I wanted to know if there is a package which allows to compute representations of a group like the definition representation, adjoint and so on (for example the Pauli matrix for $SU(2)$ if I specify that I want the definition representation of $SU(2)$). Which allow also to compute the structure constants of a particular group and, given a set of matrix, to test if these matrix form a representation of a particular group.

  • 1
    $\begingroup$ Have you looked at the guide here? $\endgroup$
    – Jens
    Apr 11, 2014 at 16:33
  • $\begingroup$ Perhaps related: How to generate a matrix group?. This doesn't answer the question whether there exists a separate package, though. $\endgroup$
    – Jens
    Apr 11, 2014 at 16:42
  • $\begingroup$ @Jens: for the symmetric group, I miss the character table and the link to partitions and Young Tableaux. $\endgroup$
    – Wouter
    Apr 11, 2014 at 18:46
  • $\begingroup$ @Wouter Your comment reminded me to look into the Cmobinatorica package - so I now added that as an answer. $\endgroup$
    – Jens
    Apr 11, 2014 at 21:49
  • 2
    $\begingroup$ Thank you for your feedback. The "How to generate a matrix group" question is really interesting, I'm strongly thinking to implement it so thank you for that. But for your other answers, it's not really what I want... I'm working with unitary matrix such as $SU(n)$ but in Mathematica there is no command for example to compute the structure constants of the Lie algebra corresponding to $SU(n)$ neither in Combinatorica. I found a package called LieART which does things close to what I want to do but with a little work behind but it seems promising to me :) However thank you ! $\endgroup$
    – KoObO
    Apr 14, 2014 at 14:43

2 Answers 2

  1. I presume you mean the representation of the generators of su(2) (i.e. of the algebra) rather than a presentation of the group.

  2. Representations of the SU(2) group in any dimension can be obtained from WignerD. Something like

    SU2repj = Table[
     WignerD[{j, m1, m2}, a, b, c],
     {m1, j, -j, -1}, {m2, j, -j, -1}
    ] /. j -> 2

    will generate the $5\times5$ matrix for angular momentum $j=2$.

  3. A representation of the algebra in the same dimension as that of the group can be obtained by taking the derivative w/r to the appropriate parameter of the group matrix element, and setting all other group parameters to 0, i.e.

    Lymatj = -i D[SU2repj, b] /. {a -> 0, b -> 0, c -> 0}
    • This would only give you representations of $L_z$ and $L_y$ since the usual parametrization of SU(2) matrices in terms of Euler angles contains exponentials of $L_z$ and $L_y$ only. $L_x$ can be obtained by commutation of the matrices for $L_y$ and $L_z$, respectively.
  4. I am not aware of Mathematica codes for systematically computing matrix elements of generators for other groups, be they unitary or otherwise. For the special unitary groups there is a method based on Gelf'and-Tsetlin patterns, but as far as I know it has not been implemented in Mathematica. Searching for "Gelf'and Tsetlin bases" usually results in multiple hits describing the G-T algorithm.

Edit: there is a package called LieART by Robert Feger and Thomas W. Kephart available on arxiv as https://arxiv.org/abs/1206.6379

  • $\begingroup$ Thanks Oleksandr R. for editing my post correctly. Somehow I couldn't get the tabs to work (Chrome on Mac OS X) and go to the correct "code" mode. $\endgroup$
    – user14281
    May 10, 2014 at 23:27
  • $\begingroup$ @Jens: beware as the built-in D does not agree in its parametrization with the more common parametrization of the canonical text by Varshalovich et al. $\endgroup$
    – user14281
    Dec 25, 2016 at 17:26
  • $\begingroup$ I know this may be a little late, but it is worth noting that there is now a new and improved version of LieART out by Robert Feger, Thomas Kephart, and Robert Saskowski $\endgroup$
    – arow257
    Aug 10, 2020 at 21:02

There is an add-on package that's worth mentioning in this context, mainly to address Wouter's comment: in the Combinatorica package, you find some group-theory related commands that are not part of the System context to which the linked guide refers.

One of them is ConstructTabelau, addressing the comment by Wouter. Sometimes the built-in gems are hard to find, like WignerD for the representations of the rotation group.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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