Hi all,
Thanks for the replies. I'm on the sage-combinat-devel list now.
I don't know if it's helpful to post so much code, but here is my code
for the Demazure character and rank 2 plotting. The Demazure character
code contains the same idea as bump's code.
def demazure(ch, i):
"""
EXAMPLES:
# Check the Demazure character formula in A2
sage: A = WeylCharacterRing(['A',2])
sage: a = WeightRing(A)
sage: space = a.space()
sage: lam = space.fundamental_weights()
sage: rho = sum(list(lam))
sage: wch = A(2*rho) # character of the irred. representation
sage: ch = a(2*rho) # element in the weight ring
sage: dch = demazure(ch,[2,1,2]) # apply 3 Demazure operators
sage: sorted(wch.mlist()) == sorted(dch.mlist())
True
# Check the Demazure character formula in B2
sage: A = WeylCharacterRing(['B',2])
sage: a = WeightRing(A)
sage: space = a.space()
sage: lam = space.fundamental_weights()
sage: rho = sum(list(lam))
sage: wch = A(2*rho) # character of the irred. representation
sage: ch = a(2*rho) # element in the weight ring
sage: dch = demazure(ch,[1,2,1,2]) # apply 4 Demazure
operators
sage: sum([m for (wt,m) in dch.mlist()]) # degree should be 81
81
sage: sorted(wch.mlist()) == sorted(dch.mlist())
True
"""
a = ch.parent()
# passing a list applies the operators in the indicated order
# elements of i index simple roots
if type(i) == list:
ch_temp = copy(ch)
for j in i:
ch_temp = demazure(ch_temp, j)
return ch_temp
# otherwise i is an index for a simple root
space = a.space()
if i not in space.index_set():
raise IndexError('%s does not index a simple root' %
i.__str__())
alpha = space.simple_root(i) # if i is in space.index_set()
ch_temp=a(0)
for (wt,wtm) in ch.mlist():
n = wt.scalar(2*alpha/alpha.scalar(alpha))
if n >= 0:
ch_temp += wtm * sum([a(wt-i*alpha) for i in range(n+1)])
if n == -1: # for clarity
ch_temp += 0
if n <= -2:
ch_temp += wtm * sum([-1*a(wt+i*alpha) for i in range(1,-
n)])
return ch_temp
I agree with the design comments bump has. I think it would also be
useful to have a "degree" method in WeightRingElement:
def degree(self):
"""
The degree of the character, that is, the dimension of module.
EXAMPLES::
sage: b = WeightRing(WeylCharacterRing(['B',2]))
sage: ch = 3*b((3,1)) + 2*b((2,1)) + b((1,1))
sage: ch.degree()
6
"""
return sum(self._mdict[k] for k in self._mdict)
Here is the code for 2d plotting. There are helper functions for
computing coordinates in A2 and G2, and then the plot() method.
def sl3_2d_coords(v):
"""
INPUT:
v = 3d vector lying in the plane x+y+z=0
OUTPUT:
2d vector of coordinates with respect to basis vectors:
u1 = vector((1/2, -1/2, 0))
u2 = vector((1/2*sqrt(1/2)*sqrt(2/3), 1/2*sqrt(1/2)*sqrt(2/3),
-sqrt(1/2)*sqrt(2/3)))
"""
u1 = vector((1/2, -1/2, 0))
u2 = vector((1/2*sqrt(1/2)*sqrt(2/3), 1/2*sqrt(1/2)*sqrt(2/3),
-sqrt(1/2)*sqrt(2/3)))
return vector((v*u1, v*u2))
def g2_2d_coords(v):
"""
INPUT:
v = 3d vector lying in the plane x+y+z=0
OUTPUT:
2d vector of coordinates with respect to basis vectors:
u1 = vector((0, 1, -1))
u2 = vector((1, -1/2, -1/2))
"""
u1 = vector((0, 1, -1))
u2 = vector((1, -1/2, -1/2))
return vector((v*u1, v*u2))
def plot(self, plot_roots=True, arrow_style=None, point_style=None,
mult_scale=None):
"""
Plot the character if cartan type is rank 2.
EXAMPLES::
sage: A = WeylCharacterRing(['B',2])
sage: a = WeightRing(A)
sage: lam = a.space().fundamental_weights()
sage: M = demazure(a(4*lam[1]+4*lam[2]), [1,2])
sage: M.plot()
"""
# plot parameters
if arrow_style == None:
arrow_style={'rgbcolor':(0,0,1),'width':
1,'linestyle':'dotted'}
if point_style == None:
point_style={'rgbcolor':(1,0,0)}
if mult_scale == None:
mult_scale = lambda m: m^2*10
# Type A2
if tuple(self.cartan_type()) == ('A',2):
A2 = WeylCharacterRing(['A',2])
roots = A2.space().roots()
# compute coordinates of the roots
root_coords = [ sl3_2d_coords(vector(A2.coerce_to_sl(r)))
for r in roots ]
# compute coordinates of the weights and store with
multiplicity
wt_coords =
[ (tuple(sl3_2d_coords(vector(A2.coerce_to_sl(wt)))),m)
for (wt,m) in self.mlist() ]
# Type B2: alpha[2] is short
elif tuple(self.cartan_type()) == ('B',2):
B2 = WeylCharacterRing(['B',2])
roots = B2.space().roots()
# Note: we reflect about the line y=x here to be consistent
with
# the std. picture of B2 where alpha[2] points east and
# alpha[1] points northwest
root_coords = [ vector((r.coefficient(1), r.coefficient(0)))
for r in roots ]
wt_coords = [ ((wt.coefficient(1), wt.coefficient(0)), m)
for (wt,m) in self.mlist() ]
# Type G2: alpha[1] is short
elif tuple(self.cartan_type()) == ('G',2):
G2 = WeylCharacterRing(['G',2])
roots = G2.space().roots()
root_coords =
[ g2_2d_coords(vector(map(r.coefficient,range(3))))
for r in roots ]
wt_coords =
[ (tuple(g2_2d_coords(vector(map(wt.coefficient,range(3))))), m)
for (wt,m) in self.mlist() ]
else:
raise NotImplementedError('Plotting is not implemented for
this type')
root_plot = sum(plot(v,**arrow_style) for v in root_coords)
wt_plot = sum([point(wt, size=mult_scale(m), **point_style)
for (wt,m) in wt_coords])
if plot_roots:
return wt_plot+root_plot
else:
return wt_plot
I should add more examples and flesh out the docstrings, of course. I
need to clean up my plotting code a bit as well.
I'd be happy to start a ticket to add the following methods to
WeightRingElement:
WeightRingElement.demazure_character(w) # w = Weyl group element
WeightRingElement._demazure_character(i) # helper method; i = index of
simple root
WeightRingElement.plot( ... ) # plot method for types A2, B2, G2, A1 x
A1
WeightRingElement.degree() # as above
I can wait until #7922 is reviewed, or defer to someone else with more
experience here.
--
Benjamin Jones
[email protected]
On Jan 31, 3:17 am, "Nicolas M. Thiery" <[email protected]>
wrote:
> Dear Adrien, Anne, Benjamin, Dan, Vivianne, Nicolas, ...
>
> We should make sure to keep Benjamin in CC, since I am not sure he is
> on sage-combinat-devel (I bounced his message which was originally on
> sage-devel).
>
>
>
>
>
>
>
>
>
> On Sun, Jan 30, 2011 at 04:07:09PM -0800, bump wrote:
> > I too needed some code to compute with Demazure characters using the
> > weight ring. I agree that this is useful independent of the
> > Demazure characters as implemented in the crystal code.
>
> > What we did was just to make a standalone program that did what we
> > needed, but it would make a good enhancement to sage.
> > I think the right way would be as a method of the weight ring.
> > So the syntax would be
>
> > ch.demazure_character(w)
>
> > where ch is a weight ring element and w is an element of the Weyl
> > group.
> > There could be a helper method
>
> > ch._demazure_character(i)
>
> > that would produce the operator corresponding to the i-th simple
> > reflection. It would do something like this:
>
> > def _demazure(self, i):
> > a = self._parent.space().simple_coroots()[i]
> > ret = self._parent(0)
> > d = self._mdict
> > for t in d.keys():
> > k = t.inner_product(a)
> > if k >= 0:
> > ret += d[t]*sum(self._parent(t-j*a) for j in range(k+1))
> > else:
> > ret += -d[t]*sum(self._parent(t+j*a) for j in range(1,k))
> > return ret
>
> > This helper method would then be called by the method demazure,
> > which would use the reduced_word method of Weyl group elements
> > to to produce a sequence of called to the helper method.
>
> This looks very related to what Vivianne (and Adrien and Nicolas) is
> implementing around Schubert polynomials and the like. Vivianne: can
> you already compute demazure characters with your code?
> (WeylCharacterRing is basically the group algebra of the ambient
> lattice)? If yes, could you give an example here?
>
> By the way some suggestions for the code above:
> - .space() -> .basis().indices() ?
> - self._parent -> self.parent()
> - d = self._mdict; d.keys() -> self.support()
> - sum -> self.sum()
> - P(i) -> self.monomial(i) (where P is self._parent above)
>
> Ooooops, but all those comments are assuming that #7922 is applied,
> which I still haven't reviewed ... Ok, that going up my TODO pile ...
>
> Cheers,
> Nicolas
> --
> Nicolas M. Thi�ry "Isil" <[email protected]>http://Nicolas.Thiery.name/
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