Finite groups of Lie type

Applications for exploring finite groups of Lie type and their unipotent structure. These apps are powered by a TypeScript port of PyCox by Meinolf Geck and of parts of the GAP3/Chevie package by Jean Michel.

• Unipotent Characters — display the unipotent characters of a reductive group with generic degrees, fake degrees, Frobenius eigenvalues, and Lusztig family labels.
• Unipotent Classes — enumerate the unipotent conjugacy classes of a reductive group, together with the component groups $A(u)$, Bala–Carter–Dynkin labels, dimensions of Springer fibres, and Springer correspondence data.
• ICC Table — compute the intersection-cohomology multiplicity polynomials $P_{X_i,\,L_j}(q)$ for a Springer series, displayed as a matrix over the polynomial ring $\mathbb{Z}[q]$.
• X-Table — values of character sheaves $X_i$ on local systems $L_j$ or unipotent classes; also shows the companion $Y$-matrix.
• Green Table — Green functions $Q_{\phi,j}(q)$: values of relative-Weyl-group characters on local systems or unipotent classes.
• Unipotent Values — values of unipotent characters on local systems or unipotent classes.
• Deligne–Lusztig Character — expand the Deligne–Lusztig character $R_T(1)$ in the basis of unipotent characters, specifying the torus $T$ either by conjugacy class or by a Coxeter word.
• Green Polynomial Entry — look up a single Green polynomial $Q_{\phi,j}(q)$ by selecting a row and column from the Green table.
• Unipotent Value Entry — look up a single unipotent character value $R_{\phi,L}(q)$ by selecting a row and column from the unipotent-values table.

📄️Unipotent Classes

Enumerate unipotent classes of a reductive group with Dynkin labels, component groups, and Springer series

📄️ICC Table

Compute intersection-cohomology multiplicity polynomials P_{X,L}(q) for a Springer series

📄️Unipotent Characters

Display the unipotent characters of a reductive group with generic degrees, fake degrees, Frobenius eigenvalues, and Lusztig family labels

📄️X-Table

Values of character sheaves on local systems or unipotent classes

📄️Green Table

Green functions Q_{φ,j}(q) — values of relative-Weyl-group characters on local systems or unipotent classes

📄️Unipotent Values

Values of unipotent characters on local systems or unipotent classes

📄️Green Polynomial Entry

Look up a single Green polynomial $Q_{\phi,j}(q)$ by choosing a row and column from the Green table

📄️Deligne–Lusztig Character

Explore the Deligne–Lusztig character $R_T(1)$ of a reductive group, expanded in the basis of unipotent characters

📄️Unipotent Value Entry

Look up a single unipotent character value by choosing a row and column from the unipotent-values table