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Representation Theory

Pick a group and a dual real form once with the shared pickers below — mirroring the "Dual Group" choice from the original atlas-gui — then either run each mini app individually or use Run all mini apps to compute every section on this page at once. Blocks, Kazhdan–Lusztig polynomials, and W-graphs can take a long time to compute for larger groups, so nothing runs automatically: use the individual Run buttons if you only need a specific section, or Run all if you want everything.

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Mode:

Family

n

Real form

Dual real form

Atlas expression

Mode: Real groupFamily: SLₙ(ℂ)Selection: SU(1,0)

Type A, simply connected.

Compact form.

SU(1,0)

Block of representations

Build the block of representations for the selected real form paired with the chosen dual real form. See the atlas docs on block.

Run:

Results

Choose a group and run the atlas command to see structured results.

All KL polynomials

Compute every nonzero Kazhdan–Lusztig–Vogan polynomial for the block. See the atlas docs on KL blocks.

Run:

Results

Choose a group and run the atlas command to see structured results.

Distinct KL polynomials

List only the distinct Kazhdan–Lusztig polynomials appearing in the block. See the atlas docs on partial KL blocks.

Run:

Results

Choose a group and run the atlas command to see structured results.

W-graph

Compute the W-graph attached to the block. See the atlas docs on KL blocks.

Run:

Results

Choose a group and run the atlas command to see structured results.

W-cells

Compute the W-cells attached to the block. See the atlas docs on KL blocks.

Run:

Results

Choose a group and run the atlas command to see structured results.

Block sizes

Tabulate block sizes across all real forms and dual real forms in the inner class. See the atlas docs on block sizes.

Run:

Results

Choose a group and run the atlas command to see structured results.