# Hecke Monoids#

- sage.monoids.hecke_monoid.HeckeMonoid()#
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.

See also

`CoxeterGroups.ParentMethods.simple_projections`

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