Maple Programs

The proof of Theorem 5.3 in Enumerative properties of shifted Dyck partitions relies on computer calculations of the parabolic Kazhdan-Lusztig polynomials for the exceptional Hermitian symmetric pairs. The code for these calculations is contained in the file pWKL. The programs run on Maple, and use the Maple packages coxeter/weyl (Version 2.3) and posets (Version 2.2), as well as some of the coxeter/weyl examples, developed by John Stembridge.

Here is a sample run of the programs, assuming that the coxeter/weyl examples are all contained in a file called Coxex.

> read Coxeter;

coxeter and weyl 2.3v loaded. Run 'withcoxeter()' or 'withweyl()' to use abbreviated names.

> withcoxeter();

[cox_matrix, interior_pt, perm_char, name_of, perm_rep, exponents, diagram, highest_root, restrict, base, multperm, size, reduce, induce, co_base, class_size, longest_elt, root_coords, vec2fc, index, iprod, orbit_size, char_poly, degrees, orbit, irr_chars, reflect, num_refl, cartan_matrix, length_gf, rank, presentation, stab_chain, perm2word, cprod, pos_roots, class_rep, cox_number]

> read Coxex:

> read posets;

posets 2.2v loaded. Run 'withposets()' to use abbreviated names.

> withposets();

Warning: new definition for lattice
Warning: new definition for char_poly
[chain, lattice, modular, omega, filter, ranked, plot_poset, Posets, closure, W, subposet, subinterval, zeta, extensions, J, rm_isom, covers, rand_poset, autgroup, mobius, char_poly, distributive, connected, meet, canon, antichains, atomic, Lattices, isom, dual]

> read pWKL;

> A:=coset_reps({1,2,3,4,5},E6);

A := [[], [6], [5, 6], [4, 5, 6], [2, 4, 5, 6], [3, 4, 5, 6], [3, 2, 4, 5, 6], [1, 3, 4, 5, 6], [1, 3, 2, 4, 5, 6], [4, 3, 2, 4, 5, 6], [4, 1, 3, 2, 4, 5, 6], [5, 4, 3, 2, 4, 5, 6], [3, 4, 1, 3, 2, 4, 5, 6], [5, 4, 1, 3, 2, 4, 5, 6], [6, 5, 4, 3, 2, 4, 5, 6], [5, 3, 4, 1, 3, 2, 4, 5, 6], [6, 5, 4, 1, 3, 2, 4, 5, 6], [4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [6, 5, 3, 4, 1, 3, 2, 4, 5, 6], [2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [6, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [6, 2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [5, 6, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [5, 6, 2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [4, 5, 6, 2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [3, 4, 5, 6, 2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6], [1, 3, 4, 5, 6, 2, 4, 5, 3, 4, 1, 3, 2, 4, 5, 6]]

> B:=map(Winverse,A);

B := [[], [6], [6, 5], [6, 5, 4], [6, 5, 4, 2], [6, 5, 4, 3], [6, 5, 4, 2, 3], [6, 5, 4, 3, 1], [6, 5, 4, 2, 3, 1], [6, 5, 4, 2, 3, 4], [6, 5, 4, 2, 3, 1, 4], [6, 5, 4, 2, 3, 4, 5], [6, 5, 4, 2, 3, 1, 4, 3], [6, 5, 4, 2, 3, 1, 4, 5], [6, 5, 4, 2, 3, 4, 5, 6], [6, 5, 4, 2, 3, 1, 4, 3, 5], [6, 5, 4, 2, 3, 1, 4, 5, 6], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4], [6, 5, 4, 2, 3, 1, 4, 3, 5, 6], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 6, 5], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3], [6, 5, 4, 2, 3, 1, 4, 3, 5, 4, 2, 6, 5, 4, 3, 1]]

> Bru:={}:

> for u in B do for v in B do if Wlength(v,E6)=Wlength(u,E6)+1 and WweakBruhat(u,v,E6)=1 then Bru:=Bru union {[u,v]}: else fi: od: od:

> WJRHSP(op(1,B),op(18,B),q,E6,{1,2,3,4,5});

-q + 1