77.10 ChevieCharInfo

ChevieCharInfo( W )

returns information about the irreducible characters of the finite Coxeter group W. The result is a record with the components: charparams contains the result of CharParams(W), charnames contains the corresponding names returned by CharName, and a, A, b, B, contain the results of LowestPowerGenericDegrees(W), HighestPowerGenericDegrees(W), LowestPowerFakeDegrees(W), HighestPowerFakeDegrees(W), respectively. Moreover, the components positionId and positionSgn contain the position of the trivial and the sign character, respectively, in the character table of W. If W is irreducible, there is an additional component positionRefl which contains the position of the character of the reflection representation.

    gap>  ChevieCharInfo( CoxeterGroup( "G", 2 ) );
    rec(
      charparams := [ [ [ 1, 0 ] ], [ [ 1, 6 ] ], [ [ 1, 3, "'" ] ], 
          [ [ 1, 3, "''" ] ], [ [ 2, 1 ] ], [ [ 2, 2 ] ] ],
      charnames := 
       [ "phi_{1,0}", "phi_{1,6}", "phi_{1,3}'", "phi_{1,3}''", 
          "phi_{2,1}", "phi_{2,2}" ],
      a := [ 0, 6, 1, 1, 1, 1 ],
      A := [ 0, 6, 5, 5, 5, 5 ],
      b := [ 0, 6, 3, 3, 1, 2 ],
      B := [ 0, 6, 3, 3, 5, 4 ],
      positionId := 1,
      positionSgn := 2,
      positionRefl := 5 )

If W is irreducible of type F_4 or of type E_n (n=6,7,8) then there is an additional component kondo or frame, respectively, which gives the labeling of the characters as determined by Kondo and Frame.

    gap> W := CoxeterGroup( "E", 6 );;
    gap> ChevieCharInfo( W ).frame;
    [ "1_p", "1_p'", "10_s", "6_p", "6_p'", "20_s", "15_p", "15_p'", 
      "15_q", "15_q'", "20_p", "20_p'", "24_p", "24_p'", "30_p", "30_p'", 
      "60_s", "80_s", "90_s", "60_p", "60_p'", "64_p", "64_p'", "81_p", 
      "81_p'" ]