# Representation of symmetric group

Posted 2 years 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?
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Posted 2 years ago
 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 2 years 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..
Posted 2 years ago
 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 /@ % 
Posted 2 years ago
 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 2 years ago
 THanks again. This looks very useful. Where are you finding all this good stuff?
Posted 2 years ago
 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
Posted 2 years ago
 And if you are interested: S3 is isomorphic to C3v, and its regular representation R is reduced according toR = A1 + A2 + 2 E
Posted 2 years 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
Posted 2 years ago
 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 2 years 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.
Posted 2 years ago
 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