atlas
0.6

Table of Contents

  • General Information
    • About Us
    • About the Software
  • Getting Started
    • Download and Install
      • Linux (and Solaris)
        • Compiling the Sofware from Source
        • Using Docker
        • File Input and Output in Docker
      • Mac
        • Compiling the Sofware from Source
        • Using Docker
        • File Input and Output in Docker
      • Windows
        • Using Docker
        • Compiling the Sofware from Source
    • Run atlas
      • Get atlas running
      • Load scripts
      • Quit atlas
    • Help with git
      • Primary commands
      • Branches
      • Other helpful commands
      • Getting unstuck
      • More on git objects
      • Tagging and branches
      • Git references
  • Tutorial
    • Preliminaries
      • Getting started using the software
        • The basic.at file
      • Help files
        • atlas.help
        • atlas-functions.help
        • scripts.help
      • Basic Commands
        • Helpful unix tools
        • Basic atlas Operations
        • n-tuples
      • Vectors
        • Operations and coordinates
      • Matrices
        • Operations on Matrices
        • Basic Linear Algebra operations
      • Functions and Loops
        • Arguments and Outputs
        • Loops
        • Functions defined using loops
      • Rational Matrices
      • Local Variables
        • The ; syntax
        • Use of ; and local variables in for statements
    • Structure Theory
      • Root Data and Lie Types
      • Real Groups
      • atlas choice of coordinates
      • Change of Coordinates
      • Cartan Subgroups
      • Real Forms
        • Cartan Classes Accross Real Forms
        • Inner Classes and Outer Involutions
    • Coordinates and Characters
      • Occurrence Matrices
      • Parameters
        • Introduction
        • Parameters for \(SL(2,\mathbb R)\)
      • Parameters for Real Tori
        • The characters of \(S^1\)
        • The characters of \({\mathbb R}^{\times}\)
        • Characters of \({\mathbb C}^{\times}\).
      • The Character Differential
      • Trivial Representation of \(SL(2,R)\)
    • First Examples of Representations.
      • Introduction
      • Example \(G=SL(2,\mathbb R)\).
      • Example \(G=PSL(2,\mathbb R)\)
      • Example \(G=Sp(4,\mathbb R)\)
      • Example \(G=SO(3,2)\)
      • More parameter commands
        • Principal series commands
        • all_parameters_gamma
        • all_parameters
      • Cuspidal Data for Representations
      • Lowest \(K\)-types of a Representation
      • Representations with Zero Infinitesimal Character
      • Translation Principle
      • Blocks of Representations
    • \(K\) orbits on \(G/B\).
      • Introduction on \(K\) orbits on \(G/B\)
        • \(KGB\) elements
        • Parametrization Theorem
        • Parameter Set
      • Example: \(SL(2,\mathbb R)\)
        • Moral of the Story
      • More on \(K\backslash G/B\)
        • Effect on \((\mathfrak g , K_x)\) modules
        • Involutions and Conjugacy classes of Cartan subgroups
        • KGB ordering
      • \(K\backslash G/B\) for other Cartan subgroups
    • Discrete Series
      • Background
      • Example \(SL(2,R)\)
      • Example \(Sp(4,R)\)
      • Example \(Sp(6,\mathbb R)\)
    • General Parameters
      • Introduction
        • Composition series and character formulas
      • Principal Series and Discrete Series revisited
      • General Parameters \(Sp(4,\mathbb R )\)
        • Cuspidal Data
    • Parabolic Induction
      • Real Parabolic Induction
        • Defining a Real Parabolic Subgroup
        • Real Induction
      • Cohomological Parabolic Induction
        • Defining a \(\theta\)-Stable Parabolic Subalgebra
        • Theta-Stable Induction
      • \(A_{\mathfrak q}(\lambda)\) Construction
  • atlas Library
    • A Brief Introduction
    • The Axis Language
      • Design Principles
      • Types
        • Introduction
        • Primitive Types
        • Composite Types
      • Array
        • Array Displays
        • Array Selection
        • Array Slicing
        • Other Array-Like Types
      • Tuple Displays and Pattern Matching
      • Implicit Conversions
      • Defining and Modifying Identifiers
      • Defining Functions and Operators
      • Overloading v.s. Redefining, and Conflicts
      • Control Structures
        • Sequenced expressions
        • Conditional expressions and case expressions
        • Loops
      • Recursion
      • Splitting Commands
      • Output, Input, and Redirection
      • Miscellaneous Commands
      • Syntax Summary
        • Command level
        • Quaternary expressions: ‘let’, anonymous functions, casts, sequences
        • Tertiary expressions: assignments
        • Secondary expressions: formulas
        • Primary expressions: subscriptions, function calls, atomic expressions
        • Closed expressions: displays and groupings, conditionals, loops
        • Identifier patterns
        • Types
    • Basic Operators
    • Built-In atlas Functions
      • Basic Functions
        • flex
        • convolve
        • stack_rows
        • error
        • int_format
        • to_string
        • print
        • prints
        • ascii
        • ascii
        • null
        • null
      • Matrix Manipulating Functions
        • id_mat
        • diagonal
        • swiss_matix_knife
        • echelon
        • diagonalize
        • adapted_basis
        • kernel
        • eigen_lattice
        • row_saturate
        • inv_fact
        • Smith_basis
        • Smith
        • invert
        • mod2_section
        • subspace_normal
      • Lie Group Basics
        • Lie_type
        • Lie_type
        • Cartan_matrix
        • Cartan_matrix
        • Cartan_matrix_type
        • rank
        • semisimple_rank
        • #
        • %
        • Smith_Cartan
        • filter_units
        • ann_mod
        • replace_gen
        • involution
        • involution
        • root_datum
        • root_datum
        • root_datum
        • root_datum
        • quotient_basis
        • simply_connected
        • adjoint
        • dual
        • derived_info
        • mod_central_torus_info
        • rank
        • semisimple_rank
      • Roots and Weights
        • simple_roots
        • simple_coroots
        • posroots
        • poscoroots
        • roots
        • coroots
        • root_coradical
        • coroot_radical
        • fundamental_weight
        • fundamental_coweight
        • nr_of_posroots
        • root_index
        • coroot_index
        • integrality_datum
        • integrality_points
      • KGB Related
        • KGB
        • %
        • Cartan_class
        • involution
        • length
        • status
        • status
        • cross
        • Cayley
        • twist
        • torus_factor
        • torus_bits
        • KGB_elt
        • =
        • block
        • %
        • #
        • element
        • index
        • dual
        • status
      • Parameter, KL, print, etc.
        • param
        • %
        • real_form
        • infinitesimal_character
        • is_standard
        • is_zero
        • is_final
        • dominant
        • =
        • cross
        • Cayley
        • inv_Cayley
        • cross
        • Cayley
        • twist
        • orientation_nr
        • reducibility_points
        • print_block
        • block
        • partial_block
        • length
        • KL_block
        • partial_KL_block
      • More Advanced Functions
        • classify_involution
        • inner_class
        • twisted_involution
        • inner_class
        • inner_class
        • =
        • nr_of_real_forms
        • nr_of_dual_real_forms
        • nr_of_Cartan_classes
        • block_sizes
        • form_names
        • dual_form_names
        • occurrence_matrix
        • dual_occurrence_matrix
        • real_form
        • form_number
        • quasisplit_form
        • components_rank
        • count_Cartans
        • =
        • KGB_size
        • base_grading_vector
        • Cartan_order
        • dual_real_form
        • dual_quasisplit_form
        • Cartan_class
        • Cartan_class
        • most_split_Cartan
        • involution
        • Cartan_info
        • real_forms
        • dual_real_forms
        • square_classes
        • fiber_partition
        • real_form
        • central_fiber
        • initial_torus_bits
    • User Defined Functions
      • 2i12.at
        • 2i12.at Function References
        • 2i12.at Function Index
      • 2i12s.at
        • 2i12s.at Function References
        • 2i12s.at Function Index
      • 2r21s.at
        • 2r21s.at Function References
        • 2r21s.at Function Index
      • A1.at
        • A1.at Function References
        • A1.at Function Index
      • all.at
        • all.at Function References
        • all.at Function Index
      • all_parameters.at
        • all_parameters.at Function References
        • all_parameters.at Function Index
      • aql.at
        • aql.at Function References
        • aql.at Function Index
      • basic.at
        • basic.at Function References
        • basic.at Function Index
      • bigMatrices.at
        • bigMatrices.at Function References
        • bigMatrices.at Function Index
      • center.at
        • center.at Function References
        • center.at Function Index
      • coherent.at
        • coherent.at Function References
        • coherent.at Function Index
      • complementary_series.at
        • complementary_series.at Function References
        • complementary_series.at Function Index
      • complex.at
        • complex.at Function References
        • complex.at Function Index
      • complex_nilpotent_orbits.at
        • complex_nilpotent_orbits.at Function References
        • complex_nilpotent_orbits.at Function Index
      • conjugacy_classes.at
        • conjugacy_classes.at Function References
        • conjugacy_classes.at Function Index
      • convert_c_form.at
        • convert_c_form.at Function References
        • convert_c_form.at Function Index
      • coordinates.at
        • coordinates.at Function References
        • coordinates.at Function Index
      • coxeter.at
        • coxeter.at Function References
        • coxeter.at Function Index
      • cross_W_orbit.at
        • cross_W_orbit.at Function References
        • cross_W_orbit.at Function Index
      • cyclotomic.at
      • deform.at
        • deform.at Function References
        • deform.at Function Index
      • dual.at
        • dual.at Function References
        • dual.at Function Index
      • elliptic_elements.at
        • elliptic_elements.at Function References
        • elliptic_elements.at Function Index
      • ext_deform.at
        • ext_deform.at Function References
        • ext_deform.at Function Index
      • ext_signs.at
        • ext_signs.at Function References
        • ext_signs.at Function Index
      • extended.at
        • extended.at Function References
        • extended.at Function Index
      • extended_cayley.at
        • extended_cayley.at Function References
        • extended_cayley.at Function Index
      • extended_cross.at
        • extended_cross.at Function References
        • extended_cross.at Function Index
      • extended_misc.at
        • extended_misc.at Function References
        • extended_misc.at Function Index
      • extended_types.at
        • extended_types.at Function References
        • extended_types.at Function Index
      • extParamPol.at
        • extParamPol.at Function References
        • extParamPol.at Function Index
      • finite_dimensional.at
        • finite_dimensional.at Function References
        • finite_dimensional.at Function Index
      • finite_dimensional_signature.at
        • finite_dimensional_signature.at Function References
        • finite_dimensional_signature.at Function Index
      • galois.at
        • galois.at Function References
        • galois.at Function Index
      • generate_groups.at
        • generate_groups.at Function References
        • generate_groups.at Function Index
      • genuine.at
        • genuine.at Function References
        • genuine.at Function Index
      • gl4H.at
        • gl4H.at Function References
        • gl4H.at Function Index
      • group_operations.at
        • group_operations.at Function References
        • group_operations.at Function Index
      • groups.at
        • groups.at Function References
        • groups.at Function Index
      • hecke.at
        • hecke.at Function References
        • hecke.at Function Index
      • hermitian.at
        • hermitian.at Function References
        • hermitian.at Function Index
      • hermitian_debug.at
        • hermitian_debug.at Function References
        • hermitian_debug.at Function Index
      • induction.at
        • Parabolic induction:
        • \(A_q(\lambda)\) construction:
        • Good/Fair conditions:
      • induction_sp4.at
        • induction_sp4.at Function References
        • induction_sp4.at Function Index
      • inverse.at
        • inverse.at Function References
        • inverse.at Function Index
      • iterate_deform.at
        • iterate_deform.at Function References
        • iterate_deform.at Function Index
      • jantzen.at
        • jantzen.at Function References
        • jantzen.at Function Index
      • K.at
        • K.at Function References
        • K.at Function Index
      • K_highest_weights.at
        • K_highest_weights.at Function References
        • K_highest_weights.at Function Index
      • K_types.at
        • K_types.at Function References
        • K_types.at Function Index
      • kl.at
        • kl.at Function References
        • kl.at Function Index
      • KL_polynomial_matrices.at
        • KL_polynomial_matrices.at Function References
        • KL_polynomial_matrices.at Function Index
      • lattice.at
        • lattice.at Function References
        • lattice.at Function Index
      • lietypes.at
        • lietypes.at Function References
        • lietypes.at Function Index
      • matrix.at
        • matrix.at Function References
        • matrix.at Function Index
      • misc.at
        • misc.at Function References
        • misc.at Function Index
      • modules.at
        • modules.at Function References
        • modules.at Function Index
      • new_blocks.at
        • new_blocks.at Function References
        • new_blocks.at Function Index
      • nilpotent.at
        • nilpotent.at Function References
        • nilpotent.at Function Index
      • nonintegral.at
        • nonintegral.at Function References
        • nonintegral.at Function Index
      • parabolics.at
        • parabolics.at Function References
        • parabolics.at Function Index
      • parameters.at
        • parameters.at Function References
        • parameters.at Function Index
      • partitions.at
        • partitions.at Function References
        • partitions.at Function Index
      • polynomial.at
        • polynomial.at Function References
        • polynomial.at Function Index
      • print_K_types.at
        • print_K_types.at Function References
        • print_K_types.at Function Index
      • rational_polynomial.at
        • rational_polynomial.at Function References
        • rational_polynomial.at Function Index
      • ratmat.at
        • ratmat.at Function References
        • ratmat.at Function Index
      • representations.at
        • representations.at Function References
        • representations.at Function Index
      • sort.at
        • sort.at Function References
        • sort.at Function Index
      • sp4.at
        • sp4.at Function References
        • sp4.at Function Index
      • stable.at
        • stable.at Function References
        • stable.at Function Index
      • std_decs.at
        • std_decs.at Function References
        • std_decs.at Function Index
      • synthetic.at
        • synthetic.at Function References
        • synthetic.at Function Index
      • synthetic_aux.at
        • synthetic_aux.at Function References
        • synthetic_aux.at Function Index
      • test_K.at
        • test_K.at Function References
        • test_K.at Function Index
      • test_unitarity.at
        • test_unitarity.at Function References
        • test_unitarity.at Function Index
      • tests.at
        • tests.at Function References
        • tests.at Function Index
      • tits.at
        • tits.at Function References
        • tits.at Function Index
      • torus.at
        • torus.at Function References
        • torus.at Function Index
      • translate.at
        • translate.at Function References
        • translate.at Function Index
      • twist.at
        • twist.at Function References
        • twist.at Function Index
      • Unipotent_Packets_All_Types.at
        • Unipotent_Packets_All_Types.at Function References
        • Unipotent_Packets_All_Types.at Function Index
      • unitary.at
        • unitary.at Function References
        • unitary.at Function Index
      • Vogan-dual.at
        • Vogan-dual.at Function References
        • Vogan-dual.at Function Index
      • W_K.at
        • W_K.at Function References
        • W_K.at Function Index
      • W_orbit.at
        • W_orbit.at Function References
        • W_orbit.at Function Index
      • W_Reps_Mod.at
        • W_Reps_Mod.at Function References
        • W_Reps_Mod.at Function Index
      • W_reps_type_BC.at
        • W_reps_type_BC.at Function References
        • W_reps_type_BC.at Function Index
      • Wdelta.at
        • Wdelta.at Function References
        • Wdelta.at Function Index
      • weyl_character_formula.at
        • weyl_character_formula.at Function References
        • weyl_character_formula.at Function Index
      • Weylgroup.at
        • Weylgroup.at Function References
        • Weylgroup.at Function Index
      • wreps_type_C.at
        • wreps_type_C.at Function References
        • wreps_type_C.at Function Index
  • Videos
    • Online Training Videos
      • Online Training Session 1, Part A
        • 1. Introduction
        • 2. Basics
        • 3. First Command
        • 4. atlas Scripts
        • 5. More Commands and Shortcuts
        • 6. Redefining Variables, Vector Operations
        • 7. Operator $, %, and booleans
        • 8. Matrix Operations
        • 9. Loading Files
        • 10. Inverse
        • 11. File Output
        • 12. Solving Matrix Equations
        • 13. Functions, Types, etc.
        • 14. Loops
      • Online Training Session 1, Part B
        • 1. Questions from Part A
        • 2. Data Types, Root Datum
        • 3. More about Lie Groups
        • 4. Fundamental Weight Coordinates, \(Sp(8,\mathbb{R})\)
        • 5. Simple Roots of \(SL(5,\mathbb{R})\)
        • 6. More on Coordinates
        • 7. The atlas-functions.help File
        • 8. Get Lie Groups Information
        • 9. Real Forms
        • 10. Inner Class
      • Online Training Session 2, Part A
        • 1. New Features on Website
        • 2. Questions from Session 1
        • 3. Cartans
        • 4. Parameters
        • 5. Example: \(SL(2,\mathbb{R})\)
        • 6. Characters of Tori
        • 7. \(T = \mathbb{R}^x\)
        • 8. \(T = \mathbb{C}^x\)
        • 9. Differential
        • 10. Trivial Representation of \(SL(2,\mathbb{R})\)
        • 11. Preview: KGB Orbits
      • Online Training Session 2, Part B
        • 1. Summary of Session 2 Part A
        • 2. Minimal Principle Series of Split Groups
        • 3. More Functions
        • 4. \(G=PSL(2,\mathbb{R})\)
        • 5. \(G=Sp(4,\mathbb{R})\)
        • 6. Characters on Disconnected Part
        • 7. Lowest K Types
        • 8. \(G = SO(3,2)\)
        • 9. More on \(Sp(4,\mathbb{R})\)
        • 10. \(E8\)
      • Online Training Session 3, Part A
        • 1. Welcome Back
        • 2. More on Software Syntax
        • 3. More Principle Series Examples
        • 4. Representations Attached to Given Cartans
        • 5. Slides: KGB Parameters
        • 6. Slides: Finiteness of KGB
        • 7. Slides: \(SL(2,\mathbb{R})\) Example
        • 8. Slides: continued
      • Online Training Session 3, Part B
        • 1. Questions
        • 2. More Slides
        • 3. Examples Using the Software
        • 4. \(Sp(4,\mathbb{R})\)
        • 5. \(Sp(6,\mathbb{R})\)
        • 6. KGB on other Cartans
        • 7. Preview for Next Time
      • Online Training Session 4, Part A
        • 1. Welcome Back
        • 2. New atlas Version
        • 3. Review of \(SL(2,\mathbb{R})\)
        • 4. \(G = PGL(2,\mathbb{R})\)
        • 5. \(Sp(4,\mathbb{R})\)
        • 6. \(Sp(4,\mathbb{R})\), KGBElt x=5
        • 7. : More on \(Sp(4,\mathbb{R})\)
        • 8. Representations of \(Sp(4,\mathbb{R})\)
        • 9. print_block
      • Online Training Session 4, Part B
        • 1. \(SO(4,2)\)
        • 2. \(SO(5,1)\)
        • 3. \(Sp(4,\mathbb{R})\)
        • 4. \(SL(2,\mathbb{R})\)
        • 5. \(Sp(4,\mathbb{R})\)
        • 6. \(SO(4,4)\)
        • 7. \(PSO(4,4)\), \(Spin(4,4)\)
        • 8. \(Sp(8,\mathbb{R})\)
    • Miscellaneous Videos
      • Atlas Startup Options
      • Change Coordinates
  • Trouble Shooting
    • Installation – Trouble Shooting
      • The readline package
      • ctanglex error
      • Compiler error
  • Web Interface
atlas
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  • atlas Library »
  • The Axis Language
  • View page source

The Axis Language¶

  • Design Principles
  • Types
    • Introduction
    • Primitive Types
    • Composite Types
  • Array
    • Array Displays
    • Array Selection
    • Array Slicing
    • Other Array-Like Types
  • Tuple Displays and Pattern Matching
  • Implicit Conversions
  • Defining and Modifying Identifiers
  • Defining Functions and Operators
  • Overloading v.s. Redefining, and Conflicts
  • Control Structures
    • Sequenced expressions
    • Conditional expressions and case expressions
    • Loops
  • Recursion
  • Splitting Commands
  • Output, Input, and Redirection
  • Miscellaneous Commands
  • Syntax Summary
    • Command level
    • Quaternary expressions: ‘let’, anonymous functions, casts, sequences
    • Tertiary expressions: assignments
    • Secondary expressions: formulas
    • Primary expressions: subscriptions, function calls, atomic expressions
    • Closed expressions: displays and groupings, conditionals, loops
    • Identifier patterns
    • Types
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