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Generating a certain list of non-commuting polynomials with Sage

asked 1 year ago

klaaa gravatar image

updated 1 year ago

I have a set of variables x1,...,xn and y1,...,ym for n,m>=1.

Now I can build all quadratic monomials of the form xiyj and yjxi (but we do not have xiyj=yjxi as we calcualte in the non-commutative polynomial ring). But something like xixj is not allowed as after an xi there must come an yj and after an yi there must come an xi.

Now I want with Sage the list of all possible relations of the form w1±w2±w3 such that all wi are different quadratic relations that all start either with a xi or a yj.

For example for n=2 and m=1, possible relations are (I hope I did not forget any relation) : x1y1,x1y1x2y2,x1y2+x2y2,x2y1,y1x1,y1x2,y1x1y1x2,y1x1+y1x2.

I am not sure how to do this in an easy way with Sage, but maybe someone knows a simple trick.

Thanks for any help.

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Please provide a code defining xi and yj.

Max Alekseyev gravatar imageMax Alekseyev ( 1 year ago )

I omitted a formal definition because it is a bit complicated. A formal definition would be to take the connected quiver Q with two points 1 and 2 and arrows xi from 1 to 2 and arrows yi from 2 to 1. Then one would have restrictions in the post. One can also work over a non-commutative polynomial ring with the relations xixj=0 and yiyj=0. I am not sure whether there is code to work with quivers like this in sage.

klaaa gravatar imageklaaa ( 1 year ago )

For this purpose one could also define xi and yj just as strings probably or formal non-commutative variables in sage to get the needed output. I might use the output mainly to continue to work in GAP, where quivers are available.

klaaa gravatar imageklaaa ( 1 year ago )

I understand that you want to work in a ring R where the addition is commutative biut the multiplication is not ; that is :

  • x,yR2, x+y=y+x

  • x,yR2, xyyx

Is that right ?

Also : do you consider quadratic monomials such as x2,xR ?

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 1 year ago )

Your example lists only moinomials and binomials, whereas your definition would also accept trinomials such as x1y1+x2y1+y1x1 (etc...) and quadrinomials such as x1y1+x2y1+y1x1+y1x2 (etc).

Could you either :

  • clarify your definition, or

  • complete your example ?

Thanks !

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 1 year ago )

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answered 1 year ago

Max Alekseyev gravatar image

Is this what you want?

n=2
m=3
x = [f'x_{i+1}' for i in range(n)]
y = [f'y_{i+1}' for i in range(m)]

xy = [xi+yj for xi in x for yj in y]   # x first, then y

yx = [yj+xi for xi in x for yj in y]    # y first, then x

for t in Tuples((-1..1),len(xy)):
    r = ' '.join([('+' if s>0 else '-')+mon for s,mon in zip(t,xy) if s])
    print(r)

for t in Tuples((-1..1),len(yx)):
    r = ' '.join([('+' if s>0 else '-')+mon for s,mon in zip(t,yx) if s])
    print(r)
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Asked: 1 year ago

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Last updated: Oct 13 '23