Available Tools
ABV Packets for G₂
A-packetsCompute unramified and dihedral G2 ABV packets in both root embeddings using Singular and runtime vanishing cycles.
→ABV Packets for GLₙ
A-packetsCompute non-archimedean GL(n) ABV packets locally in the browser.
→Adams conjecture
A-packetsLists theta lifts of elements in a local A packet.
→Arthur Packets
A-packetsLists elements in a local A packet.
→Depth-Zero L-Packets
A-packetsExplore tame L-packets, normalized enhancements, ordered finite compact types, and induction data.
→Depth-Zero Representation Inspector
A-packetsRead the enhanced parameter of a depth-zero representation and follow its explicit construction from finite compact types and segment inductions.
→Derivatives and Socles
A-packetsComputes derivatives and socles.
→Extended multisegment inspector
A-packetsDisplays details regarding extended multisegments.
→L-Packets
A-packetsLists elements in a local L-packet.
→Representation inspector
A-packetsFinds all local A-packets containing a given representation.
→Stadler extended multisegments
A-packetsDisplays details regarding stadler extended multisegments.
→Atlas Terminal
AtlasUse the Atlas of Lie Groups and Representations interpreter directly from your browser, compiled to WebAssembly.
→Legacy Atlas Group Explorer
AtlasThe original all-in-one Atlas wizard, kept for reference alongside the newer focused single-purpose Atlas apps.
→Representation Theory
AtlasChoose a group and a dual real form once, then run mini apps individually or all at once.
→Structure Theory
AtlasChoose a real (or complex) group once and run structure-theory mini apps individually or all at once.
→Aubert Dual
Aubert DualComputes the Aubert-Zelevinsky dual for classical groups.
→Franke Filtration
Automorphic representationsComputes the Franke filtration for GL_n.
→Bruhat Order
Coxeter groupsVisualize the Bruhat order of any Coxeter group.
→Coxeter Group Explorer
Coxeter groupsInspect the root system, matrices, length distribution, conjugacy classes, subgroups, and character tables for any Coxeter group.
→Kazhdan–Lusztig Polynomials
Coxeter groupsCompute Kazhdan–Lusztig polynomials, cells, and Duflo involutions for Coxeter groups.
→Longest Weyl Parabolic
Coxeter groupsCompute the longest elements of a Weyl group and a parabolic subgroup, and display them as matrices in the standard group model for types A, B, C, D.
→Root SL2 Explorer
Coxeter groupsAllows one to visualize SL2 embeddings for Chevalley groups.
→Bruhat Decomposition
Finite groupsComputes the Bruhat decomposition g = u1 dw u2 of a matrix entered in LaTeX.
→Character values
Finite groupsValues of a given character evaluated at all conjugacy classes.
→Conjugacy class
Finite groupsValues of characters evaluated at a given conjugacy class.
→Conjugacy classes list
Finite groupsList of conjugacy classes of the general linear group.
→Gelfand-Graev multiplicities
Finite groupsCompute generalized Gelfand-Graev multiplicities for representations of the general linear group.
→GGG multiplicities by partition
Finite groupsCompute generalized Gelfand-Graev multiplicities for all representations of the general linear group, given a partition.
→Green polynomials
Finite groupsComputes Green polynomials for .
→Representations list
Finite groupsList of irreducible representations of the general linear group.
→Theta elevator
Theta correspondenceComputes theta lifts for classical groups.
→Theta lifts
Theta correspondenceLists theta lifts of a given representation.
→Deligne–Lusztig Character
UncategorizedExplore the Deligne–Lusztig character of a reductive group, expanded in the basis of unipotent characters.
→Green Polynomial Entry
UncategorizedLook up a single Green polynomial by choosing a row and column from the Green table.
→Green Table
UncategorizedGreen functions Q_{φ,j}(q) — values of relative-Weyl-group characters on local systems or unipotent classes.
→ICC Table
UncategorizedCompute intersection-cohomology multiplicity polynomials P_{X,L}(q) for a Springer series.
→Intertwining operator decomposition
UncategorizedDecomposes the degenerate principal series intertwining operator into simple reflection intertwining operators.
→Root exchange algorithm
UncategorizedDemonstrates the root exchange algorithm for (k,c) models.
→Standard module calculator
UncategorizedCompute the Jordan–Hölder series of a standard module or the standard module expansion of a Zelevinsky irreducible for GL_n.
→Unipotent Characters
UncategorizedDisplay the unipotent characters of a reductive group with generic degrees, fake degrees, Frobenius eigenvalues, and Lusztig family labels.
→Unipotent Classes
UncategorizedEnumerate unipotent classes of a reductive group with Dynkin labels, component groups, and Springer series.
→Unipotent Value Entry
UncategorizedLook up a single unipotent character value by choosing a row and column from the unipotent-values table.
→Unipotent Values
UncategorizedValues of unipotent characters on local systems or unipotent classes.
→X-Table
UncategorizedValues of character sheaves on local systems or unipotent classes.
→Unipotent Representation Explorer
Unipotent representationsExplore DRC diagrams, local systems, counting formulas, Barbasch-Vogan duality, and Springer correspondence for classical groups.
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