Software demos

Software demos

Worked atlas sessions showing what the software can compute.

Root Systems: the Dynkin diagram, Cartan matrix, fundamental weights and coweights, positive roots and Weyl group order for root systems of types A–G, with a choice of labelling and the matrices relating the standard and weight coordinates. It runs in your browser.

Spherical Unitary Explorer: test whether a spherical representation of a classical group is unitary, using an algorithm due to Dan Barbasch. It runs in your browser, with nothing to install.

Deformation Algorithm: step through the deformation algorithm for the c-invariant form on the spherical principal series of SL(2,ℝ), Sp(4,ℝ) or split G₂, from ν = kρ down to the tempered terms. All data were computed in atlas.

Hodge Levels on Sp(4,ℝ): the levels of the Hodge filtration on the K-types of the spherical principal series of Sp(4,ℝ), and how they move as ν passes the reducibility points. Data from atlas’s hodge_branch_std.

FPP Facet Explorer: the facets of the fundamental parallelepiped that carry a unitary spherical representation, with their closure relations, for SL(2,ℝ), Sp(4,ℝ), split G₂, split F₄ and split E₇.

W-cell Graphs: the graph on W-cells computed by W_cell_graph in cell_graph.at, for the block of the trivial representation of SL(2,ℝ), Sp(4,ℝ), and split G₂, F₄, E₆ and E₇. Each cell is labelled by its special nilpotent orbit, and rows are Gelfand–Kirillov dimension.

More demos will appear here.