Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Hi,

I am looking for a software package that will allow me to experiment with the irreducible representations of lie groups (for example, $SL(2,p)$) over the complex field and over finite fields. That is, I would like to get the corresponding matrices for group elements. Thanks, Shachar

share|improve this question
    
The CHEVIE package for GAP3 or MAPLE can be useful. math.rwth-aachen.de/~CHEVIE Unfortunately, the GAP4 version is still pending. –  Leandro Vendramin Jan 18 '12 at 19:29

1 Answer 1

up vote 9 down vote accepted

For representations over the complex field, I know that GAP does a good job. (I'm not sure if it can do modular representations as well, but I wouldn't be surprised.)

Here is some example code to get you started:

G:=SL(2,3);;
reps:=IrreducibleRepresentations(G);;
Elements(G);
List(G,g->g^reps[5]);

This prints the elements of the group $SL(2,3)$:

[ [ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ], 
  [ [ 0*Z(3), Z(3)^0 ], [ Z(3), Z(3)^0 ] ], 
  [ [ 0*Z(3), Z(3)^0 ], [ Z(3), Z(3) ] ], 
  [ [ 0*Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ], 
  [ [ 0*Z(3), Z(3) ], [ Z(3)^0, Z(3)^0 ] ], 
  [ [ 0*Z(3), Z(3) ], [ Z(3)^0, Z(3) ] ], 
  [ [ Z(3)^0, 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ], 
  [ [ Z(3)^0, 0*Z(3) ], [ Z(3)^0, Z(3)^0 ] ], 
  [ [ Z(3)^0, 0*Z(3) ], [ Z(3), Z(3)^0 ] ], 
  [ [ Z(3)^0, Z(3)^0 ], [ 0*Z(3), Z(3)^0 ] ], 
  [ [ Z(3)^0, Z(3)^0 ], [ Z(3)^0, Z(3) ] ], 
  [ [ Z(3)^0, Z(3)^0 ], [ Z(3), 0*Z(3) ] ], 
  [ [ Z(3)^0, Z(3) ], [ 0*Z(3), Z(3)^0 ] ], 
  [ [ Z(3)^0, Z(3) ], [ Z(3)^0, 0*Z(3) ] ], 
  [ [ Z(3)^0, Z(3) ], [ Z(3), Z(3) ] ], 
  [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ],
  [ [ Z(3), 0*Z(3) ], [ Z(3)^0, Z(3) ] ], 
  [ [ Z(3), 0*Z(3) ], [ Z(3), Z(3) ] ], 
  [ [ Z(3), Z(3)^0 ], [ 0*Z(3), Z(3) ] ],
  [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ], 
  [ [ Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ], 
  [ [ Z(3), Z(3) ], [ 0*Z(3), Z(3) ] ],
  [ [ Z(3), Z(3) ], [ Z(3)^0, 0*Z(3) ] ], 
  [ [ Z(3), Z(3) ], [ Z(3), Z(3)^0 ] ] 
 ]

followed by complex matrices representing each element:

[ [ [ 0, -1 ], [ 1, 0 ] ],
  [ [ E(3)^2, -E(3) ], [ 1, 0 ] ], 
  [ [ -E(3), -E(3)^2 ], [ 1, 0 ] ], 
  [ [ 0, 1 ], [ -1, 0 ] ], 
  [ [ E(3), E(3)^2 ], [ -1, 0 ] ], 
  [ [ -E(3)^2, E(3) ], [ -1, 0 ] ], 
  [ [ 1, 0 ], [ 0, 1 ] ],
  [ [ E(3), E(3)^2 ], [ 0, 1 ] ], 
  [ [ E(3)^2, -E(3) ], [ 0, 1 ] ], 
  [ [ 1, 0 ], [ E(3), E(3)^2 ] ], 
  [ [ -E(3)^2, E(3) ], [ E(3), E(3)^2 ] ], 
  [ [ 0, -1 ], [ E(3), E(3)^2 ] ], 
  [ [ 1, 0 ], [ -E(3)^2, E(3) ] ], 
  [ [ 0, 1 ], [ -E(3)^2, E(3) ] ], 
  [ [ -E(3), -E(3)^2 ], [ -E(3)^2, E(3) ] ], 
  [ [ -1, 0 ], [ 0, -1 ] ], 
  [ [ -E(3)^2, E(3) ], [ 0, -1 ] ], 
  [ [ -E(3), -E(3)^2 ], [ 0, -1 ] ], 
  [ [ -1, 0 ], [ E(3)^2, -E(3) ] ], 
  [ [ E(3), E(3)^2 ], [ E(3)^2, -E(3) ] ], 
  [ [ 0, -1 ], [ E(3)^2, -E(3) ] ], 
  [ [ -1, 0 ], [ -E(3), -E(3)^2 ] ], 
  [ [ 0, 1 ], [ -E(3), -E(3)^2 ] ], 
  [ [ E(3)^2, -E(3) ], [ -E(3), -E(3)^2 ] ] 
]

Change "5" to other numbers to see different representations. Also, note that the GAP symbol

Z(p)

denotes a generator of the multiplicative group of the finite field $\mathbb{F}_p$. Similarly,

E(k)

denotes a primitive $k^{\mbox{th}}$ root of unity.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.