠❛t✐q❡s❦✐❤ ✐❞❡❅ ✐✱ ③❞♦r♦✈⑦✤ ♥❛ ❞♦❧❣✐❡ ❣♦❞② ✐ sq❛st⑦✤ ✈ ❧✐q♥♦❅ ✐ ✙✐③♥✐✳ Dear Misha, on behalf of all participants many thanks for all what you did for the development of algebraic combinatorics, for all the stimulating and encouraging discussions, good health and further success and all the best for the next 10 years

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (28/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks

Final (personal) remark

❉♦r♦❣♦❅ ✐ ▼✐①❛✱ ❱♦t ✉✙❡ ❜♦❧❡❡ s♦r♦❦❛ ❧❡t ♠② ❦♦❧❧❡❣✐ ✐ ❞r✉③⑦✤✳ ■ ✤ ❤♦q✉ s❦❛③❛t⑦ t❡❜❡ s♣❛s✐❜♦ ③❛ ♥❛①❡ ✉s♣❡①♥♦❡ s♦tr✉❞♥✐q❡st✈♦✳ ❍♦q✉ ♣♦✙❡❧❛t⑦ t❡❜❡ ✈s❡❣♦ ♥❛✐❧✉q①❡❣♦ ✈ ❜✉❞✉✇❡♠✱ ✤r❦✐❤ ♠❛t❡♠❛t✐q❡s❦✐❤ ✐❞❡❅ ✐✱ ③❞♦r♦✈⑦✤ ♥❛ ❞♦❧❣✐❡ ❣♦❞② ✐ sq❛st⑦✤ ✈ ❧✐q♥♦❅ ✐ ✙✐③♥✐✳ Dear Misha, on behalf of all participants many thanks for all what you did for the development of algebraic combinatorics, for all the stimulating and encouraging discussions, good health and further success and all the best for the next n years (n ∈ N)

Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (28/29)

Schur-rings (S-rings) Automorphism groups of S-rings Final (personal) remarks Villanova, June, 2014,

• M. Klin, R. Pöschel, Automorphism groups of Schur rings (29/29)