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### 3 Reducible Representations

In this chapter we introduce some functions which deal with a complex reducible representation $$R$$ of a finite group $$G$$.

#### 3.1 Constituents of Representations

##### 3.1-1 ConstituentsOfRepresentation
 ‣ ConstituentsOfRepresentation( rep ) ( function )

called with a representation rep of a group $$G$$. This function returns a list of irreducible representations of $$G$$ which are constituents of rep, and their corresponding multiplicities. For example, if rep is a representation of $$G$$ affording a character $$X$$ such that $$X = mY + nZ$$, where $$Y$$ and $$Z$$ are irreducible characters of $$G$$, and $$m$$ and $$n$$ are the corresponding multiplicities, then ConstituentsOfRepresentation returns $$[[m, S]$$, $$[n, T]]$$ where $$S$$ and $$T$$ are irreducible representations of $$G$$ affording $$Y$$ and $$Z$$, respectively. This function call can be quite expensive when $$G$$ is a large group.

##### 3.1-2 IsReducibleRepresentation
 ‣ IsReducibleRepresentation( rep ) ( function )

If rep is a representation of a group $$G$$ then IsReducibleRepresentation returns true if rep is a reducible representation of $$G$$.

#### 3.2 Block Representations

##### 3.2-1 EquivalentBlockRepresentation
 ‣ EquivalentBlockRepresentation( rep ) ( function )
 ‣ EquivalentBlockRepresentation( list ) ( function )

If rep is a reducible representation of a group $$G$$, this function returns a block diagonal representation of $$G$$ equivalent to rep. If list $$= [[m1, R1]$$, $$[m2, R2]$$, ... , $$[mt, Rt]]$$ is a list of irreducible representations $$R1$$, $$R2$$, ... , $$Rt$$ of $$G$$ with multiplicities $$m1$$, $$m2$$, ... , $$mt$$, then EquivalentBlockRepresentation returns a block diagonal representation of $$G$$ containing the blocks $$R1$$, $$R2$$, ... , $$Rt$$.

gap> G := AlternatingGroup( 5 );;
gap> H := SylowSubgroup( G, 2 );;
gap> chi := TrivialCharacter( H );;
gap> Hrep := IrreducibleAffordingRepresentation( chi );;
gap> rep := InducedSubgroupRepresentation( G, Hrep );;
gap> IsReducibleRepresentation( rep );
true
gap> con := ConstituentsOfRepresentation( rep );
[ [ 1, [ (1,2,3,4,5), (3,4,5) ] -> [ [ [ 1 ] ], [ [ 1 ] ] ] ],
[ 1, [ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2 ],
[ 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
[ 1, -E(3), E(3), 0 ],
[ 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2 ] ],
[ [ 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2 ],
[ 0, -E(3), E(3), 1 ],
[ 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2 ],
[ 0, 0, 1, 0 ] ] ] ],
[ 2, [ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ -1, 1, 1, 1, -1 ],
[ 0, 0, 0, 0, 1 ],
[ -1, 0, 0, 1, -1 ],
[ 0, 0, 1, 0, 0 ],
[ 0, -1, 0, -1, 1 ] ],
[ [ 0, 0, 0, 0, 1 ],
[ 0, -1, -1, -1, 0 ],
[ 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 1, 0 ],
[ -1, 0, 0, 1, -1 ] ] ] ] ]
gap> EquivalentBlockRepresentation( con );
[ (1,2,3,4,5), (3,4,5) ] ->
[ [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, E(3), -1/3*E(3)-2/3*E(3)^2, 0, 1/3*E(3)-1/3*E(3)^2, 0,
0, 0, 0, 0,  0, 0, 0, 0, 0 ],
[ 0, 1, -4/3*E(3)+1/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, -E(3), E(3), 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, -1/3*E(3)+1/3*E(3)^2, 1, 1/3*E(3)+2/3*E(3)^2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, -1, 1, 1, 1, -1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, -1, 0, -1, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, -1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, 1 ] ],
[ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 1, -2/3*E(3)-1/3*E(3)^2, 0, 2/3*E(3)+1/3*E(3)^2, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, -E(3), E(3), 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, -4/3*E(3)-2/3*E(3)^2, E(3), -2/3*E(3)-1/3*E(3)^2, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, -1, -1, -1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, -1 ] ] ]

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