71.33 GoodNodes

GoodNodes(e|H, mu)
GoodNodes(e|H, mu, r)

Given a partition and an integer e, Kleshchev [K] defined the notion of good node for each residue r (0 le r). When e is prime and mu is e--regular, Kleshchev showed that the good nodes describe the restriction of the socle of D(mu) in the symmetric group case. Brundan~[B] has recently generalized this result to the Hecke algebra.

By definition, there is at most one good node for each residue r, and this node is a removable node (in the diagram of mu). The function GoodNodes returns a list of the rows of mu which end in a good node; the good node of residue r (if it exists) is the (r+1)--st element in this list. In the second form, the number of the row which ends with the good node of residue r is returned; or false if there is no good node of residue r.

gap> GoodNodes(5,[5,4,3,2]);
[ false, false, 2, false, 1 ]
gap> GoodNodes(5,[5,4,3,2],0);
false
gap> GoodNodes(5,[5,4,3,2],4);
1