Coverage for local/lib/python2.7/site-packages/sage/monoids/hecke_monoid.py : 100%

# -*- coding: utf-8 -*- 

""" 

Hecke Monoids 

""" 

#***************************************************************************** 

# Copyright (C) 2015 Nicolas M. Thiéry <nthiery at users.sf.net> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.misc.cachefunc import cached_function 

from sage.sets.finite_set_maps import FiniteSetMaps 

@cached_function 

def HeckeMonoid(W): 

r""" 

Return the `0`-Hecke monoid of the Coxeter group `W`. 

INPUT: 

- `W` -- a finite Coxeter group 

Let `s_1,\ldots,s_n` be the simple reflections of `W`. The 0-Hecke 

monoid is the monoid generated by projections `\pi_1,\ldots,\pi_n` 

satisfying the same braid and commutation relations as the `s_i`. 

It is of same cardinality as `W`. 

.. NOTE:: 

This is currently a very basic implementation as the submonoid 

of sorting maps on `W` generated by the simple projections of 

`W`. It's only functional for `W` finite. 

.. SEEALSO:: 

- :class:`CoxeterGroups` 

- :class:`CoxeterGroups.ParentMethods.simple_projections` 

- :class:`IwahoriHeckeAlgebra` 

EXAMPLES:: 

sage: from sage.monoids.hecke_monoid import HeckeMonoid 

sage: W = SymmetricGroup(4) 

sage: H = HeckeMonoid(W); H 

0-Hecke monoid of the Symmetric group of order 4! as a permutation group 

sage: pi = H.monoid_generators(); pi 

Finite family {1: ..., 2: ..., 3: ...} 

sage: all(pi[i]^2 == pi[i] for i in pi.keys()) 

True 

sage: pi[1] * pi[2] * pi[1] == pi[2] * pi[1] * pi[2] 

True 

sage: pi[2] * pi[3] * pi[2] == pi[3] * pi[2] * pi[3] 

True 

sage: pi[1] * pi[3] == pi[3] * pi[1] 

True 

sage: H.cardinality() 

24 

""" 

ambient_monoid = FiniteSetMaps(W, action="right") 

pi = W.simple_projections(length_increasing=True).map(ambient_monoid) 

H = ambient_monoid.submonoid(pi) 

H.rename("0-Hecke monoid of the %s"%W) 

return H