Script Reference
W_reps.at
Representations of the Weyl group: W-graphs, W-cells, and the associated combinatorial data.
Mathematical background
The W-graph of a block encodes the action of the Hecke algebra, with vertices labeled by \(\tau\)-invariants and edges labeled by \(\mu\)-values. The strongly connected components of the oriented W-graph are the W-cells.
Definitions
| Name | Signature | Description |
|---|---|---|
| operator | (W_rep (dimension,operators), WeylElt w) = mat: | |
| operator | (W_rep pi) = (WeylElt -> mat): | |
| trivial_W | (RootDatum rd) = W_rep: | |
| character | (WeylClassTable tab, W_rep pi) = [int]: | |
| is_isomorphic | (WeylClassTable tab, W_rep pi, W_rep sigma) = bool: | |
| matrix_of_inner_products | (WeylClassTable tab) = ([[int]] characters) mat: | matrix of inner products of characters of representations |
| unique | (WeylClassTable tab,[W_rep] list) = [W_rep]: | |
| root_datum | (WCell (,(rd,),)) = RootDatum: rd | |
| # | (WGraph (,nodes)) = int: #nodes | |
| graph_action | (WGraph (rd,nodes),int s) = sparse_mat: | matrix for s of graph, of coherent continuation action on irreducibles. } set graph_action (WGraph (rd,nodes),int s) = sparse_mat: |
| graph_action | (WGraph graph,[int] w) = mat: | matrix of action of product of simple reflections on a cell |
| graph_action | (WGraph graph,WeylElt w) = mat: | matrix of action of WeylElt on a cell |
| cell_action | (WCell cell,int s) = sparse_mat: | matrix of action of i^th simple reflection on a cell |
| cell_action | (WCell cell,[int] w) = mat: | matrix of action of product of simple reflections on a cell |
| cell_action | (WCell cell,WeylElt w) = mat: | matrix of action of WeylElt on a cell |
| vertex_and_W_cells | (Param p) = (int,[WCell]): | |
| W_cells_of | (Param p) = [WCell]: | |
| cell_character | (WeylClassTable Wct,WCell cell) = [int]: | character of representation of W on cell |
| cell_characters | (WeylClassTable Wct,[WCell] cells) = [[int]]: | list of characters of representation on list of cells |
| cells_table | (WeylClassTable Wct,[WCell] cells) = mat: | list of characters of representation on list of cells |
| cells_table_augmented | (WeylClassTable Wct, [WCell] cells) = mat: | |
| cell_representation | (WeylClassTable Wct,WCell cell) = W_rep: | |
| cell_representations | (WeylClassTable Wct,[WCell] cells) = [W_rep]: | |
| induce_character | (WeylClassTable Wct_G,WeylClassTable Wct_L,[int] pi_L) = | |
| smallest_degree | (WeylClassTable Wct, [int] character) = int: | smallest k so that |character| has factor in common with S^k(reflection) |