The "wcells" command prints out the cells of the W-graph of the block, as well
as the induced graph on the collection of cells. Each cells is output
abstractly, renumbering its vertices from zero, but for each vertex the number
of the corresponding block element is also indicated. The abstract structure
is easier for reading and comparing individual cells (although there is
nothing canonical about the numbering of vertices, so isomorphic cells might
differ by a permutation of the vertex labels).
The output of the command that can be written to a file has three parts.
The first part is introduced by the line "// Cells and their vertices."
What follows is a list of cells, numbered #0 to #m-1, one per line. A cell is
represented by a line like
#2={2,3}
This means that cell #2 consists of block elements 2 and 3.
The second part is introduced by the line "// Induced graph on cells."
What follows is a line for each cell, listing all the cells "below it"
in the Kazhdan-Lusztig ordering. A typical line is
#3:->#1,#2.
This means that cell #3 has elements linked by directed edges to
elements of cells #1 and #2. A consequence is that the
"associated variety" for representations in cell #1 must strictly
contain the associated variety for representations in cell #3.
The third and last part is introduced by the line "// Individual cells."
Cell #i is introduced by a line "// cell #i:" What follows is
a list of elements in the cell, numbered 0 to n-1, one per line. The
number is followed in brackets by the number of the corresponding
block element. Next comes the tau invariant, enclosed in braces,
a subset of {1,...,r} (with r the number of simple roots).
Following the tau invariant for vertex j is a collection of non-negative
integers k (or pairs (k,m), with m a positive integer). The integer k
is shorthand for the pair (k,1). The presence of a pair means that
there is an oriented edge of the W-graph joining the present vertex j
of the cell to vertex k, having multiplicity m. (The multiplicity is
a coefficient of highest possible power of q in the Kazhdan-Lusztig
polynomial indexed by j and k.)
The presence of the edge from j to k means that two conditions are
satisfied: first, that tau(j) is not contained in tau(k); and second,
that the Kazhdan-Lusztig polynomial indexed by j and k has highest
possible degree.
The Weyl group representation attached to the cell may be described as
follows. It has a Z basis {L_j} indexed by the cell elements j. If
root i is in the tau invariant for element j, then
s_i(L_j) = -L_j.
If i is not in the tau invariant, then
s_i(L_j) = L_j + sum_{elements k, i in tau(k)} m_{kj} * L_k.
where m_{kj} denotes the multiplicity of the edge from k to j.
That is, m*L_k appears in the sum here if i is in the tau-invariant of
L_k, and the pair (j,m) appears in the list for row k.
For example, here is the output of wcells for the simply connected
group SU(2,1) and the block of the trivial representation:
// Cells and their vertices.
#0={0}
#1={1,4}
#2={2,5}
#3={3}
// Induced graph on cells.
#0:.
#1:->#0.
#2:->#0.
#3:->#1,#2.
// Individual cells.
// cell #0:
0[0]: {}
// cell #1:
0[1]: {2} --> 1
1[4]: {1} --> 0
// cell #2:
0[2]: {1} --> 1
1[5]: {2} --> 0
// cell #3:
0[3]: {1,2}
Note that cells #1 and #2 are isomorphic, but the numbering must be reversed
in order to match the tau invariants. So the cell on block elements {1,4} is
isomorhpic to the one on block elements {2,5} under the bijection that maps
1 -> 5 and 4 -> 2.
If we look at cell #1, we may conclude that the Weyl group representation
attached to it is given by
s_1(L_0) = L_0 + L_1 s_2(L_0) = -L_0
s_1(L_1) = -L_1 s_2(L_1) = L_1 + L_0.