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Representation of symmetric group

Posted 1 month ago
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I can't seem to find any information/algorithms on generating matrix representations of the symmetric group. Can someone point me in the right direction?

11 Replies

Did you perhaps mean this?

aa = {a, b, c};
pp = Permutations[aa]
res = Table[
   Normal[SparseArray[# -> 1 & /@ 
      Table[{j, Position[pp[[k]], aa[[j]]][[1, 1]]}, {j, 1, Length[aa]}]]], {k, 1, Length[pp]}];
MatrixForm /@ res
Posted 1 month ago

Thanks for the useful lead. I was hoping for a canned Mathematica code that would, for example, generate the matrices of the regular representation of say S3. I guess I am getting lazy in my old age. I did the job myself by brute force, but I suspect that a variation of your code would do the trick more elegantly..

Like this perhaps?

aa = {a, b, c};
pp = Permutations[aa]
res = Table[
   Normal[SparseArray[# -> 1 & /@ 
      Table[{j, Position[pp[[k]], aa[[j]]][[1, 1]]}, {j, 1, 
        Length[aa]}]]], {k, 1, Length[pp]}];

 (* find matrix-elements for g[ i ]* g[k]  *)

rr[k_] := Module[{},
  tt = Table[
    Position[res, res[[k]].res[[j]]][[1, 1]], {j, 1, Length[res]}];
  mm = Normal[SparseArray[# -> 1 & /@ Transpose[{Range[Length[res]], tt}]]] ]  

rr /@ Range[Length[res]];
MatrixForm /@ %

Or more compact

n = 3;
ge = Permutations[Range[n]]  (* the elements of S3  *)

(* Multiplication of group-elements *)

gp[a_, b_] := Module[{},
gL = Length[a];
Table[a[[b[[j]]]], {j, 1, gL}]]   

(* Make Matrix for regular representation *)

gm[a_] := Module[{},
  gLg = Length[ge];
  mm = Table[0, {i, gLg}, {j, gLg}];
  j = 1;
  While[
   j <= gLg,
   gxg = gp[a, ge[[j]]];
   mm[[j, Position[ge, gxg][[1, 1]]]] = 1;
   j = j + 1
   ];
  mm
  ]

And finally

MatrixForm /@ (gm /@ ge)
Posted 1 month ago

THanks again. This looks very useful. Where are you finding all this good stuff?

Hmmm, once I used to program a lot concerning group theory. One of the problems was enumerating substitutional isomers of a given molecule. See J.Chem.Inf.Comput.Sci. 2000, 40, 956-966. In fact I cannot say "where to find this stuff", I just remembered what I was doing then.

Kind regards, HD

And if you are interested: S3 is isomorphic to C3v, and its regular representation R is reduced according to

R = A1 + A2 + 2 E

Posted 1 month ago

My question "where do you find..." was using the "ask first think later" protocol. " which is always dumb. I am using S3 as an example in a pedagogical paper on identical particles in quantum mechanics. After looking at your codes I finally realized that the Permutations function does all I need for my example. Thanks again for your help

example in a pedagogical paper on identical particles in quantum mechanics

That sounds interesting. Would you mind to send me a copy of your paper?

h.dolhaine@gmx.de

Posted 1 month ago

I will certainly send you a copy when I have it ready. However, on each iteration I have been cutting back on the group theoretical details in order to make the paper more useful to those teaching lower-level courses in quantum mechanics.

I would like to forward a text concernig atomic orbitals to you (but for the time being it is in german). Would you give me your email?

h.dolhaine@gmx.de

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