algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety Iwahori-Hecke algebra; symmetric group; Newton divided differences; Yang- Baxter relations; Euler-Poincaré characteristic; flat manifolds; irreducible representations; hook partitions; Kazhdan-Lustig graphs; quantum spin chain; quantum superalgebra; symmetry algebra 