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LieART -- A Mathematica Application for Lie Algebras and Representation Theory

Robert Feger, Thomas W. Kephart

arXiv:1206.6379v2math-phhep-phhep-th

TL;DR

LieART addresses the need for Mathematica-based Lie-algebra representation computations used in particle physics and GUT model building. It implements tensor products, branching, weight-system calculations, and related operations across classical and exceptional algebras, using Weyl reflections and physicist-friendly notation. Running time ranges from less than a second to hours depending on algebra and representation complexity, while the package also provides extensive tables.

  • Problem

    Particle-physics and GUT model building require tensor products, branching, and related representation computations, while existing tables do not cover every needed result and no comparable Mathematica implementation existed when the project began.

  • Method

    LieART uses Weyl reflection groups for weight-system and related computations, Klimyk’s formula or Young tableaux for tensor products, and projection matrices for subalgebra decomposition.

  • Results

    LieART provides computations for irreps of all classical and exceptional Lie algebras, with dimensional-name input, textbook-style output, and extensive tables of properties, tensor products, and branching rules.

  • Takeaways & Limitations

    LieART brings fast, memory-efficient Lie-algebra representation computations and extended lookup tables to Mathematica for particle-physics applications.

  • Takeaways & Limitations

    In rare high-dimensional, high-rank cases, an insufficient maximum Dynkin digit can assign an incorrect prime to a displayed dimensional name, although computations remain correct.

Abstract

from arXiv · show

We present the Mathematica application LieART (Lie Algebras and Representation Theory) for computations frequently encountered in Lie Algebras and representation theory, such as tensor product decomposition and subalgebra branching of irreducible representations. LieART can handle all classical and exceptional Lie algebras. It computes root systems of Lie algebras, weight systems and several other properties of irreducible representations. LieART's user interface has been created with a strong focus on usability and thus allows the input of irreducible representations via their dimensional name, while the output is in the textbook style used in most particle-physics publications. The unique Dynkin labels of irreducible representations are used internally and can also be used for input and output. LieART exploits the Weyl reflection group for most of the calculations, resulting in fast computations and a low memory consumption. Extensive tables of properties, tensor products and branching rules of irreducible representations are included in the appendix.

Program Summary

LieART addresses common Lie-algebra representation computations in physics, using Weyl reflections and specialized decomposition algorithms across classical and exceptional algebras. It prioritizes physicist-friendly input and output while retaining Dynkin labels internally.

  • Availability: LieART is distributed under the GNU Lesser General Public License and implemented in Mathematica for cross-platform systems.The listed supported computer architectures include x86, x86_64, and PowerPC.
  • Purpose: LieART supports tensor-product decomposition and subalgebra decomposition for irreducible representations relevant to gauge theories and GUT model building.These tasks arise from Lagrangian mass and interaction terms and spontaneous symmetry breaking.
  • Solution method: Weight systems are generated using Weyl reflection groups, while tensor products use Klimyk’s formula except for SU(N), which uses Young tableaux.SU(N) subalgebra decompositions use projection matrices generated from maximal-subalgebra algorithms.
  • Representation handling: Irreps are internally represented by unique Dynkin labels, while TraditionalForm displays dimensional names preferred by physicists.Dimensional names can require primes when multiple irreps share a dimension.
  • Performance: Running time ranges from less than a second to hours depending on the algebra’s rank and type and the representations’ dimensionality.Memory usage likewise depends strongly on the algebra and representation dimensionality.

1. Introduction

LieART extends existing Lie-algebra tables and algorithms with a Mathematica implementation for representation computations used in particle physics and GUT model building. It combines broad algebraic coverage with physicist-oriented notation and extensive lookup tables.

  • Motivation: Lie groups and their representations are used across particle physics, condensed matter, gravity, and string theory.The classical groups and exceptional groups have all appeared with varying frequency in particle physics.
  • Motivation: Existing tables support GUT construction, but the authors note that model building sometimes requires results beyond what is tabulated.The paper therefore provides extended tables and software for further computations.
  • Related work: At project inception, no comparable implementation existed for Mathematica, despite multiple computer implementations in other languages.The authors position LieART as a Mathematica platform for established representation-theory algorithms.
  • Contribution: LieART makes tensor products and subalgebra branching available for classical and exceptional Lie algebras in Mathematica.It originated as an in-house solution for computerized SU(N) grand-unified-model scans.
  • Contribution: The application uses Weyl reflection groups for fast, memory-economical computations and supports dimensional-name input with textbook-style output.Dynkin labels remain available for internal representation and optional input or output.
  • Tables: The appendix supplies extended tables of representation properties, tensor products, and branching rules that can also be used without LieART.The authors plan to maintain and further extend these tables.

2. Download and Installation

LieART is downloaded as a tar.gz archive and installed through Mathematica’s application mechanism or manually when automatic installation is insufficient. Its documentation and tables are integrated into Mathematica’s help system.

  • Download: The LieART Mathematica application is hosted by Hepforge and distributed as a tar.gz archive.The project home page provides the download location.
  • Installation: Automatic installation uses Mathematica’s front end, selecting Application as the item type and the archive as the source.Users can choose an individual-user or system-wide installation.
  • Manual installation: Mathematica 7 automatic installation installs only LieART.m without documentation, so manual installation is suggested when documentation integration is needed.The same manual procedure is suggested for Mathematica 8–10 if automatic installation problems occur.
  • Manual installation: Manual installation places LieART in the appropriate AddOns/Applications directory and requires restarting Mathematica for help integration.The location differs between user-only and system-wide installations.
  • Documentation: LieART documentation includes a Quick Start Tutorial, function descriptions, and tables of representation properties, tensor products, and branching rules.The documentation is accessible through Mathematica’s help system or by searching for LieART.
  • LaTeX support: The package includes a LaTeX style file for displaying irreps, roots, and weights properly in publications.The file lieart.sty is located in the project tree’s latex/ subdirectory.

3. Quick Start

The quick start introduces LieART’s representation syntax, tensor-product decomposition, subalgebra branching, and Young-tableau tools through examples. Irreps can use Dynkin labels or dimensional names, with results displayed in physicist-friendly notation.

  • Representation input: LieART represents irreps internally by Dynkin labels and can also accept dimensional names when the algebra is specified.The algebra is required because dimensional names can be ambiguous across classical algebras.
  • Representation input: Dynkin-label input uses an algebra class and label, with classical classes A, B, C, and D and exceptional classes E6, E7, E8, F4, and G2.For classical algebras, the label length determines the precise algebra.
  • Output: TraditionalForm displays dimensional names by default, whereas StandardForm displays textbook Dynkin labels.The output format can be changed through $DefaultOutputForm.
  • Product algebras: ProductIrrep gathers irreps of simple factors for product algebras such as SU(3)⊗SU(2)⊗U(1).Individual factors are supplied as arguments to ProductIrrep.
  • Tensor products: DecomposeProduct decomposes tensor products of several irreps, including products entered through Mathematica’s product and power syntax.Examples include SU(4) products and 27⊗27⊗27 of E6, whose output is 1 + 2(78) + 3(650) + 2925 + 3003 + 2(5824).
  • Tensor products: Product irreps can also be decomposed componentwise, as illustrated for products of SU(3) and SU(2) representations.The package represents sums internally with IrrepPlus and IrrepTimes and can convert results to lists using IrrepList.
  • Subalgebra decomposition: DecomposeIrrep branches an irrep to a specified subalgebra, and its Listable form handles multiple irreps together.The examples include SU(5) representations branching to SU(3)⊗SU(2)⊗U(1).
  • Subalgebra decomposition: BasicIrreps[SU6] produces branching rules for all totally antisymmetric SU(6) irreps to SU(3)⊗SU(3)⊗U(1).The displayed results include the 6, 15, and 20 representations with their associated charges.

4. Theoretical Background and Implementation

LieART implements Lie-algebra theory through roots, weights, Weyl-group orbits, Dynkin classifications, and basis transformations, supporting representation computations across classical and exceptional algebras.

  • LieART provides a self-contained theoretical and implementation overview covering algebra properties, roots, weights, Weyl orbits, representations, and decompositions.The section introduces relevant functions alongside the theory they implement.
  • Algebra structure: The rank equals the number of simultaneously diagonalizable, mutually commuting Cartan generators and is available through Rank[expr].The remaining generators satisfy eigenvalue equations whose eigenvalues form root vectors.
  • Roots and Dynkin diagrams: Simple roots are the indecomposable positive roots, and their number equals the rank; specifying them determines the Lie algebra.Dynkin diagrams encode relations among simple roots, including their lengths and angles.
  • Classification: Simple Lie algebras comprise four classical families, A_n, B_n, C_n, and D_n, plus five exceptional algebras, E6, E7, E8, F4, and G2.LieART represents classical algebras as Algebra[algebraClass][n] and exceptional algebras by their named labels.
  • Bases: LieART uses orthogonal coordinates because the Weyl reflection group acts naturally on the root space, while also supporting α- and ω-bases.The Cartan matrix mediates between the simple-root α-basis and fundamental-weight ω-basis; orthogonal-space dimension m need not equal rank n.
  • Weight systems: The weight-system algorithm groups weights into Weyl orbits represented by unique dominant weights, assigning each orbit a common multiplicity.This Weyl-group approach is presented as a performance-oriented alternative to recursively subtracting simple roots and computing multiplicities directly.

Index.

LieART represents irreducible representations through Dynkin labels while providing dimensional names, indices, congruency classes, and algorithms for decomposition and representation-name assignment. Its naming procedure distinguishes equal-dimensional irreps but is constrained by brute-force search limits and can affect computational efficiency.

  • Representation properties: The dimension, index, and congruency class help distinguish irreps that share a dimensional name.Congruency classes generalize SU(N) n-ality and are implemented through CongruencyClass[irrep].
  • Representation names: Dynkin labels uniquely specify an irrep together with its Lie algebra, while dimensional names provide shorter physicist-oriented labels.LieART internally uses unique Dynkin labels and can represent them through Irrep[algebraClass][label].
  • Representation names: LieART assigns primes to unrelated equal-dimensional irreps by sorting them according to ascending index.Related irreps share the same index and are instead distinguished using conjugation or SO(8) symmetry conventions.
  • Representation names: Conjugate complex irreps are labeled with overbars according to their congruency classes, while primes may also appear when equal-dimensional irreps are unrelated.SO(8) requires additional distinctions because its symmetry can produce more than two related irreps of the same dimension.
  • Naming limitations: Prime determination uses brute-force generation of same-dimensional irreps and is limited by the maximum Dynkin digit searched.A larger Dynkin-digit bound can improve coverage but rapidly increases generated labels, runtime, and memory use.
  • Naming limitations: Increasing $MaxDynkinDigit from 3 to 4 resolves the failure to identify the 70′ of SU(5).The error indicates either that the dimensional name is absent or that a required Dynkin digit exceeds the search bound.
  • Tensor products: Klimyk’s formula assigns multiplicities through weight multiplicities and Weyl signs, yielding the stated decomposition 8 + 10.The worked example shows the unit contribution cancels when the terms are summed.
  • Tensor products: SU(N) tensor products use Young tableaux because the algorithm performs better in CPU time and memory consumption than the generic procedure.Other classical and exceptional algebras use weight addition followed by filtering of irreps.

5. Benchmarks

The benchmarks measure tensor-product and subalgebra decompositions, emphasizing the greater CPU and memory demands of exceptional algebras. Timings include Mathematica kernel computation but exclude front-end display time.

  • Exceptional algebras require more CPU and memory than classical algebras of equal rank because their Weyl reflection groups have high orders.
  • Benchmark timings use Mathematica’s Timing[] and exclude the time required to display results in the front end.Dimensional-name determination can itself be time-consuming and is included in display-related overhead rather than kernel computation.
  • E6 tensor products: 27^n decompositions in E6 were benchmarked for n = 2, . . . , 8, with reported runtimes ranging from 0.143171 to 1.53349 seconds.
  • E6 tensor products: 78^n decompositions in E6 were benchmarked for n = 2, . . . , 7, with reported runtimes ranging from 12.9962 to 93.8079 seconds.
  • E8 tensor products: 248^n decompositions in E8 were benchmarked for n = 2, . . . , 7, with reported runtimes ranging from 31.9855 to 7636.29 seconds.
  • Subalgebra decomposition: The 600600 representation of E6 was decomposed into SU(3) × SU(3) × SU(3) as an example of subalgebra decomposition of a large irrep.

6. LATEX Package

The supplemental LieART LaTeX package formats irreps, roots, and weights for publication-style output. Its commands cover dimensional names, conjugates, primes, SO(8) subscripts, Dynkin labels, roots, and weights.

  • LieART’s lieart.sty package defines LaTeX commands for properly displaying irreps, roots, and weights.
  • Table 6.1 documents commands for dimensional names, conjugated irreps, prime counts, SO(8) subscripts, Dynkin labels, roots, and weights.

7. Conclusions and Outlook

LieART brings Lie-algebra and representation-theory computations to Mathematica, combining decomposition functions, Weyl-group algorithms, usability features, and extensive tabulated data. Future work targets additional branching rules, while large tables remain computationally demanding.

  • LieART provides Mathematica functions for tensor-product decomposition and irreducible-representation branching rules across classical and exceptional Lie algebras.
  • Weyl reflection groups make many LieART applications fast and memory efficient, while dimensional-name input and textbook-style output emphasize usability.
  • LieART reproduces and extends tabulated data on irreducible representations, tensor products, and branching rules.
  • Only a selection of common branching rules is currently implemented, and future versions are planned to add more.
  • Tables for high-rank algebras or higher-dimensional irreducible representations require large CPU time and high memory consumption.

A. Tables

The appendix provides extensive tables of representation properties, tensor products, and subalgebra branching rules for many classical and all exceptional Lie algebras. The tables are generated with LieART and can be extended by users.

  • Representation properties: Section A.1 lists Dynkin labels, dimensional names, indices, congruency classes, and singlet counts in subalgebra branchings.
  • Tensor products and branching: Sections A.2 and A.3 provide tensor-product tables and subalgebra branching rules for many classical and all exceptional Lie algebras.
  • Table generation: The supplemental Tables.m package created the appendix tables, which are also available as Mathematica notebooks in LieART’s integrated documentation.
  • Table generation: Users can extend the tables through LieART’s table-generating functions up to the limits of their computer power.
  • Table A.1 is identified as a table of tables.

A.1. Properties of Irreducible Representations

This section provides appendix tables of irreducible representations for a range of SU and SO algebras, including several continued tables.

  • Tables A.2–A.12 list irreducible representations for SU(2) through SU(12).
  • Several entries are explicitly marked as continued tables, indicating that their representation listings span multiple table segments.
  • Tables A.13–A.22 list irreducible representations for SO(7) through SO(14), SO(18), and SO(22).
  • Tables A.23 and A.25 provide irreducible-representation tables for SO(26) and related higher-rank SO algebras.

A.1.3. Sp(N)

This section provides tables of irreducible representations for the symplectic algebras Sp(4) through Sp(14). Several algebras use continuation tables to accommodate the listed representations.

  • Sp(N): Tables A.24 and A.25 list irreducible representations for Sp(4) and Sp(6), with continuation tables for both algebras.

A.1.4. Exceptional Algebras

This section provides irreducible-representation tables for the exceptional algebras E6, E7, E8, F4, and G2. F4 and G2 each have continuation tables.

  • Exceptional algebras: Tables A.30–A.32 list irreducible representations for E6, E7, and E8.
  • Exceptional algebras: Tables A.33 and A.34 list irreducible representations for F4 and G2, respectively, with continuation tables for both.

A.2. Tensor Products

This appendix section supplies tensor-product tables for classical and exceptional algebras, alongside explicit decompositions for selected products. The entries enumerate product representations and their decomposed components.

  • Explicit decompositions: The explicit entries include decompositions such as 105 × 21 = 189 + 280 + 840′′ + 896 and 160v × 160v = 1 + 2(28) + 2(35v) + 35c + 35s + 294v + 2(300) + 3(350) + 3(567v) + 567c + 567s + 840′ s + 840′.
  • SU(N) tensor products: Tables A.40–A.45 provide tensor products for SU(7) through SU(12).
  • Sp(N) tensor products: Tables A.57–A.62 list tensor products for Sp(4) through Sp(14).
  • Exceptional-algebra tensor products: Tables A.63–A.67 provide tensor products for E6, E7, E8, F4, and G2.

A.3. Branching Rules

This section compiles branching rules for classical groups from SU(2) through SU(12) and SO(7) through SO(26), alongside exceptional groups E6, E7, and E8.

  • The appendix lists branching-rule tables for SU(2)–SU(12).
  • The displayed entries express higher-dimensional representations as sums of component representations, including examples such as 9 = 1 + 8v and 16 = 8s + 8c.
  • Additional entries expand dimensions such as 2772, 3900, 4004, 4158, and 4608 into sums of labeled representations.
  • Exceptional-group branching rules are tabulated for E6, E7, and E8.
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