atlas
0.6
Table of Contents
General Information
About Us
About the Software
Getting Started
Download and Install
Linux
Download the software
Installation from Source
Mac
Download the source code
Compiling atlas for the Mac using MacPorts
(1) Install MacPorts.
(2) Install the XCode C compiler.
(3) Install the MacPorts (gcc) C Compiler
(4) Install readline
(5) Edit the Makefiles
(6) Compile Fokko and atlas
Other Compilation options:
(7) Installing Fokko and atlas
Example:
Example:
Example:
Windows
Compiling
Installing under Cygwin
Pre-compiled
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 of a Complex Group
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
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 on minimal pricipal series.
Cuspidal Data for Representations
Lowest
\(K\)
-types of a Representation
Representations with Zero Infinitesimal Character
Translation Principle
Blocks of Representations
Coordinates and
\(KGB\)
elements
More Principal Series
General Parameters
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_Kparameters.at
all_Kparameters.at Function References
all_Kparameters.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
coordinates.at
coordinates.at Function References
coordinates.at Function Index
cross_W_orbit.at
cross_W_orbit.at Function References
cross_W_orbit.at Function Index
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
galois.at
galois.at Function References
galois.at Function Index
generate_groups.at
generate_groups.at Function References
generate_groups.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
herm.at
herm.at Function References
herm.at Function Index
hermitian.at
hermitian.at Function References
hermitian.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
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
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
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
Source Code Library
Web Interface
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
atlas
Docs
»
Tutorial
»
First Examples of Representations.
»
Coordinates and
\(KGB\)
elements
View page source
Coordinates and
\(KGB\)
elements
ΒΆ