Demonstration: Representation theory of monoids and Markov chains: generalized Tsetlin library (experimental)
.. MODULEAUTHOR:: Nicolas M. Thiéry <nthiery at users.sf.net>,
Requirements
This demonstration requires experimental code that has not yet been migrated from the Sage-Combinat queue to the sage-semigroups package.
In a first step, we construct a poset, its set of linear extensions, and endow this set with the promotion action:
sage: P = Poset([[1,2,3,4], [[1,2], [3,4]]], linear_extension=True) sage: view(P) sage: L = P.linear_extensions(); L The set of all linear extensions of Finite poset containing 4 elements... sage: L.cardinality() 6 sage: list(L) [[1, 3, 2, 4], ...] sage: G = L.markov_chain_digraph(labeling="source") sage: view(G)
sage: M = G.transition_monoid(); M The transition monoid of Looped multi-digraph on 6 vertices sage: M.is_r_trivial() False sage: M.is_l_trivial() True sage: M = G.transition_monoid(category=LTrivialMonoids()) sage: V = G.transition_module(monoid=M).algebra(QQ); V sage: V.character() 6*C[()] + C[(1, 2, 3, 4)] + 3*C[(2,)] + 2*C[(2, 4)] + 3*C[(4,)] sage: V.composition_factors() 2*S[()] + S[(1, 2, 3, 4)] + S[(2,)] + S[(2, 4)] + S[(4,)]
One can read off the eigenvalues of the generators of the monoid and of the transition matrix!