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# Obtaining signed permutation in the bruhat poset in another form

(I edited the question, to make it more clear)

When I input the Bruhat poset of type Bn in Sage as follows

W = WeylGroup("B2", prefix="s")
P = W.bruhat_poset()
display(plot(P))


the elements look like s2*s1*s2.

Question: Is there a way to represent (in the picture of the poset in Sage) the elements in the form of signed permutation as for example findstat does (see for example http://www.findstat.org/StatisticsDat... ) but so that - is replaced by 0 and brackets and commas are ommited? So for example all signed permutations for $n=2$ would look as follows in this notation:

12
102
012
0102
21
201
021
0201


(The motivation is that one can use the code in the thread https://ask.sagemath.org/question/562... to obtain the quiver algebra in GAP with names one can regognize later).

Thanks for any help

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## Comments

We have

sage: W=ColoredPermutations(4,2)
sage: W.category()
Category of well generated finite irreducible complex reflection groups


but this has no Bruhat order method.

( 2021-03-19 20:49:19 +0200 )edit

And we have

sage: W=CoxeterGroup(['B',3])
sage: W.category()
Category of finite irreducible coxeter groups
sage: W.an_element()
[ 0  1 -a]
[ 1  1 -a]
[ 0  a -1]


which you may want to try.

( 2021-03-19 20:50:47 +0200 )edit

## 1 Answer

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You can use

sage: W = WeylGroup("B2", prefix="s")
sage: w=W.an_element()
sage: w.to_permutation()
(2, -1)


And relabel the poset using this map.

EDIT

sage: def new_label(w):
....:     return ''.join(str(u) for u in w.to_permutation()).replace('-','0')
sage: W.bruhat_poset().relabel(new_label)
Finite poset containing 8 elements

more

## Comments

Thanks, but it seems that in this form the code of the other thread can not be applied to the poset. I will do some more tests.

( 2021-03-19 21:05:06 +0200 )edit

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Asked: 2021-03-19 19:46:59 +0200

Seen: 177 times

Last updated: Mar 24 '21