In this chapter we consider automorphisms of single piece groupoids; then homogeneous discrete groupoids; and finally homogeneous groupoids. We also consider matrix representations and groupooid actions.
‣ GroupoidAutomorphismByObjectPerm( gpd, imobs ) | ( operation ) |
‣ GroupoidAutomorphismByGroupAuto( gpd, gpiso ) | ( operation ) |
‣ GroupoidAutomorphismByNtuple( gpd, imrays ) | ( operation ) |
‣ GroupoidAutomorphismByRayShifts( gpd, imrays ) | ( operation ) |
We first describe automorphisms of a groupoid \(G\) where \(G\) is the direct product of a group \(g\) and a complete digraph with \(n\) objects.. The automorphism group is generated by three types of automorphism:
given a permutation \(\pi\) of the \(n\) objects, we define
\[ \alpha_{\pi} : G \to G,~ (g : o_i \to o_j) \mapsto (g : o_{\pi i} \to o_{\pi j}); \]
given an automorphism \(\theta\) of the root group \(g\), we define
\[ \alpha_{\theta} : G \to G,~ (g : o_i \to o_j) \mapsto (\theta g : o_i \to o_j); \]
given \(L = [g_1,g_2,g_3,\ldots,g_n] \in g^n\) we define
\[ \alpha_L : G \to G,~ (g : o_i \to o_j) \mapsto (g_i^{-1}gg_j : o_i \to o_j). \]
If \(g_1 = 1_g\), then for all \(j\) the rays \((r_j : o_1 \to o_j)\) are shifted by \(g_j\;\): so they map to \((r_jg_j : o_1 \to o_j)\). So the operation GroupoidAutomorphismByRayShifts is the special case of GroupoidAutomorphismByNtuple when \(g_1=1\).
gap> perm1 := [-13,-12,-14];; gap> aut1 := GroupoidAutomorphismByObjectPerm( Ha4, perm1 );; gap> Display( aut1 ); homomorphism to single piece groupoid: Ha4 -> Ha4 root group homomorphism: (1,2,3) -> (1,2,3) (2,3,4) -> (2,3,4) object map: [ -14, -13, -12 ] -> [ -13, -12, -14 ] ray images: [ (), (), () ] gap> d := Arrow( Ha4, (1,3,4), -12, -13 ); [(1,3,4) : -12 -> -13] gap> d1 := ImageElm( aut1, d ); [(1,3,4) : -14 -> -12] gap> gensa4 := GeneratorsOfGroup( a4 );; gap> alpha2 := GroupHomomorphismByImages( a4, a4, gensa4, [(2,3,4), (1,3,4)] );; gap> aut2 := GroupoidAutomorphismByGroupAuto( Ha4, alpha2 );; gap> Display( aut2 ); homomorphism to single piece groupoid: Ha4 -> Ha4 root group homomorphism: (1,2,3) -> (2,3,4) (2,3,4) -> (1,3,4) object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ] ray images: [ (), (), () ] gap> d2 := ImageElm( aut2, d1 ); [(1,2,4) : -14 -> -12] gap> L3 := [(1,2)(3,4), (1,3)(2,4), (1,4)(2,3)];; gap> aut3 := GroupoidAutomorphismByNtuple( Ha4, L3 );; gap> Display( aut3 ); homomorphism to single piece groupoid: Ha4 -> Ha4 root group homomorphism: (1,2,3) -> (1,4,2) (2,3,4) -> (1,4,3) object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ] ray images: [ (), (1,4)(2,3), (1,3)(2,4) ] gap> d3 := ImageElm( aut3, d2 ); [(2,3,4) : -14 -> -12] gap> L4 := [(), (1,3,2), (2,4,3)];; gap> aut4 := GroupoidAutomorphismByRayShifts( Ha4, L4 );; gap> Display( aut4 ); homomorphism to single piece groupoid: Ha4 -> Ha4 root group homomorphism: (1,2,3) -> (1,2,3) (2,3,4) -> (2,3,4) object map: [ -14, -13, -12 ] -> [ -14, -13, -12 ] ray images: [ (), (1,3,2), (2,4,3) ] gap> d4 := ImageElm( aut4, d3 ); [() : -14 -> -12] gap> h4 := Arrow( Ha4, (2,3,4), -12, -13 );; gap> aut1234 := aut1*aut2*aut3*aut4;; gap> Display( aut1234 ); homomorphism to single piece groupoid: Ha4 -> Ha4 root group homomorphism: (1,2,3) -> (1,4,3) (2,3,4) -> (1,2,3) object map: [ -14, -13, -12 ] -> [ -13, -12, -14 ] ray images: [ (), (2,3,4), (1,3,4) ] gap> d4 = ImageElm( aut1234, d ); true gap> inv1234 := InverseGeneralMapping( aut1234 );; gap> Display( inv1234 ); homomorphism to single piece groupoid: Ha4 -> Ha4 root group homomorphism: (1,2,3) -> (1,4,3) (2,3,4) -> (2,4,3) object map: [ -14, -13, -12 ] -> [ -12, -14, -13 ] ray images: [ (), (1,3,2), (1,3,4) ]
‣ GroupoidInnerAutomorphism( gpd, arrow ) | ( operation ) |
‣ GroupoidInnerAutomorphismNormalSubgroupoid( gpd, subgpd, arrow ) | ( operation ) |
Given an arrow \(a = (c : p \to q) \in G\) with \(p \neq q\), the inner automorphism \(\alpha_a\) of \(G\) by \(a\) is the mapping \(g \mapsto g^a\) where conjugation of arrows is defined in section 4.5. It is easily checked that if \(L_a = [1,\ldots,1,c^{-1},1,\ldots,1,c,1,\ldots,1]\), with \(c^{-1}\) in position \(p\) and \(c\) in position \(q\), then
\[ \alpha_a ~=~ \alpha_{(p,q)} * \alpha_{L_a}. \]
Similarly, when \(p=q\), then \(\alpha_a = \alpha_{L_a}\) where now \( L_a = [1,\ldots,1,c,1,\ldots,1]\), with \(c\) in position \(p\).
gap> inn1 := GroupoidInnerAutomorphism( Ha4, h4 );; gap> Display( inn1 ); homomorphism to single piece groupoid: Ha4 -> Ha4 root group homomorphism: (1,2,3) -> (1,2,3) (2,3,4) -> (2,3,4) object map: [ -14, -13, -12 ] -> [ -14, -12, -13 ] ray images: [ (), (2,4,3), (2,3,4) ] gap> d5 := ImageElm( inn1, d4 ); [(2,3,4) : -14 -> -13]
Conjugation may also be applied to certain normal subgroupoids of \(G\). Firstly, let \(N\) be the wide subgroupoid of \(G\) determined by a normal subgroup \(n\) of the root group. Then, provided the group element of \(a\) is in \(n\), the inner automorphism by \(a\) may be applied to \(N\).
gap> Nk4 := SubgroupoidBySubgroup( Ha4, k4 );; gap> SetName( Nk4, "Nk4" ); gap> e4 := Arrow( Ha4, (1,2)(3,4), -14, -13 );; gap> inn2 := GroupoidInnerAutomorphismNormalSubgroupoid( Ha4, Nk4, e4 );; gap> Display( inn2 ); homomorphism to single piece groupoid: Nk4 -> Nk4 root group homomorphism: (1,2)(3,4) -> (1,2)(3,4) (1,3)(2,4) -> (1,3)(2,4) object map: [ -14, -13, -12 ] -> [ -13, -14, -12 ] ray images: [ (), (), (1,2)(3,4) ]
Secondly, if \(H\) is a homogeneous, discrete subgroupoid of \(G\) and if the group element of \(a\) is in the common vertex groups, then the inner automorphism may be applied to \(H\).
gap> Ma4 := MaximalDiscreteSubgroupoid( Ha4 );; gap> SetName( Ma4, "Ma4" ); gap> inn3 := GroupoidInnerAutomorphism( Ha4, Ma4, e4 );; gap> Display( inn3 ); homogeneous discrete groupoid mapping: [ Ma4 ] -> [ Ma4 ] images of objects: [ -13, -14, -12 ] object homomorphisms: GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,4,2), (1,4,3) ] ) GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,4,2), (1,4,3) ] ) GroupHomomorphismByImages( a4, a4, [ (1,2,3), (2,3,4) ], [ (1,2,3), (2,3,4) ] )
Let \(S\) be a wide subgroupoid with rays of a standard groupoid \(G\).
An automorphism \(\alpha\) of the root group \(H\) extends to the whole of \(S\) with the rays fixed by the automorphism: \((r^{-1}_ihr_j : o_i \to o_j) \mapsto (r^{-1}_i (\alpha h)r_j : o_i \to o_j)\).
An automorphism of \(G\) obtained by permuting the objects may map \(S\) to a different subgroupoid. So we construct an isomorphism \(\iota\) from \(S\) to a standard groupoid \(T\), construct \(\alpha\) permuting the objects of \(T\), and return \(\iota*\alpha*\iota^{-1}\).
For an automorphism by ray shifts we require that the shifts are elements of the root group of \(S\).
gap> ## (1) automorphism by group auto gap> a6 := GroupHomomorphismByImages( k4, k4, > [ (1,2)(3,4), (1,3)(2,4) ], [ (1,3)(2,4), (1,4)(2,3) ] );; gap> aut6 := GroupoidAutomorphismByGroupAuto( Kk4, a6 ); groupoid homomorphism : Kk4 -> Kk4 [ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14], [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ], [ [(1,3)(2,4) : -14 -> -14], [(1,4)(2,3) : -14 -> -14], [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ] ] gap> a := Arrow( Kk4, (1,3)(2,4), -12, -12 );; gap> ImageElm( aut6, a ); [(1,4)(2,3) : -12 -> -12] gap> b := Arrow( Kk4, (1,4,2), -12, -13 );; gap> ImageElm( aut6, b ); [(1,2,3) : -12 -> -13] gap> ## (2) automorphism by object perm gap> aut7 := GroupoidAutomorphismByObjectPerm( Kk4, [-13,-12,-14] ); groupoid homomorphism : Kk4 -> Kk4 [ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14], [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ], [ [(1,4)(2,3) : -13 -> -13], [(1,2)(3,4) : -13 -> -13], [(2,3,4) : -13 -> -12], [(1,4,3) : -13 -> -14] ] ] gap> ImageElm( aut7, a ); [(1,3)(2,4) : -14 -> -14] gap> ImageElm( aut7, b ); [(1,3)(2,4) : -14 -> -12] gap> ## (3) automorphism by ray shifts gap> aut8 := GroupoidAutomorphismByRayShifts( Kk4, > [ (), (1,4)(2,3), (1,3)(2,4) ] ); groupoid homomorphism : Kk4 -> Kk4 [ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14], [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ], [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14], [(1,2,3) : -14 -> -13], [(1,2)(3,4) : -14 -> -12] ] ] gap> ImageElm( aut8, a ); [(1,3)(2,4) : -12 -> -12] gap> ImageElm( aut8, b ); [(1,2,3) : -12 -> -13] gap> ## (4) combine these three automorphisms gap> aut678 := aut6 * aut7 * aut8; groupoid homomorphism : Kk4 -> Kk4 [ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14], [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ], [ [(1,2)(3,4) : -13 -> -13], [(1,3)(2,4) : -13 -> -13], [(1,4,3) : -13 -> -12], [(1,3,2) : -13 -> -14] ] ] gap> ImageElm( aut678, a ); [(1,4)(2,3) : -14 -> -14] gap> ImageElm( aut678, b ); [(1,4)(2,3) : -14 -> -12] gap> ## (5) conjgation by an arrow gap> e8 := Arrow( Kk4, (1,3)(2,4), -14, -12 );; gap> aut9 := GroupoidInnerAutomorphism( Kk4, e8 ); groupoid homomorphism : Kk4 -> Kk4 [ [ [(1,2)(3,4) : -14 -> -14], [(1,3)(2,4) : -14 -> -14], [(1,3,4) : -14 -> -13], [(1,4)(2,3) : -14 -> -12] ], [ [(1,2)(3,4) : -12 -> -12], [(1,3)(2,4) : -12 -> -12], [(1,4,2) : -12 -> -13], [(1,4)(2,3) : -12 -> -14] ] ]
‣ AutomorphismGroupOfGroupoid( gpd ) | ( operation ) |
‣ NiceObjectAutoGroupGroupoid( gpd, aut ) | ( operation ) |
As above, let \(G\) be the direct product of a group \(g\) and a complete digraph with \(n\) objects. The AutomorphismGroup \(\Aut(G)\) of \(G\) is isomorphic to the quotient of \(S_n \times A \times g^n\) by a subgroup isomorphic to \(g\), where \(A\) is the automorphism group of \(g\) and \(S_n\) is the symmetric group on the \(n\) objects. This is one of the main topics in [AW10].
If \(H\) is the union of \(k\) groupoids, all isomorphic to \(G\), then \(\Aut(H)\) is isomorphic to \(S_k \ltimes \Aut(G)\).
The function NiceObjectAutoGroupGroupoid takes a groupoid and a subgroup of its automorphism group and retuns a nice monomorphism from this automorphism group to a pc-group, if one is available. The current implementation is experimental. Note that ImageElm at present only works on generating elements.
gap> AHa4 := AutomorphismGroupOfGroupoid( Ha4 ); Aut(Ha4) gap> Agens := GeneratorsOfGroup( AHa4);; gap> Length( Agens ); 8 gap> NHa4 := NiceObject( AHa4 );; gap> MHa4 := NiceMonomorphism( AHa4 );; gap> Size( AHa4 ); ## (3!)x24x(12^2) 20736 gap> SetName( AHa4, "AHa4" ); gap> SetName( NHa4, "NHa4" ); gap> ## either of these names may be returned gap> names := [ "(((A4 x A4 x A4) : C2) : C3) : C2", > "(C2 x C2 x C2 x C2 x C2 x C2) : (((C3 x C3 x C3) : C3) : (C2 x C2))" ];; gap> StructureDescription( NHa4 ) in names; true gap> ## cannot test images of Agens because of random variations gap> ## Now do some tests! gap> mgi := MappingGeneratorsImages( MHa4 );; gap> autgen := mgi[1];; gap> pcgen := mgi[2];; gap> ngen := Length( autgen );; gap> ForAll( [1..ngen], i -> Order(autgen[i]) = Order(pcgen[i]) ); true
The inner automorphism subgroup \(\mathrm{Inn}(G)\) of the automorphism group of \(G\) is the group of inner automorphisms \(\wedge a : b \mapsto b^a\) for \(a \in G\). It is not the case that the map \(G \to \mathrm{Inn}(G), a \mapsto \wedge a\) preserves multiplication. Indeed, when \(a=(o,g,p), b=(p,h,r) \in G\) with objects \(p,q,r\) all distict, then
\[ \wedge(ab) ~=~ (\wedge a)(\wedge b)(\wedge a) ~=~ (\wedge b)(\wedge a)(\wedge b). \]
(Compare this with the permutation identity \((pq)(qr)(pq) = (pr) = (qr)(pq)(qr)\).) So the map \(G \to \mathrm{Inn}(G)\) is of type IsMappingWithObjectsByFunction.
In the example we convert the automorphism group AGa4 into a single object groupoid, and then define the inner automorphism map.
gap> AHa40 := Groupoid( AHa4, [0] ); single piece groupoid: < Aut(Ha4), [ 0 ] > gap> conj := function(a) > return ArrowNC( Ha4, true, GroupoidInnerAutomorphism(Ha4,a), 0, 0 ); > end;; gap> inner := MappingWithObjectsByFunction( Ha4, AHa40, conj, [0,0,0] );; gap> a1 := Arrow( Ha4, (1,2,3), -14, -13 );; gap> inner1 := ImageElm( inner, a1 );; gap> a2 := Arrow( Ha4, (2,3,4), -13, -12 );; gap> inner2 := ImageElm( inner, a2 );; gap> a3 := a1*a2; [(1,3)(2,4) : -14 -> -12] gap> inner3 := ImageElm( inner, a3 ); [groupoid homomorphism : Ha4 -> Ha4 [ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13], [() : -14 -> -12] ], [ [(1,3,4) : -12 -> -12], [(1,2,4) : -12 -> -12], [(1,3)(2,4) : -12 -> -13], [() : -12 -> -14] ] ] : 0 -> 0] gap> (inner3 = inner1*inner2*inner1) and (inner3 = inner2*inner1*inner2); true true
‣ GroupoidAutomorphismByGroupAutos( gpd, auts ) | ( operation ) |
Homogeneous, discrete groupoids are the second type of groupoid for which a method is provided for AutomorphismGroupOfGroupoid. This is used in the XMod package for constructing crossed modules of groupoids. The two types of generating automorphism are GroupoidAutomorphismByGroupAutos, which requires a list of group automorphisms, one for each object group, and GroupoidAutomorphismByObjectPerm, which permutes the objects. So, if the object groups \(g\) have automorphism group \(\Aut(g)\) and there are \(n\) objects, the autmorphism group of the groupoid has size \(n!|\Aut(g)|^n\).
gap> Dd8 := HomogeneousDiscreteGroupoid( d8, [ -13..-10] ); homogeneous, discrete groupoid: < d8, [ -13 .. -10 ] > gap> aut10 := GroupoidAutomorphismByObjectPerm( Dd8, [-12,-10,-11,-13] ); groupoid homomorphism : morphism from a homogeneous discrete groupoid: [ -13, -12, -11, -10 ] -> [ -12, -10, -11, -13 ] object homomorphisms: IdentityMapping( d8 ) IdentityMapping( d8 ) IdentityMapping( d8 ) IdentityMapping( d8 ) gap> gend8 := GeneratorsOfGroup( d8 );; gap> g1 := gend8[1];; gap> g2 := gend8[2];; gap> b1 := IdentityMapping( d8 );; gap> b2 := GroupHomomorphismByImages( d8, d8, gend8, [g1, g2*g1 ] );; gap> b3 := GroupHomomorphismByImages( d8, d8, gend8, [g1^g2, g2 ] );; gap> b4 := GroupHomomorphismByImages( d8, d8, gend8, [g1^g2, g2^(g1*g2) ] );; gap> aut11 := GroupoidAutomorphismByGroupAutos( Dd8, [b1,b2,b3,b4] ); groupoid homomorphism : morphism from a homogeneous discrete groupoid: [ -13, -12, -11, -10 ] -> [ -13, -12, -11, -10 ] object homomorphisms: IdentityMapping( d8 ) GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,6,7,8), (5,8)(6,7) ] ) GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (5,7) ] ) GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (6,8) ] ) gap> ADd8 := AutomorphismGroupOfGroupoid( Dd8 ); <group with 4 generators> gap> Size( ADd8 ); ## 4!*8^4 98304 gap> genADd8 := GeneratorsOfGroup( ADd8 );; gap> Length( genADd8 ); 4 gap> w := GroupoidAutomorphismByGroupAutos( Dd8, [b2,b1,b1,b1] );; gap> x := GroupoidAutomorphismByGroupAutos( Dd8, [b3,b1,b1,b1] );; gap> y := GroupoidAutomorphismByObjectPerm( Dd8, [ -12, -11, -10, -13 ] );; gap> z := GroupoidAutomorphismByObjectPerm( Dd8, [ -12, -13, -11, -10 ] );; gap> ok := ForAll( genADd8, a -> a in[ w, x, y, z ] ); true gap> NADd8 := NiceObject( ADd8 );; gap> MADd8 := NiceMonomorphism( ADd8 );; gap> w1 := ImageElm( MADd8, w );; gap> x1 := ImageElm( MADd8, x );; gap> y1 := ImageElm( MADd8, y );; gap> z1 := ImageElm( MADd8, z );; gap> u := z*w*y*x*z; groupoid homomorphism : morphism from a homogeneous discrete groupoid: [ -13, -12, -11, -10 ] -> [ -11, -13, -10, -12 ] object homomorphisms: IdentityMapping( d8 ) GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,6,7,8), (5,8)(6,7) ] ) IdentityMapping( d8 ) GroupHomomorphismByImages( d8, d8, [ (5,6,7,8), (5,7) ], [ (5,8,7,6), (5,7) ] ) gap> u1 := z1*w1*y1*x1*z1; (1,2,4,3)(5,17,23,11,6,18,24,16)(7,19,25,15,9,21,27,13)(8,20,26,14,10,22,28,12) gap> imu := ImageElm( MADd8, u );; gap> u1 = imu; true
‣ AutomorphismGroupoidOfGroupoid( gpd ) | ( attribute ) |
If \(G\) is a single piece groupoid with automorphism group \(\Aut(G)\), and if \(H\) is the union of \(k\) pieces, all isomorphic to \(G\), then the automorphism group of \(H\) is the wreath product \(S_k \ltimes \Aut(G)\). However, we find it more convenient to construct the automorphism groupoid of \(H\). This is a single piece groupoid \(\AUT(H)\) with \(k\) objects - the object lists of the pieces of \(H\) - and root group \(\Aut(G)\). Isomorphisms between the root groups of the \(k\) pieces may be applied to the generators of \(\Aut(G)\) to construct automorphism groups of these pieces, and then isomorphisms between these automorphism groups. We then construct \(\AUT(H)\) using GroupoidByIsomorphisms.
In the special case that \(H\) is homogeneous, there is no need to construct a collection of automorphism groups. Rather, the rays of \(\AUT(H)\) are given by IsomorphismNewObjects. For the example we use HGd8 constructed in subsection HomogeneousGroupoid (4.1-5).
gap> HGd8 := HomogeneousGroupoid( Gd8, > [ [-39,-38,-37], [-36,-35,-34], [-33,-32,-31] ] );; gap> SetName( HGd8, "HGd8" ); gap> AHGd8 := AutomorphismGroupoidOfGroupoid( HGd8 ); Aut(HGd8) gap> ObjectList( AHGd8 ); [ [ -39, -38, -37 ], [ -36, -35, -34 ], [ -33, -32, -31 ] ] gap> RaysOfGroupoid( AHGd8 ){[2..3]}; [ groupoid homomorphism : [ [ [(5,6,7,8) : -39 -> -39], [(5,7) : -39 -> -39], [() : -39 -> -38], [() : -39 -> -37] ], [ [(5,6,7,8) : -36 -> -36], [(5,7) : -36 -> -36], [() : -36 -> -35], [() : -36 -> -34] ] ], groupoid homomorphism : [ [ [(5,6,7,8) : -39 -> -39], [(5,7) : -39 -> -39], [() : -39 -> -38], [() : -39 -> -37] ], [ [(5,6,7,8) : -33 -> -33], [(5,7) : -33 -> -33], [() : -33 -> -32], [() : -33 -> -31] ] ] ] gap> obgp := ObjectGroup( AHGd8, [ -36, -35, -34 ] );; gap> Size( obgp ); ## 3!*8^3 3072
Suppose that gpd is the direct product of a group \(G\) and a complete digraph, and that \(\rho : G \to M\) is an isomorphism to a matrix group \(M\). Then, if rep is the isomorphic groupoid with the same objects and root group \(M\), there is an isomorphism \(\mu\) from gpd to rep mapping \((g : i \to j)\) to \((\rho g : i \to j)\).
When gpd is a groupoid with rays, a representation can be obtained by restricting a representation of its parent.
gap> reps := IrreducibleRepresentations( a4 );; gap> rep4 := reps[4]; Pcgs([ (2,4,3), (1,3)(2,4), (1,2)(3,4) ]) -> [ [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], [ [ -1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, -1 ] ], [ [ 1, 0, 0 ], [ 0, -1, 0 ], [ 0, 0, -1 ] ] ] gap> Ra4 := Groupoid( Image( rep4 ), Ga4!.objects );; gap> ObjectList( Ra4 ) = [ -15 .. -11 ]; true gap> gens := GeneratorsOfGroupoid( Ga4 ); [ [(1,2,3) : -15 -> -15], [(2,3,4) : -15 -> -15], [() : -15 -> -14], [() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ] gap> images := List( gens, > g -> Arrow( Ra4, ImageElm(rep4,g![2]), g![3], g![4] ) ); [ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -15 -> -15], [[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -15 -> -15], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -14], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -13], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -12], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -11] ] gap> mor := GroupoidHomomorphismFromSinglePiece( Ga4, Ra4, gens, images ); groupoid homomorphism : [ [ [(1,2,3) : -15 -> -15], [(2,3,4) : -15 -> -15], [() : -15 -> -14], [() : -15 -> -13], [() : -15 -> -12], [() : -15 -> -11] ], [ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -15 -> -15], [[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -15 -> -15], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -14], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -13], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -12], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -15 -> -11] ] ] gap> IsMatrixGroupoid( Ra4 ); true gap> a := Arrow( Ha4, (1,4,2), -12, -13 ); [(1,4,2) : -12 -> -13] gap> ImageElm( mor, a ); [[ [ 0, 0, 1 ], [ -1, 0, 0 ], [ 0, -1, 0 ] ] : -12 -> -13] gap> rmor := RestrictedMappingGroupoids( mor, Ha4 ); groupoid homomorphism : [ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13], [() : -14 -> -12] ], [ [[ [ 0, 0, -1 ], [ 1, 0, 0 ], [ 0, -1, 0 ] ] : -14 -> -14], [[ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] : -14 -> -14], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -14 -> -13], [[ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] : -14 -> -12] ] ] gap> ParentMappingGroupoids( rmor ) = mor; true
Recall from sections 4.5 and GroupoidInnerAutomorphism (6.1-2) the notion of conjugation in a groupoid, and the associated inner automorphisms.
It was mentioned there that the map \(\wedge : G \to \Aut(G),~ a \to \wedge a\), is not a groupoid homomorphism. It is in fact a groupoid action which we now define. Let \(\{p,q,r,u,v\}\) be distinct objects in \(G\) and let:
\(a_1 = (c_1 : p \to q),~~ a_2 = (c_2 : q \to r),~~ a_3 = (c_3 : q \to p),~~ a_4 = (c_4 : u \to v)\),
\(b_1 = (d_1 : p \to p),~~ b_2 = (d_2 : p \to p),~~ b_3 = (d_3 : q \to q),~~ b_4 = (c_3c_1 : q \to q)\) be arrows in \(G\). Then the following conjugation identities must be satisfied:
\(\wedge(a_1a_2) ~=~ (\wedge a_1)*(\wedge a_2)*(\wedge a_1) ~=~ (\wedge a_2)*(\wedge a_1)*(\wedge a_2)\),
\(\wedge(b_1a_1) ~=~ (\wedge b_1)*(\wedge a_1)*(\wedge b_1)^{-1}\),
\(\wedge(a_1b_3) ~=~ (\wedge b_3)^{-1}*(\wedge a_1)*(\wedge b_3)\),
\(\wedge(b_1b_2) ~=~ (\wedge b_1)*(\wedge b_2)\),
\(\wedge(a_1a_3) ~=~ (\wedge a_1)*(\wedge a_3)*(\wedge b_4)\),
\((\wedge a_1)*(\wedge a_4) ~=~ (\wedge a_4)*(\wedge a_1)\).
In the following example we check the first of these identities in one particular case.
gap> c1 := Arrow( Ha4, (1,2)(3,4), -14, -13);; gap> innc1 := GroupoidInnerAutomorphism( Ha4, c1 ); groupoid homomorphism : Ha4 -> Ha4 [ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13], [() : -14 -> -12] ], [ [(1,4,2) : -13 -> -13], [(1,4,3) : -13 -> -13], [() : -13 -> -14], [(1,2)(3,4) : -13 -> -12] ] ] gap> c2 := Arrow( Ha4, (1,4,2), -13, -12);; gap> innc2 := GroupoidInnerAutomorphism( Ha4, c2 ); groupoid homomorphism : Ha4 -> Ha4 [ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13], [() : -14 -> -12] ], [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [(1,4,2) : -14 -> -12], [(1,2,4) : -14 -> -13] ] ] gap> c12 := c1 * c2; [(2,4,3) : -14 -> -12] gap> innc12 := GroupoidInnerAutomorphism( Ha4, c12 ); groupoid homomorphism : Ha4 -> Ha4 [ [ [(1,2,3) : -14 -> -14], [(2,3,4) : -14 -> -14], [() : -14 -> -13], [() : -14 -> -12] ], [ [(1,4,2) : -12 -> -12], [(2,3,4) : -12 -> -12], [(2,3,4) : -12 -> -13], [(2,4,3) : -12 -> -14] ] ] gap> [ innc1 * innc2 * innc1 = innc12, innc2 * innc1 * innc2 = innc12 ]; [ true, true ]
‣ GroupoidActionByConjugation( gpd ) | ( operation ) |
‣ IsGroupoidAction( map ) | ( category ) |
‣ ActionMap( act ) | ( attribute ) |
The operation GroupoidInnerAutomorphism, which produces the conjugation action of \(G\) on itself, does satisfy the conjugation identities and so provides a standard example of an action.
An action is a record with fields Source, Range and ActionMap.
The examples repeat those in section GroupoidInnerAutomorphism (6.1-2): firstly with a groupoid acting on itself.
gap> act1 := GroupoidActionByConjugation( Ha4 ); <general mapping: Ha4 -> Aut(Ha4) > gap> IsGroupoidAction( act1 ); true gap> amap1 := ActionMap( act1 );; gap> amap1( h4 ) = inn1; true
Secondly with an action on a single piece, normal subgroupoid.
gap> act2 := GroupoidActionByConjugation( Ha4, Nk4 ); <general mapping: Ha4 -> Aut(Nk4) > gap> IsGroupoidAction( act2 ); true gap> amap2 := ActionMap( act2 );; gap> amap2( e4 ) = inn2; true
Thirly with an action on a homogeneous, discrete subgroupoid.
gap> act3 := GroupoidActionByConjugation( Ha4, Ma4 ); <general mapping: Ha4 -> Aut(Ma4) > gap> IsGroupoidAction( act3 ); true gap> amap3 := ActionMap( act3 );; gap> amap3( e4 ) = inn3; true
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