Here is of Jeff's screen interaction with atlas from the 4/27/21 seminar atlas> set cells=W_cells_of (Sp(4,R).trivial) { More or less the command "W_cells_of(parameter p)" produces a list of all the W_cells in the block of the parameter p. Sometimes the output is a pair (int,[Wcell]), where int is the number of p in its block, and [Wcell] is the list of cells. } Variable cells: (int,[WCell]) atlas> cells[0] Error in expression cells[0] at :154:0-8 Cannot subscript value of type (int,[WCell]) with index of type int Expression analysis failed Evaluation aborted. {Problem was that Jeff expected a list of cells (of which it would be legal to take the first one cells[0]); but he got a pair (int,[Wcell])} atlas> set (,cells)=W_cells_of (Sp(4,R).trivial) Variable cells: [WCell] (overriding previous instance, which had type (int,[WCell])) { Now he replaced the pair (int,[Wcell] with its second term, the list he wants } atlas> cells[0] Value: ([0],(simply connected root datum of Lie type 'C2',[([],[])]),[[[(0,1)]],[[(0,1)]]]) atlas> cells Value: [([0],(simply connected root datum of Lie type 'C2',[([],[])]),[[[(0,1)]],[[(0,1)]]]),([1],(simply connected root datum of Lie type 'C2',[([],[])]),[[[(0,1)]],[[(0,1)]]]),([2,5,7],(simply connected root datum of Lie type 'C2',[([0],[(1,1)]),([1],[(0,1),(2,1)]),([0],[(1,1)])]),[[[(0,-1)],[(1,1),(0,1),(2,1)],[(2,-1)]],[[(0,1),(1,1)],[(1,-1)],[(2,1),(1,1)]]]),([3,6,8],(simply connected root datum of Lie type 'C2',[([0],[(1,1)]),([1],[(0,1),(2,1)]),([0],[(1,1)])]),[[[(0,-1)],[(1,1),(0,1),(2,1)],[(2,-1)]],[[(0,1),(1,1)],[(1,-1)],[(2,1),(1,1)]]]),([4,9,11],(simply connected root datum of Lie type 'C2',[([0],[(1,1)]),([1],[(0,1),(2,1)]),([0],[(1,1)])]),[[[(0,-1)],[(1,1),(0,1),(2,1)],[(2,-1)]],[[(0,1),(1,1)],[(1,-1)],[(2,1),(1,1)]]]),([10],(simply connected root datum of Lie type 'C2',[([0,1],[])]),[[[(0,-1)]],[[(0,-1)]]])] atlas> whattype cells Type: [WCell] atlas> set (labels,graph,ops)=cells[0] Variable labels: A_class Variable graph: WGraph Variable ops: [sparse_mat] {A WCell is a triple (A_class,WGraph,[sparse_mat]. Here A_class is the list of block element numbers making up the nodes of the WGraph; and [sparse_mat] is one matrix (size the number of nodes in the WGraph) for each simple reflection in W.} ) atlas> graph Value: (simply connected root datum of Lie type 'C2',[([],[])]) atlas> set (labels,graph,ops)=cells[3] Variable labels: A_class (overriding previous instance, which had type A_class) Variable graph: WGraph (overriding previous instance, which had type WGraph) Variable ops: [sparse_mat] (overriding previous instance, which had type [sparse_mat]) atlas> graph Value: (simply connected root datum of Lie type 'C2',[([0],[(1,1)]),([1],[(0,1),(2,1)]),([0],[(1,1)])]) { The list is the list of the (three) nodes of the Wgraph ([0],[(1,1)]) ([1],[(0,1),(2,1)]) ([0],[(1,1)]) Node 0 has tau invariant (first list) consisting of the simple root 0; it is linked to one other node (second list) 1 by an edge of weight 1. Node 1 has tau invariant consisting of the simple root 1; it is linked to nodes 0 and 2 by edges of weight 1. Node 2 has tau invariant consisting of the simple root 0; it is linked to node 1 by an edge of weight 1. atlas> labels Value: [3,6,8] { node 0 is block element 3; node 1 is block element 6; and node 2 is block element 8 } atlas> graph atlas> ops Value: [[[(0,-1)],[(1,1),(0,1),(2,1)],[(2,-1)]],[[(0,1),(1,1)],[(1,-1)],[(2,1),(1,1)]]] This is a list of two sparse matrices [[(0,-1)],[(1,1),(0,1),(2,1)],[(2,-1)]] [[(0,1),(1,1)],[(1,-1)],[(2,1),(1,1)]] describing the actions of the two simple reflections on the basis of irreducibles in the cell. { Sorry, I think there was more stuff, but this is all I could find. DV }