Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Invariance groups of functions, orbit equivalence and group actions Reinhard P¨ oschel Institut f¨ ur Algebra Technische Universit¨ at Dresden Workshop on Algebraic Graph Theory Plzeˇ n, October 3, 2016 R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (1/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Joint work with • Eszter Horv´ ath, • G´ eza Makay, • Tamas Waldhauser*. 2012: Invariance groups of finite functions and orbit equivalence of permutation groups, arXiv:1210.1015. *invited talk at AAA85 (2013, Luxembourg) We acknowledge helpful discussions with • Erik Friese, • Keith Kearnes, • Erkko Lehtonen, • P 3 (P´ eter P´ al P´ alfy), • S´ andor Radeleczki, • Sven Reichard. R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (2/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Outline Invariance groups The closure under a general point of view: group actions and a Galois connection Characterizations of the closure Concrete results Another closure and Galois connection R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (3/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations The orginal problem: Symmetry groups of Boolean functions The symmetry group or invariance group of a Boolean function f : 2 n �→ 2 ( 2 := { 0 , 1 } ) is f ⊢ := { σ ∈ Sym( n ) | ∀ x 1 , . . . , x n ∈ 2 : f ( x 1 σ , . . . , x n σ ) = f ( x 1 , . . . , x n ) } . Notation: σ : i �→ i σ (action of σ ∈ Sym( n ) on i ∈ { 1 , . . . , n } ) x σ = ( x 1 , . . . , x n ) σ := ( x 1 σ , . . . , x n σ ) (action of σ on x ∈ 2 n ) Thus f ⊢ = { σ ∈ Sym( n ) | ∀ x ∈ 2 n : f ( x σ ) = f ( x ) } Problem: Which groups are such symmetry groups? A. Kisielewicz , Symmetry groups of Boolean functions and constructions of permutation groups . J. of Algebra 1998, (1998), 379–403. R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (4/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Examples and Non-Examples Problem: Which groups are symmetry groups of Boolean functions? What means are ? Does it mean isomorphic? Then we have Every group G ≤ Sym( n ) is isomorphic to the symmetry group of a Boolean function. Proof. recall the result of Frucht (1939) Every group is isomorphic to the automorphism group of a graph. and notice the 1-1-correspondence: f : 2 n �→ 2 � H = ( n , { E ∈ P ( n ) | f ( χ E ) = 1 } ) (hypergraph) Example: � � f ⊢ = Aut ∼ = A 3 R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (5/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Examples and Non-Examples Problem: Which groups are symmetry groups of Boolean functions? What means are ? Does it mean equality ? Then there are groups which are not symmetry groups of a Boolean function, e.g., for the alternating group A n ≤ Sym( n ) there is no f with f ⊢ = A n . Suppose that f ⊢ = A 3 for some f : 2 n → 2 . Then f must be constant on the orbits of A 3 acting on 2 3 : 000 �→ a 100 , 010 , 001 �→ b 011 , 101 , 110 �→ c 111 �→ d However, such a function is totally symmetric, i.e., f ⊢ = S 3 � = A 3 . R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (6/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Generalization Instead of f : 2 n → 2 consider functions f : k n → m invariance group f ⊢ = { σ ∈ Sym( n ) | ∀ x 1 , . . . , x n ∈ k : f ( x 1 σ , . . . , x n σ ) = f ( x 1 , . . . , x n ) } Definition • A group G ≤ Sym( n ) is called ( k , m )-representable if there exists a function f : k n → m such that G = f ⊢ . • A group G ≤ Sym( n ) is called ( k , ∞ )-representable if it is ( k , m )-representable for some m ∈ N + . Thus G is (2 , 2)-representable iff it is the invariance group of a Boolean function. R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (7/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Pseudo-Boolean functions A group G ≤ Sym( n ) is (2 , m )-representable iff it is the invariance group of a pseudo-Boolean function f : 2 n → m . ( � hypergraph on n vertices with colored edges ( m − 1 colors)) Clote, Kranakis 1991: If G is the invariance group of a pseudo-Boolean function, then G is the invariance group of a Boolean function. Kisielewicz 1998: False! The Klein four-group V is a counterexample; moreover, it is the only counterexample that one could “easily” find. Conjecture (Kisielewicz) There are infinitely many groups that are (2 , ∞ ) -representable but not (2 , 2) -representable. An equivalent result is given in Dalla Volta, Siemons (2012), Corollary 5.3, however its proof contains a gap. R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (8/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Example: Klein’s four-group The Klein four-group V = { id , (12)(34) , (13)(24) , (14)(23) } ≤ Sym(4) is not (2 , 2)-representable but it is the intersection of the invariance groups of two Boolean functions: � � � � � � V = Aut = Aut ∩ Aut R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (9/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Invariance groups and orbit closed groups Clote, Kranakis 1991: The following are equivalent for any group G ≤ Sym( n ): (i) G is the invariance group of a pseudo-Boolean function (i.e., G is (2 , ∞ ) -representable), (ii) G is the intersection of invariance groups of Boolean functions (i.e., G = F ⊢ for a set F of Boolean functions), (iii) G is orbit closed. Definition Two subgroups of Sym( n ) are orbit equivalent if they have the same orbits on P ( n ). The orbit closure of G is the greatest element of its orbit equivalence class. Note: action ( 2 n , Sym( n )) � action ( P ( n ) , Sym( n )). a := ( a 1 , . . . , a n ) ∈ 2 n ← → � a := { i ∈ { 1 , . . . , n } | a i = 1 } ∈ P ( n ) R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (10/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Example: primitive groups Inglis; Cameron, Neumann, Saxl; Siemons,Wagner 1984-85: Almost all primitive groups are orbit closed. Seress 1997: All primitive subgroups of Sym( n ) are orbit closed (equivalently, (2 , ∞ )-representable) except for A n and C5, AGL(1,5), PGL(2,5), AGL(1,8), AGL(1,8), AGL(1,9), ASL(2,3), PSL(2,8), PGL(2,8) and PGL(2,9). Horv´ ath, Makay, P¨ oschel, Waldhauser 2014: Every primitive permutation group except for A n ( n ≥ 4) is (3 , ∞ )-representable. R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (11/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Outline Invariance groups The closure under a general point of view: group actions and a Galois connection Characterizations of the closure Concrete results Another closure and Galois connection R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (12/36)
Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations Example: functions f : k n → m Recall: invariance group for f : k n → m f ⊢ = { σ ∈ Sym( n ) | ∀ x 1 , . . . , x n ∈ k : f ( x 1 σ , . . . , x n σ ) = f ( x 1 , . . . , x n ) } f ( x σ ) = f ( x ) for action σ ∈ Sym( n ) on x ∈ k n The following are equivalent for any group G ≤ Sym( n ): (i) G is ( k , ∞ )-representable. (ii) G is the invariance group of a function f : k n → N . (iii) G is the intersection of invariance groups of functions k n → 2 . (iv) G is the intersection of invariance groups of functions k n → k . (v) G is Galois closed with respect to the Galois connection induced by ⊢ . here ⊢ ⊆ O ( n ) × Sym( n ) is a binary relation between n -ary functions (on k k = { 0 , . . . , k − 1 } ) and permutations from Sym( n ). R. P¨ oschel, Invariance groups of functions, orbit equivalence and group actions (13/36)
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