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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  • KGB Related
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KGB Related¶

Function Argument(s) -> Result(s)
KGB (RealFrom,int->KGBElt)
% (KGBElt->RealForm,int)
Cartan_class (KGBElt->CartanClass)
involution (KGBElt->mat)
length (KGBElt->int)
status (int,KGBElt->int)
status (vec,KGBElt->int)
cross (int,KGBElt->KGBElt)
Cayley (int,KGBElt->KGBElt)
twist (KGBElt->KGBElt)
torus_factor (KGBElt->ratvec)
torus_bits (KGBElt->vec)
= (KGBElt,KGBElt)
block (RealForm,RealForm->Block)
% (Block->RealForm,RealForm)
# (Block->int)
element (Block,int->KGBElt,KGBElt)
index (Block,KGBElt,KGBElt->int)
dual (Block->Block)
status (int,Block,int->)

KGB¶

(RealFrom,int->KGBElt): select a KGB element x among those of a real form

The call KGB(i,rf) selects the element in line i of the print_KGB(rf) output.

%¶

(KGBElt->RealForm,int): real form and number; inverse of KGB@(RealForm,int)

Cartan_class¶

(KGBElt->CartanClass): the Cartan class for the KGB element

involution¶

(KGBElt->mat): the involution of X^* associated to the KGB element

length¶

(KGBElt->int): length of the element within its KGB set

status¶

(int,KGBElt->int): status of generator on KGB element, scale 0..4

Encoding 0: Complex descent, 1: imaginary compact, 2: real, 3: imaginary non-compact, 4: Complex ascent

status¶

(vec,KGBElt->int): status of any root on KGB element, scale 0..4

The value status(alpha,x) gives the status of the root alpha at x; the encoding of statuses is the same as for the function status@(int,KGBElt)

cross¶

(int,KGBElt->KGBElt): cross action for (posrootnr,KGB element)

Returns result of cross action by root reflection of the given KGB element

Cayley¶

(int,KGBElt->KGBElt): Cayley transform for (posrootnr,KGB element)

Returns either the Cayley transform or an inverse Cayley transform of the KGB element through the given positive root, or returns that element itself when neither is defined. If defined, one can find which it is using status, and whether there is a second inverse Cayley transform can be found out by applying cross action to the result; it either returns the same result, which signifies a single-value inverse Cayley, or else the second value

twist¶

(KGBElt->KGBElt): twist of KGB element x, useful for Hermitian dual

This is defined by conjugation of x by the distinguished involution delta

torus_factor¶

(KGBElt->ratvec): coweight for KGB element, twice that in print_X

This is a \(\theta^t\) stable rational vector v with integral evaluations on all imaginary roots; interpreted modulo the image of 1+theta^t acting on \(X_*\), so in particular modulo \((2\\mathbb{Z})^n\). For an imaginary root \(\alpha\) the scalar product \(langle v + \check\rho_i , \alpha \rangle\) determines whether \(\alpha\) is compact (if even) or noncompact (if odd). Together with the inner class and the invlution, this value completely characterises the KGB element (and its real form), and it can be reconstructed using the KGB_elt function below.

torus_bits¶

(KGBElt->vec): torus part of KGB element as vector of 0,1 values

A vector of size the rank that distinguishes between different KGB elements at the same involution, in the format used internally. From this the value torus_factor(x) is computed using base_grading_vector(rf)-torus_bits(x), for the real form rf of x, which value is then symmetrized for theta^t.

KGB_elt¶

(RealForm,mat,ratvec): KGB element defined by rational coweight

This function finds a KGB element x such that real_form(x), involution(x), and torus_factor(x) match the values (rf,M,v) given as arguments. The interpretation for the (involution) matrix M and rational (coweight) vector v are as for real_form@(InnerClass,mat,ratvec); moreover rf should be equal to the real form returned (for this inner class) by real_form(ic,M,v), and the difference v-base_grading_vector(rf) must be an integer vector. There is then at most one KGB element x for rf such that torus_factor(x) is congruent to v modulo the image of ^M+1 (i.e., M+1 applied on right), which x is then returned by this function (if no such x is found, an error is signaled).

=¶

(KGBElt,KGBElt): equality of elements of a same KGB set

block¶

(RealForm,RealForm->Block): construct tradiational atlas block

%¶

(Block->RealForm,RealForm): decompose block, inverse of ‘block’

#¶

(Block->int): number of elements of block

element¶

(Block,int->KGBElt,KGBElt): KGB and dual KGB values for block element

index¶

(Block,KGBElt,KGBElt->int): index in block, from KGB, dual KGB components This is the inverse of the map defined by element@(Block,int): given a compatible pair of a KGB element x and a dual KGB element y, it returns the index in the block of the corresponding element. For efficiency reasons the function requires the containing block to be supplied as first argument; if needed, block(real_form(x),dual_real_form(real_form(y))) computes the block

dual¶

(Block->Block): dual block, with real form and dual real form swapped

status¶

(int,Block,int->): status at a block element of a simple reflection

For s the index of a reflection, and i the index of an element of block b, status(s,b,i) gives according to the codes 0:C-, 1:ic, 2:r1, 3:r2, 4:C+, 5: rn, 6:i1, 7:i2. Note that the descents s have status(b,i,s)<4

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