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Demonstration: Calculations with character rings of the biHecke monoid (experimental)ΒΆ


This demonstration requires experimental code that has not yet been migrated from the Sage-Combinat queue to the sage-semigroups package.

sage: %hide
sage: pretty_print_default()
sage: attach /home/nthiery/work/frg/Articles/Hivert_Schilling_Thiery_HeckeMonoid/main.sage
sage: M = BiHeckeMonoid(["A",3])
sage: G = M.character_ring()
sage: E = G.E(); T = G.T(); P = G.P()

sage: M0 = M.fix_w0_monoid()
sage: G0 = M0.character_ring()
sage: S0 = G0.S(); P0 = G0.P()

sage: for e in P0.basis():
....:     print "Ind(",e, ")=",P(G.induce_from_M0(S0(e))) # indirect doctest
Ind( P[1234] )= P[1234]
Ind( P[2134] )= P[2134] + P[2314] + P[2341] + P[2413] + P[4213]
Ind( P[2314] )= P[2314] + P[2341]
Ind( P[3214] )= P[3214] + P[3241] + P[3421]
Ind( P[2341] )= P[2341]
Ind( P[3241] )= P[3241] + P[3421]
Ind( P[3421] )= P[3421]
Ind( P[4321] )= P[4321]
Ind( P[2431] )= P[2431] + P[4231]
Ind( P[4231] )= P[4231]
Ind( P[1324] )= P[1324] + P[1342] + P[3124] + P[3142] + P[3241] + P[3412] + P[4132]
Ind( P[3124] )= P[3124] + P[3142] + P[3241] + P[3412]
Ind( P[1342] )= P[1342] + P[3142] + P[3412] + P[4132]
Ind( P[3142] )= P[3142] + P[3412]
Ind( P[3412] )= P[3412]
Ind( P[4312] )= P[4312]
Ind( P[1432] )= P[1432] + P[4132] + P[4312]
Ind( P[4132] )= P[4132] + P[4312]
Ind( P[1243] )= P[1243] + P[1423] + P[2413] + P[2431] + P[4123]
Ind( P[2143] )= P[2143] + P[2413] + P[2431] + P[4213] + P[4231]
Ind( P[2413] )= P[2413] + P[2431] + P[4213] + P[4231]
Ind( P[4213] )= P[4213] + P[4231]
Ind( P[1423] )= P[1423] + P[4123]
Ind( P[4123] )= P[4123]

Behind the scene, it uses the cutting poset (to convert between translation modules to simple modules), the character formula for projective modules of J-Trivial monoids, the property that simple modules for M0 are induced to translation modules for M, etc. Plus inversion by matrix of linear morphism between finite dimensional vector spaces. It also uses the expansion of the character of a projective module for the BiHecke monoid in term of simple module, but this one is hard-coded for type A3 (currently too expensive to recalculate it). The framework supports q-characters; but few of the rules above are implemented for them, since we do not know them yet