Green Table

Displays the Green functions $Q_{wF}$, corresponding to GreenTable(uc, W) in GAP3/Chevie. These are defined by $Q_{wF} = \sum_{u,\phi} \tilde{\phi}(wF)\, \tilde{X}_{u,\phi}$, where the sum ranges over all local systems $(u, \phi)$ and $\tilde{X}_{u,\phi}$ are the (normalised) characteristic functions of the intersection cohomology complexes. The rows are indexed by conjugacy classes of the relative Weyl groups $W_\mathbf{G}(\mathbf{L})$ attached to the cuspidal local systems; the columns are the local systems $L_j$ (default) or unipotent classes $u_j$. For the principal Springer series, where $\mathbf{L} = \mathbf{T}$ is a maximal torus, the function $Q_{wF}$ coincides with the restriction to the unipotent elements of the Deligne–Lusztig character $R^{\mathbf{G}}_{\mathbf{T}_w}(1)$. A Y-table toggle shows the companion matrix used in the Lusztig–Shoji algorithm.

Dynkin TypeABCDEFG

Rank

Characteristic02