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Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation groups, permutation patterns,and Galois connections
Erkko Lehtonen and Reinhard Poschel
Technische Universitat DresdenInstitute of Algebra
AAA92Arbeitstagung Allgemeine Algebra
Workshop on General AlgebraPraha 27.5.2016
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (1/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Outline
Permutation patterns
The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)
The Galois closures gComp(n) G and Galois kernels gPat(`)H
Remarks and references
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (2/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Outline
Permutation patterns
The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)
The Galois closures gComp(n) G and Galois kernels gPat(`)H
Remarks and references
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (3/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
5 43 21
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
3
5 421
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
31
5 42
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
1
3
5 42
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
1
53
42
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
1
53
2
4
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
21
53
4
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
53
21 4
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
53 2 41
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
53 2 π41
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
53 2 π41
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoided
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Some motivationWhich number sequences (permutations) can be sorted by a stack?
53 2 π41
Proposition
A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.
i.e., such “patterns” abc (like 352) must be avoidedAAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:
π1 . . . πn := (π(1), . . . , π(n)).
e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4
graphical representation:
π = 31524 ∈ S5
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:
π1 . . . πn := (π(1), . . . , π(n)).
e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4
graphical representation:
π = 31524 ∈ S5
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:
π1 . . . πn := (π(1), . . . , π(n)).
e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4
graphical representation:
π = 31524 ∈ S5
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:
π1 . . . πn := (π(1), . . . , π(n)).
e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4
graphical representation:π = 31524 ∈ S5
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns (Example)
π = 31524 ∈ S5
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns (Example)
π = 31524 ∈ S5
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns (Example)
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns (Example)
1 2 3
3
2
1
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns (Example)
σ = 231 ∈ S3
1 2 3
3
2
1
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns (Example)
σ ≤ πσ is pattern of π
π involves σ
π = 31524 ∈ S5
σ = 231 ∈ S3
1 2 3
3
2
1
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns (Example)
352
231
231 = red(352)
reduced form
substring
π = 31524 ∈ S5
1 2 3
3
2
1
1 2 3 4 5
1
5
4
3
2
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns
` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `
such that σ = red(u)(σ is `-pattern of π, or π involves σ)
π avoids σ :⇐⇒ σ � π.
Pat(`) π := {σ ∈ S` | σ ≤ π}
The pattern involvement relation ≤ is a partial order on the setP :=
⋃n≥1 Sn of all finite permutations.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns
` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `
such that σ = red(u)(σ is `-pattern of π, or π involves σ)
π avoids σ :⇐⇒ σ � π.
Pat(`) π := {σ ∈ S` | σ ≤ π}
The pattern involvement relation ≤ is a partial order on the setP :=
⋃n≥1 Sn of all finite permutations.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns
` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `
such that σ = red(u)(σ is `-pattern of π, or π involves σ)
π avoids σ :⇐⇒ σ � π.
Pat(`) π := {σ ∈ S` | σ ≤ π}
The pattern involvement relation ≤ is a partial order on the setP :=
⋃n≥1 Sn of all finite permutations.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Permutation patterns
` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `
such that σ = red(u)(σ is `-pattern of π, or π involves σ)
π avoids σ :⇐⇒ σ � π.
Pat(`) π := {σ ∈ S` | σ ≤ π}
The pattern involvement relation ≤ is a partial order on the setP :=
⋃n≥1 Sn of all finite permutations.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Outline
Permutation patterns
The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)
The Galois closures gComp(n) G and Galois kernels gPat(`)H
Remarks and references
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (8/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection Pat(`)−Comp(n)
The relation σ � π (π avoids σ) induces a Galois connectionbetween subsets of S` and Sn. The corresponding “monotoneGalois connection” (residuation) is given by the following operators(monotone w.r.t. ⊆):
For S ⊆ S`, T ⊆ Sn (` ≤ n) let
Comp(n) S := {τ ∈ Sn | Pat(`) τ ⊆ S} = {τ ∈ Sn | ∀σ′ ∈ S` \ S : σ′ � τ},
Pat(`) T :=⋃τ∈T
Pat(`) τ = S` \ {σ′ ∈ S` | ∀τ ∈ T : σ′ � τ}.
In particular we have the defining property of a monotone Galoisconnection:
Pat(`) T ⊆ S ⇐⇒ T ⊆ Comp(n) S
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (9/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection Pat(`)−Comp(n)
The relation σ � π (π avoids σ) induces a Galois connectionbetween subsets of S` and Sn. The corresponding “monotoneGalois connection” (residuation) is given by the following operators(monotone w.r.t. ⊆):
For S ⊆ S`, T ⊆ Sn (` ≤ n) let
Comp(n) S := {τ ∈ Sn | Pat(`) τ ⊆ S} = {τ ∈ Sn | ∀σ′ ∈ S` \ S : σ′ � τ},
Pat(`) T :=⋃τ∈T
Pat(`) τ = S` \ {σ′ ∈ S` | ∀τ ∈ T : σ′ � τ}.
In particular we have the defining property of a monotone Galoisconnection:
Pat(`) T ⊆ S ⇐⇒ T ⊆ Comp(n) S
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (9/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection Pat(`)−Comp(n)
The relation σ � π (π avoids σ) induces a Galois connectionbetween subsets of S` and Sn. The corresponding “monotoneGalois connection” (residuation) is given by the following operators(monotone w.r.t. ⊆):
For S ⊆ S`, T ⊆ Sn (` ≤ n) let
Comp(n) S := {τ ∈ Sn | Pat(`) τ ⊆ S} = {τ ∈ Sn | ∀σ′ ∈ S` \ S : σ′ � τ},
Pat(`) T :=⋃τ∈T
Pat(`) τ = S` \ {σ′ ∈ S` | ∀τ ∈ T : σ′ � τ}.
In particular we have the defining property of a monotone Galoisconnection:
Pat(`) T ⊆ S ⇐⇒ T ⊆ Comp(n) S
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (9/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Connection to permutation groups
Proposition
If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.
(The converse does not hold)
Sketch of the proof.
Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:
Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.
Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Connection to permutation groups
Proposition
If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.
(The converse does not hold)
Sketch of the proof.
Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:
Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.
Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Connection to permutation groups
Proposition
If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.
(The converse does not hold)
Sketch of the proof.
Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:
Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.
Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Connection to permutation groups
Proposition
If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.
(The converse does not hold)
Sketch of the proof.
Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:
Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.
Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Connection to permutation groups
Proposition
If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.
(The converse does not hold)
Sketch of the proof.
Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:
Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.
Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Connection to permutation groups
Proposition
If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.
(The converse does not hold)
Sketch of the proof.
Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:
Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.
Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection gPat(`)− gComp(n)
Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:
gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},
gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H
Pat(`) τ〉,
This gives also a monotone Galois connection since we have:
gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G
Thus
gPat(`) gComp(n) G ⊆ G (kernel operator),
H ⊆ gComp(n) gPat(`)H (closure operator),
gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),
gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection gPat(`)− gComp(n)
Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:
gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},
gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H
Pat(`) τ〉,
This gives also a monotone Galois connection since we have:
gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G
Thus
gPat(`) gComp(n) G ⊆ G (kernel operator),
H ⊆ gComp(n) gPat(`)H (closure operator),
gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),
gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection gPat(`)− gComp(n)
Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:
gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},
gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H
Pat(`) τ〉,
This gives also a monotone Galois connection since we have:
gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G
Thus
gPat(`) gComp(n) G ⊆ G (kernel operator),
H ⊆ gComp(n) gPat(`)H (closure operator),
gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),
gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection gPat(`)− gComp(n)
Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:
gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},
gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H
Pat(`) τ〉,
This gives also a monotone Galois connection since we have:
gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G
Thus
gPat(`) gComp(n) G ⊆ G (kernel operator),
H ⊆ gComp(n) gPat(`)H (closure operator),
gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),
gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection gPat(`)− gComp(n)
Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:
gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},
gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H
Pat(`) τ〉,
This gives also a monotone Galois connection since we have:
gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G
Thus
gPat(`) gComp(n) G ⊆ G (kernel operator),
H ⊆ gComp(n) gPat(`)H (closure operator),
gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),
gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
The (monotone) Galois connection gPat(`)− gComp(n)
Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:
gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},
gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H
Pat(`) τ〉,
This gives also a monotone Galois connection since we have:
gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G
Thus
gPat(`) gComp(n) G ⊆ G (kernel operator),
H ⊆ gComp(n) gPat(`)H (closure operator),
gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),
gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Outline
Permutation patterns
The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)
The Galois closures gComp(n) G and Galois kernels gPat(`)H
Remarks and references
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (12/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Question
How to characterize the Galois closures and the Galois kernels ofthe monotone Galois connection gComp(n)− gPat(`) (` < n) ?
Answer: as automorphism groups of special relations(pc-relations and pc-extended invariants, resp. )
Recall the (usual) Galois connection Aut− Inv (betweenpermutations π ∈ Sn and relations % ∈ Reln on {1, . . . , n}):
AutR := {π ∈ Sn | ∀% ∈ R : π . %} for R ⊆ Reln,
InvT := {% ∈ Reln | ∀π ∈ T : π . %} for T ⊆ Sn,
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (13/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Question
How to characterize the Galois closures and the Galois kernels ofthe monotone Galois connection gComp(n)− gPat(`) (` < n) ?
Answer: as automorphism groups of special relations(pc-relations and pc-extended invariants, resp. )
Recall the (usual) Galois connection Aut− Inv (betweenpermutations π ∈ Sn and relations % ∈ Reln on {1, . . . , n}):
AutR := {π ∈ Sn | ∀% ∈ R : π . %} for R ⊆ Reln,
InvT := {% ∈ Reln | ∀π ∈ T : π . %} for T ⊆ Sn,
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (13/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Question
How to characterize the Galois closures and the Galois kernels ofthe monotone Galois connection gComp(n)− gPat(`) (` < n) ?
Answer: as automorphism groups of special relations(pc-relations and pc-extended invariants, resp. )
Recall the (usual) Galois connection Aut− Inv (betweenpermutations π ∈ Sn and relations % ∈ Reln on {1, . . . , n}):
AutR := {π ∈ Sn | ∀% ∈ R : π . %} for R ⊆ Reln,
InvT := {% ∈ Reln | ∀π ∈ T : π . %} for T ⊆ Sn,
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (13/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-relations
For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).
Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):
hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.
Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.
For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as
%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.
% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.
For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-relations
For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).
Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):
hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.
Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.
For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as
%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.
% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.
For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-relations
For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).
Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):
hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.
Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.
For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as
%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.
% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.
For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-relations
For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).
Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):
hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.
Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.
For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as
%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.
% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.
For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-relations
For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).
Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):
hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.
Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.
For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as
%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.
% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.
For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-relations
For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).
Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):
hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.
Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.
For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as
%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.
% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.
For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Characterization of the Galois closures
The Galois closures of the closure operator gComp(n) gPat(`) canbe characterized by a single irreflexive k-ary pc-relation.
We have:
Theorem
(A) gComp(n) gPat(`)H = Aut pcInvH for H ≤ Sn.
(B) Let H be a subgroup of Sn. Then the following are equivalent:
(a) H is Galois closed, i.e., H = gComp(n) gPat(`)H,(a’) ∃G ≤ S` : H = gComp(n) G ,(b) H = Aut pcInvH,(c) ∃ k ≤ `∃ % ⊆ [n]k6= : % = %∨∧︸ ︷︷ ︸
% is pc-relation
∧ H = Aut %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (15/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Characterization of the Galois closures
The Galois closures of the closure operator gComp(n) gPat(`) canbe characterized by a single irreflexive k-ary pc-relation.
We have:
Theorem
(A) gComp(n) gPat(`)H = Aut pcInvH for H ≤ Sn.
(B) Let H be a subgroup of Sn. Then the following are equivalent:
(a) H is Galois closed, i.e., H = gComp(n) gPat(`)H,(a’) ∃G ≤ S` : H = gComp(n) G ,(b) H = Aut pcInvH,(c) ∃ k ≤ `∃ % ⊆ [n]k6= : % = %∨∧︸ ︷︷ ︸
% is pc-relation
∧ H = Aut %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (15/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Characterization of the Galois closures
The Galois closures of the closure operator gComp(n) gPat(`) canbe characterized by a single irreflexive k-ary pc-relation.
We have:
Theorem
(A) gComp(n) gPat(`)H = Aut pcInvH for H ≤ Sn.
(B) Let H be a subgroup of Sn. Then the following are equivalent:
(a) H is Galois closed, i.e., H = gComp(n) gPat(`)H,(a’) ∃G ≤ S` : H = gComp(n) G ,(b) H = Aut pcInvH,(c) ∃ k ≤ `∃ % ⊆ [n]k6= : % = %∨∧︸ ︷︷ ︸
% is pc-relation
∧ H = Aut %.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (15/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-extended invariants
Let G ≤ S`. A relation σ ⊆ [`]k (k ≤ `) is called apattern closed extended invariant (pc-extended invariant) of G if
σ∧∨ = σ and σ∧ ∈ Inv Aut γG∧.
where γG := {s | s ∈ G} - the elements of G ≤ S` = [`]`6= ⊆ [`]`
are viewed as `-tuples.
pcExtG : set of all pc-extended invariants of G .
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (16/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-extended invariants
Let G ≤ S`. A relation σ ⊆ [`]k (k ≤ `) is called apattern closed extended invariant (pc-extended invariant) of G if
σ∧∨ = σ and σ∧ ∈ Inv Aut γG∧.
where γG := {s | s ∈ G} - the elements of G ≤ S` = [`]`6= ⊆ [`]`
are viewed as `-tuples.
pcExtG : set of all pc-extended invariants of G .
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (16/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
pc-extended invariants
Let G ≤ S`. A relation σ ⊆ [`]k (k ≤ `) is called apattern closed extended invariant (pc-extended invariant) of G if
σ∧∨ = σ and σ∧ ∈ Inv Aut γG∧.
where γG := {s | s ∈ G} - the elements of G ≤ S` = [`]`6= ⊆ [`]`
are viewed as `-tuples.
pcExtG : set of all pc-extended invariants of G .
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (16/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Characterization of the Galois kernels
Now the Galois kernels of the kernel operator gPat(`) gComp(n) canbe characterized by pc-extended invariant relations:
Theorem
gPat(`) gComp(n) G = Aut pcExtG for G ≤ S`.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (17/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Characterization of the Galois kernels
Now the Galois kernels of the kernel operator gPat(`) gComp(n) canbe characterized by pc-extended invariant relations:
Theorem
gPat(`) gComp(n) G = Aut pcExtG for G ≤ S`.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (17/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Outline
Permutation patterns
The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)
The Galois closures gComp(n) G and Galois kernels gPat(`)H
Remarks and references
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (18/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Remarks
For G ⊆ S`, M.D. Atkinson and R. Beals [1999] consideredthe sequence
G ,Comp(`+1) G , . . . ,Comp(n) G ,Comp(n+1) G , . . .
in the group case, i.e., G ≤ Sn (then Comp(n) G = gComp(n) G ),and asked for the “asymptotic” behaviour of the above sequence.
Recent and much more detailed results about this sequence:Erkko Lehtonen [2015/16]
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (19/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Remarks
For G ⊆ S`, M.D. Atkinson and R. Beals [1999] consideredthe sequence
G ,Comp(`+1) G , . . . ,Comp(n) G ,Comp(n+1) G , . . .
in the group case, i.e., G ≤ Sn (then Comp(n) G = gComp(n) G ),and asked for the “asymptotic” behaviour of the above sequence.
Recent and much more detailed results about this sequence:Erkko Lehtonen [2015/16]
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (19/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
Remarks
For G ⊆ S`, M.D. Atkinson and R. Beals [1999] consideredthe sequence
gPat(1) G , . . . , gPat(`−1) G ,G ,Comp(`+1) G , . . . ,Comp(n) G ,Comp(n+1) G , . . .
in the group case, i.e., G ≤ Sn (then Comp(n) G = gComp(n) G ),and asked for the “asymptotic” behaviour of the above sequence.
Recent and much more detailed results about this sequence:Erkko Lehtonen [2015/16]
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (19/21)
Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references
References
M.D. Atkinson and R. Beals, Permuting mechanismsand closed classes of permutations. In: Combinatorics,computation & logic ’99 (Auckland), vol. 21 of Aust. Comput.Sci. Commun., Springer, Singapore, 1999, pp. 117–127.
E. Lehtonen, Permutation groups arising from patterninvolvement. arXiv:1605.05571.
E. Lehtonen and R. Poschel, Permutation groups,pattern involvement, and Galois connections. arXiv:1605.04516.
N. Ruskuc, Classes of permutations avoiding 231 or 321.Lecture given at TU Dresden, Nov. 25, 2015.
AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (20/21)
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AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (21/21)
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