LieACh

Package that extends LieART for the use of 1-dimensional representations of Lie algebras, i.e. characters. It is able to find the decomposition of 1-dim representations into characters of irreducible representations.

Installation

The following function and commands will automatically download the latest release of LieACh (and LieART) and install it:

updateLieACh[] := Module[{json, entry, download, target},
  Check[json = Import["https://api.github.com/repos/jose-a-sa/LieACh/releases", "JSON"];
    entry = Last@SortBy[json, DateObject@*Lookup["published_at"]];
    download = Lookup[First@Lookup[entry, "assets"], "browser_download_url"];
    target = FileNameJoin[{CreateDirectory[], "LieACh.paclet"}];
    If[$Notebooks, PrintTemporary@Labeled[ProgressIndicator[Appearance -> "Necklace"], 
      "Downloading LieACh...", Right], Print["Downloading LieACh..."]];
    URLSave[download, target], Return[$Failed]
  ];
  If[FileExistsQ[target], PacletManager`PacletInstall[target, ForceVersionInstall->True], $Failed]
];
updateLieACh[]
Needs["LieACh`"]

Usage

While using functions in this extension we need to specify the algebra we are working with. We use the same definitions of Algebra and AlgebraClass in LieART. For example, A2 or equivalently Algebra[A][2] is the complexified $\mathfrak{su}(3)$.

There's no need to install or load LieART before using the package. When using in Mathematica

<< LieACh`

the package will check for LieART among the installed packages, and will load it if found. Otherwise, it will automatically be downloaded from the official website. Once LieART is imported, LieACh modifies some of the internal definitions of LieART to better work with the product algebras, product irreps and product weights.

Description

Many computations in the theoretical physics and field theory lead to large polynomial expressions that help count states and/or operators. These states/operators are organized into the character semisimple representations of a Lie algebra $\mathfrak{g}$, i.e. as a formal sum of irreps (irreducible representations). In terms of characters, this can be written as $$\chi(z_1, \dots, z_r) = \sum_{\lambda\in I} m_{\lambda} \chi_{\lambda}(z_1, \dots, z_r) ~~.$$ Here we define:

• $\chi$ the full character and $\chi_{\lambda}$ the character of an irrep $R_{\lambda}$
• $r = \mathrm{rank}(\mathfrak{g})$ as the rank of the Lie algebra
• $\lambda$ is the Dinkin label that represents the highest weight of the representation $R_{\lambda}$
• $m_{\lambda} \in \mathbb{Z}$ is the multiplicity of the irrep $R_{\lambda}$

This package allows to compute characters of irreps of Lie algebras using the WeightSystem function from the LieART package. Additionally, defines a weak ordering function to help sort dominant weights of irreps and obtain the highest weights in a expression. Finally, and more importantly, CharacterDecomposition[g][expr, {vars..}] produces an Association encoding key-value pairs of $(\lambda, m_\lambda)$ for a given character $\chi$ of any semisimple representation of the algebra g.

References

• Feger, R., Kephart, T. W., & Saskowski, R. J. (2019). LieART 2.0 -- A Mathematica Application for Lie Algebras and Representation Theory. ArXiv. https://doi.org/10.1016/j.cpc.2020.107490
• Fuchs Jürgen & Schweigert C. (1997). Symmetries lie algebras and representations : a graduate course for physicists. Cambridge University Press.