algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Mots; words; trees; permutations; toric variety; Weyl chambers; semigroups; Lie algebra; Burnside problem for semigroups; symmetric group; skew tableaux; hypermaps; combinatorial theory; representations; continued fractions; differential algebra; probability measures; grammars of zigzags; complexity; finite automaton 