68.15 Operations for Brauer Characters

FrobeniusCharacterValue returns the value of the Frobenius character corresponding to the Brauer character value value, where p is the characteristic of the field.

For an invertible matrix mat over a finite field F, BrauerCharacterValue returns the Brauer character value of mat if the order of mat is coprime to the characteristic of F, and fail otherwise.

The Brauer character value of a matrix is the sum of complex lifts of its eigenvalues.

gap> g:= SL(2,4);;           # 2-dim. irreducible representation of A5
gap> ccl:= ConjugacyClasses( g );;
gap> rep:= List( ccl, Representative );;
gap> List( rep, Order );
[ 1, 2, 3, 5, 5 ]
gap> phi:= List( rep, BrauerCharacterValue );
[ 2, fail, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ]
gap> List( phi{ [ 1, 3, 4, 5 ] }, x -> FrobeniusCharacterValue( x, 2 ) );
[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2]
gap> List( rep{ [ 1, 3, 4, 5 ] }, TraceMat );
[ 0*Z(2), Z(2)^0, Z(2^2), Z(2^2)^2]

For a cyclotomic or a list of cyclotomics val and a prime integer p, SizeOfFieldOfDefinition returns p ^m where m is the smallest positive integer such that val is fixed by the Galois automorphism that raises each root of unity to its p ^m-th power.

If val is an algebraic integer then p ^m is the size of the smallest finite field in characteristic p that contains the p-modular reduction of val. If val is a Brauer character then p ^m is the size of the smallest finite field in characteristic p over which the corresponding representation lives.

For a list matrix of absolutely irreducible Brauer characters in characteristic p, and a power q of p, RealizableBrauerCharacters returns a duplicate-free list of sums of Frobenius conjugates of the rows of matrix, each irreducible over the field with q elements.

gap> irr:= Irr( CharacterTable( "A5" ) mod 2 );
[ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ), 
  Character( BrauerTable( "A5", 2 ), [ 2, -1, E(5)+E(5)^4, E(5)^2+E(5)^3 ] ), 
  Character( BrauerTable( "A5", 2 ), [ 2, -1, E(5)^2+E(5)^3, E(5)+E(5)^4 ] ), 
  Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ]
gap> List( irr, phi -> SizeOfFieldOfDefinition( phi, 2 ) );
[ 2, 4, 4, 2 ]
gap> RealizableBrauerCharacters( irr, 2 );
[ Character( BrauerTable( "A5", 2 ), [ 1, 1, 1, 1 ] ), 
  ClassFunction( BrauerTable( "A5", 2 ), [ 4, -2, -1, -1 ] ), 
  Character( BrauerTable( "A5", 2 ), [ 4, 1, -1, -1 ] ) ]

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