This document is one of More SageMath Tutorials. You may edit it on github. $$\def\NN{\mathbb{N}}$$ $$\def\ZZ{\mathbb{Z}}$$ $$\def\QQ{\mathbb{Q}}$$ $$\def\RR{\mathbb{R}}$$ $$\def\CC{\mathbb{C}}$$

# Demonstration: Sage + GAP4 + GAP3 + Chevie + Semigroupe (experimental)¶

Requirements

Todo

update to use libsemigroups

Let us create the Coxeter group W:

sage: W = CoxeterGroup(["H",4]); W


It is constructed as a group of permutations, from root data given by GAP3+Chevie (thanks to Franco’s interface):

sage: W._gap_group
CoxeterGroup("H",4)
sage: (W._gap_group).parent()
Gap3


with operations on permutations implemented in Sage:

sage: W.an_element()^3
(1,5)(2,62)(3,7)(6,9)(8,12)(11,15)(13,17)(16,20)(18,22)(21,25)(26,29)(28,31)(30,33)(32,35)(34,37)(36,39)(38,41)(42,45)(46,48)(47,49)(50,52)(55,56)(57,58)(61,65)(63,67)(66,69)(68,72)(71,75)(73,77)(76,80)(78,82)(81,85)(86,89)(88,91)(90,93)(92,95)(94,97)(96,99)(98,101)(102,105)(106,108)(107,109)(110,112)(115,116)(117,118)


and group operations implemented in GAP 4:

sage: len(W.conjugacy_classes_representatives())
34
sage: W.cardinality()
14400


Now, assume we want to do intensive computations on this group, requiring heavy access to the left and right Cayley graphs (e.g. Bruhat interval calculations, representation theory, …). Then we can use Jean-Eric Pin’s Semigroupe, a software written in C:

sage: S = semigroupe.AutomaticSemigroup(W.semigroup_generators(), W.one(),
....:                                   category = CoxeterGroups().Finite())


The following triggers the full expansion of the group and its Cayley graph in memory:

sage: S.cardinality()
14400


And we can now iterate through the elements, in length-lexicographic order w.r.t. their reduced word:

sage: sum( x^p.length() for p in S)
x^60 + 4*x^59 + 9*x^58 + 16*x^57 + 25*x^56 + 36*x^55 + 49*x^54 + 64*x^53 + 81*x^52 + 100*x^51 + 121*x^50 + 144*x^49 + 168*x^48 + 192*x^47 + 216*x^46 + 240*x^45 + 264*x^44 + 288*x^43 + 312*x^42 + 336*x^41 + 359*x^40 + 380*x^39 + 399*x^38 + 416*x^37 + 431*x^36 + 444*x^35 + 455*x^34 + 464*x^33 + 471*x^32 + 476*x^31 + 478*x^30 + 476*x^29 + 471*x^28 + 464*x^27 + 455*x^26 + 444*x^25 + 431*x^24 + 416*x^23 + 399*x^22 + 380*x^21 + 359*x^20 + 336*x^19 + 312*x^18 + 288*x^17 + 264*x^16 + 240*x^15 + 216*x^14 + 192*x^13 + 168*x^12 + 144*x^11 + 121*x^10 + 100*x^9 + 81*x^8 + 64*x^7 + 49*x^6 + 36*x^5 + 25*x^4 + 16*x^3 + 9*x^2 + 4*x + 1
sage: S[0:10]
[[], [0], [1], [2], [3], [0, 1], [0, 2], [0, 3], [1, 0], [1, 2]]
sage: S[-1]
[0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 2, 0, 1, 0, 2, 3, 2, 0, 1, 0, 1, 2, 0, 1, 0, 2, 3, 2, 0, 1, 0, 1, 2, 0, 1, 0, 2, 3, 2, 0, 1, 0, 1, 2, 0, 1, 0, 2, 3, 2, 0, 1, 0, 1, 2, 0, 1, 0, 2, 3]


The elements of S are handles to C objects from Semigroupe:

sage: x = S.an_element()
sage: x
[0, 1, 2, 3]


Products are calculated by Semigroupe:

sage: x * x
[0, 1, 0, 2, 0, 1, 3, 2]


Powering operations are handled by Sage:

sage: x^3
[0, 1, 0, 2, 0, 1, 0, 2, 3, 2, 0, 1]

sage: x^(10^10000)


Altogether, S is a full fledged Sage Coxeter group, which passes all the generic tests:

sage: TestSuite(S).run(verbose = True, skip = "_test_associativity")


And of course it works for general semigroups too, and can further compute much more information about those, like the (Knuth-Bendix completion of the) relations between the generators:

sage: S.print_relations()
aa = 1
bb = 1
cb = bc
cc = 1
db = bd
dd = 1
cac = aca
dcd = cdc
...
dcbabacbabcdcbabacbabcdcbabacbabcdcbabacbabcdc = cdcbabacbabcdcbabacbabcdcbabacbabcdcbabacbabcd


which contains the usual commutation + braid relations.

Let’s try now the 0-Hecke monoid:

sage: from sage.combinat.j_trivial_monoids import *
sage: S = semigroupe.AutomaticSemigroup(W.simple_projections(), W.one(), by_action = True)
sage: S.cardinality()
14400

sage: S.print_relations()
aa = a
bb = b
ca = ac
cc = c
db = bd
dd = d
cbc = bcb
dcd = cdc
...
ababacbabacbabcdcbabacbabcdcbabacbabcdcbabacbabcdcbabacbabcd = 0


Let us throw in more mathematical information:

sage: W = CoxeterGroup(["A",3])
sage: S = semigroupe.AutomaticSemigroup(W.simple_projections(), W.one(), by_action = True,
....:                                   category = JTrivialMonoids().Finite())

sage: S.cardinality()

sage: H = S.algebra(QQ)
sage: H.orthogonal_idempotents()