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: Representation theory of monoids and Markov chains: generalized Tsetlin library (experimental)ΒΆ

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, 2, 3, 4], [1, 3, 2, 4], [1, 3, 4, 2], [3, 1, 2, 4], [3, 1, 4, 2], [3, 4, 1, 2]]

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!