Re: [sage-combinat-devel] Weight lattices

[Anne Schilling]

Hi Bruce,

You can also do

sage: C = KirillovReshetikhinCrystal(['A',1,1],1,2)
sage: B = C.classical_decomposition()
sage: L = CombinatorialFreeModule(ZZ,B.weight_lattice_realization())

sage: T = TensorProductOfCrystals(*[C]*3)
sage: hw = [a for a in T if a.is_highest_weight(index_set=[1])]
sage: sum( L.term( sum(a.lift().weight() for a in b), 1 ) for b in hw )
B[(3, 3)] + 3*B[(4, 2)] + 2*B[(5, 1)] + B[(6, 0)]

Then you can still compute the energy function in T.

Best wishes,

Anne

On 11/24/13 8:42 AM, Bruce wrote:
> Although I do not have an answer to the question I originally asked, I can 
> now do the calculation I wanted to do using Anne's suggestion of using 
> C.classical_decomposition(). Many thanks to Nicolas
> and Anne.

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Re: [sage-combinat-devel] Weight lattices

[Bruce]

Although I do not have an answer to the question I originally asked, I can 
now do the calculation I wanted to do using Anne's suggestion of using 
C.classical_decomposition(). Many thanks to Nicolas and Anne.

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Re: [sage-combinat-devel] Weight lattices

[Bruce]

Hi Anne,

It seems my example was an over simplification.

Start with the tensor power of a finite type crystal or representation.
This is a linear combination of weights with integer coefficients.
I have two methods for replacing each integer coefficient by a polynomial.
Each polynomial has positive integer coefficients and evaluated at 1 gives 
the original
integer. One method using the LiE package and plethysm in particular.
The other polynomial uses the energy function. The actual command (which I 
simplified
above) is:

 sage: sum( L.term( a.weight(), q^(a.energy_function() ) ) for a in hw ) 

where L now has coefficients polynomials in q.

One possibility would be to use classical_decomposition as you suggest and 
then
construct the crystal isomorphisms explicitly. Then using this isomorphism 
on words I can
construct the energy function on words in the ordinary crystal.

At the moment I am restricting attention to KR crystals whose classical 
decomposition
is irreducible. I have not understood if I also need to restrict to level 0.

On Saturday, November 23, 2013 10:16:39 PM UTC, Anne Schilling wrote:
>
> Hi Bruce, 
>
> If you are only interested in the classical weight, then you can do 
>
> sage: C = KirillovReshetikhinCrystal(['A',1,1],1,2) 
> sage: B = C.classical_decomposition() 
> sage: T = TensorProductOfCrystals(*[B]*3) 
> sage: L = CombinatorialFreeModule(ZZ,B.weight_lattice_realization()) 
> sage: hw = [a for a in T if a.is_highest_weight()] 
> sage: sum( L.term( a.weight(), 1 ) for a in hw ) 
> B[(3, 3)] + 3*B[(4, 2)] + 2*B[(5, 1)] + B[(6, 0)] 
>
> Best wishes, 
>
> Anne 
>
> On 11/23/13 2:13 AM, Bruce wrote: 
> > Sorry about being difficult. 
> > 
> > Here is one command: 
> > 
> > lie.p_tensor(3,[2],'A1') 
> > 
> > Here is an alternative: 
> > 
> > C = KirillovReshetikhinCrystal(['A',1,1],1,2) 
> > L = CombinatorialFreeModule(ZZ,C.weight_lattice_realization()) 
> > T = TensorProductOfCrystals(*[C]*3) 
> > hw = [ a for a in T if a.e(1) == None ] 
> > sum( L.term( a.weight(), 1 ) for a in hw ) 
> > 
> > I would like to convince sage (in this simplified example) that these 
> are "the same". 
> > 
> > Thank you for your patience. 
> > 
> > On Saturday, November 23, 2013 7:28:26 AM UTC, Nicolas M. Thiery wrote: 
> > 
> > On Fri, Nov 22, 2013 at 08:09:25AM -0800, Bruce wrote: 
> > >Here is the result of the first calculation (using the KR 
> crystal) and its 
> > >parent: 
> > 
> > Please, not the result but the command (or a simplified version) 
> > producing the result!  Otherwise one has to reconstruct the command 
> to 
> > play with the objects :-) 
> > 
> > Cheers, 
> > Nicolas 
>

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Re: [sage-combinat-devel] Weight lattices

[Anne Schilling]

Hi Bruce,

If you are only interested in the classical weight, then you can do

sage: C = KirillovReshetikhinCrystal(['A',1,1],1,2)
sage: B = C.classical_decomposition()
sage: T = TensorProductOfCrystals(*[B]*3)
sage: L = CombinatorialFreeModule(ZZ,B.weight_lattice_realization())
sage: hw = [a for a in T if a.is_highest_weight()]
sage: sum( L.term( a.weight(), 1 ) for a in hw )
B[(3, 3)] + 3*B[(4, 2)] + 2*B[(5, 1)] + B[(6, 0)]

Best wishes,

Anne

On 11/23/13 2:13 AM, Bruce wrote:
> Sorry about being difficult.
> 
> Here is one command:
> 
> lie.p_tensor(3,[2],'A1')
> 
> Here is an alternative:
> 
> C = KirillovReshetikhinCrystal(['A',1,1],1,2)
> L = CombinatorialFreeModule(ZZ,C.weight_lattice_realization())
> T = TensorProductOfCrystals(*[C]*3)
> hw = [ a for a in T if a.e(1) == None ]
> sum( L.term( a.weight(), 1 ) for a in hw )
> 
> I would like to convince sage (in this simplified example) that these are 
> "the same".
> 
> Thank you for your patience.
> 
> On Saturday, November 23, 2013 7:28:26 AM UTC, Nicolas M. Thiery wrote:
> 
> On Fri, Nov 22, 2013 at 08:09:25AM -0800, Bruce wrote:
> >Here is the result of the first calculation (using the KR crystal) 
> and its
> >parent:
> 
> Please, not the result but the command (or a simplified version)
> producing the result!  Otherwise one has to reconstruct the command to
> play with the objects :-)
> 
> Cheers,
> Nicolas

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Re: [sage-combinat-devel] Weight lattices

[Bruce]

Sorry about being difficult.

Here is one command:

lie.p_tensor(3,[2],'A1')

Here is an alternative:

C = KirillovReshetikhinCrystal(['A',1,1],1,2)
L = CombinatorialFreeModule(ZZ,C.weight_lattice_realization())
T = TensorProductOfCrystals(*[C]*3)
hw = [ a for a in T if a.e(1) == None ]
sum( L.term( a.weight(), 1 ) for a in hw )

I would like to convince sage (in this simplified example) that these are 
"the same".

Thank you for your patience. 

On Saturday, November 23, 2013 7:28:26 AM UTC, Nicolas M. Thiery wrote:
>
> On Fri, Nov 22, 2013 at 08:09:25AM -0800, Bruce wrote: 
> >Here is the result of the first calculation (using the KR crystal) 
> and its 
> >parent: 
>
> Please, not the result but the command (or a simplified version) 
> producing the result!  Otherwise one has to reconstruct the command to 
> play with the objects :-) 
>
> Cheers, 
> Nicolas 
> -- 
> Nicolas M. Thi�ry "Isil" > 
> http://Nicolas.Thiery.name/
>  
>

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Re: [sage-combinat-devel] Weight lattices

[Nicolas M. Thiery]

On Fri, Nov 22, 2013 at 08:09:25AM -0800, Bruce wrote:
>Here is the result of the first calculation (using the KR crystal) and its
>parent:

Please, not the result but the command (or a simplified version)
producing the result!  Otherwise one has to reconstruct the command to
play with the objects :-)

Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" 
http://Nicolas.Thiery.name/

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Re: [sage-combinat-devel] Weight lattices

[Bruce]

P.S. The first calculation uses a.weight() where a is an element of (the 
tensor product of) Kirillov-Reshetikhin crystals.
The second one use lie2sage to convert output from LiE (which is a string) 
to an element of the WeylCharacterRing.


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Re: [sage-combinat-devel] Weight lattices

[Bruce]

Here is the result of the first calculation (using the KR crystal) and its 
parent:

B[-9*Lambda[0] + 9*Lambda[1]] + (q^2+q)*B[-7*Lambda[0] + 7*Lambda[1]] + 
(q^4+q^3+q^2)*B[-5*Lambda[0] + 5*Lambda[1]] + 
(q^6+q^5+q^4+q^3)*B[-3*Lambda[0] + 3*Lambda[1]] + (q^5+q^4)*B[-Lambda[0] + 
Lambda[1]]

Free module generated by Weight lattice of the Root system of type ['A', 1, 
1] over Univariate Polynomial Ring in q over Integer Ring

Here is the result of the first calculation (using the KR crystal) and its 
parent:

(A1(3) + A1(5) + A1(9))*q^3 + (A1(1) + A1(3) + A1(5) + A1(7))*q^2 + (A1(1) 
+ A1(3) + A1(5) + A1(7))*q + A1(3)

Univariate Polynomial Ring in q over The Weyl Character Ring of Type ['A', 
1] with Rational Field coefficients

It is clear (to a human being) that we have the dictionary:
B[-9*Lambda[0] + 9*Lambda[1]] is the same as A1(9) etc.

I have just noticed that the coefficients are different but that is not the 
problem (at least not yet).


On Friday, November 22, 2013 3:46:02 PM UTC, Nicolas M. Thiery wrote:
>
> Hi Bruce, 
>
> On Fri, Nov 22, 2013 at 05:04:14AM -0800, Bruce wrote: 
> >I am trying to test a conjecture by comparing the results of two 
> >calculations. One calculation works with Kirillov-Reshetikhin 
> crystals and 
> >the result is an element of the free module on the Weight Lattice of 
> the 
> >affine root system (in one example ['A',1,1]). The other calculation 
> >returns an element of the WeylCharacterRing of the (finite) root 
> system 
> >(in the same example 'A1'). The ring of coefficients in both cases is 
> the 
> >same. This uses the ambient lattice. It is trivial to compare these 
> by 
> >hand but could I please have some suggestions how to get sage to 
> compare 
> >them? 
>
> Can you send a quick sample of both? There is a conversion from the 
> weight lattice to the ambient lattice, so that should be easy, but 
> it's best to talk on a concrete example. 
>
> Cheers, 
> Nicolas 
> -- 
> Nicolas M. Thi�ry "Isil" > 
> http://Nicolas.Thiery.name/
>  
>

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Re: [sage-combinat-devel] Weight lattices

[Nicolas M. Thiery]

Hi Bruce,

On Fri, Nov 22, 2013 at 05:04:14AM -0800, Bruce wrote:
>I am trying to test a conjecture by comparing the results of two
>calculations. One calculation works with Kirillov-Reshetikhin crystals and
>the result is an element of the free module on the Weight Lattice of the
>affine root system (in one example ['A',1,1]). The other calculation
>returns an element of the WeylCharacterRing of the (finite) root system
>(in the same example 'A1'). The ring of coefficients in both cases is the
>same. This uses the ambient lattice. It is trivial to compare these by
>hand but could I please have some suggestions how to get sage to compare
>them?

Can you send a quick sample of both? There is a conversion from the
weight lattice to the ambient lattice, so that should be easy, but
it's best to talk on a concrete example.

Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" 
http://Nicolas.Thiery.name/

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[sage-combinat-devel] Weight lattices

[Bruce]

I am trying to test a conjecture by comparing the results of two 
calculations. One calculation works with Kirillov-Reshetikhin crystals and 
the result is an element of the free module on the Weight Lattice of the 
affine root system (in one example ['A',1,1]). The other calculation 
returns an element of the WeylCharacterRing of the (finite) root system (in 
the same example 'A1'). The ring of coefficients in both cases is the same. 
This uses the ambient lattice. It is trivial to compare these by hand but 
could I please have some suggestions how to get sage to compare them?

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