An implementation of the projective linear group of 2x2 matrices over the integers (extended modular group) as a Coxeter group in Sage.
Clone the Git repository:
> git clone https://github.com/mr-infty/PGL2Z.git
To use this package, change into the newly created directory (PGL2Z) and
start Sage as usual:
> cd PGL2Z
> sage
The package is available within Sage like any other Python package, i.e. can be imported like
> import pgl2z
(see usage example down below for more details).
To update to the newest version of this package, execute
> git pull
from within the cloned Git repository (i.e. from PGL2Z or any
subdirectory).
In the hope that examples can serve as a substitute to a proper documentation which hasn't been writen yet (and might never be), here are some:
sage: from pgl2z.pgl2z import PGL2Z
sage: W = PGL2Z()
sage: s1, s2, s3 = W.gens()
sage: s1.descents()
sage: from pgl2z.chamber import FundamentalChamber as C0
sage: C0.separated_from(s1*C0, C0.walls[0])
sage: H = IwahoriHeckeAlgebra(W, 0, 0, ZZ)
sage: T = H.T()
sage: T1, T2, T3 = T.algebra_generators()
Unfortunately, Iwahori-Hecke algebras in Sage aren't implemented in the Bourbaki form, i.e. defined by relations of the form
T^2 = bT + a
with parameters a,b, but instead in the less general 'split' form
0 = (T-q_1)(T-q_2) = T^2 - (q_1+q_2) + q_1 q_2
with parameters q_1,q_2. Fortunately, for the purposes of mere computations
this isn't really a problem as you can do all computations in a localization of
the generic (universal) Hecke algebra over the ring ZZ[a,b] that is of the
split form.
Concretely,
sage: S.<a,b> = ZZ['a','b']
sage: L = S.fraction_field()
sage: R.<xbar> = L['x'].quotient(x^2 - (b**2 + 4*a))
sage: q1 = (b+xbar)/2
sage: q2 = (b-xbar)/2
sage: H = IwahoriHeckeAlgebra(W, q1, q2, R)
should work, at least in any just world. However it doesn't, namely for two reasons: 1) Since Sage relies on PARI for computations with rings, it's subject to the same (silly) constraints. Namely, in a ring extension the variables of the base ring must always be bigger (in the order that's used) than the new variables being added. So, the above example should be modified to this:
sage: S.<z,y> = ZZ['z','y']
sage: L = S.fraction_field()
sage: R.<xbar> = L['x'].quotient(x^2 - (y**2 + 4*z))
sage: q1 = (y+xbar)/2
sage: q2 = (y-xbar)/2
sage: H = IwahoriHeckeAlgebra(W, q1, q2, R)
However, this still doesn't work, because: 2) The constructor of
IwahoriHeckeAlgebra uses the function is_square to check whether the
product q1*q2 is a square, and this function only supports certain rings.
TL;DR: Just forget about IwahoriHeckeAlgebra, and instantiate the derived
class IwahoriHeckeAlgebra_nonstandard instead, which is used in any way
behind the scenes for all computations.
sage: from sage.algebra.iwahori_hecke_algebra import
IwahoriHeckeAlgebra_nonstandard
sage: H = IwahoriHeckeAlgebra_nonstandard(W)
A generic Iwahori-Hecke algebra of type [ 1 3 2]
[ 3 1 -1]
[ 2 -1 1] in u,-u^-1*v^2 over Multivariate Laurent Polynomial Ring in u, v over Integer Ring
sage: T = H.T()
sage: T1, T2, T3 = T.algebra_generators()