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2021-07-11 18:21:34 +0200 edited question Obtaining admissible relations for acyclic tree quivers with Sage for QPA

Obtaining admissible relations for acyclic tree quivers with Sage for QPA Let $Q$ be a finite acyclic quiver which is a

2021-07-11 18:20:52 +0200 edited question Obtaining admissible relations for acyclic tree quivers with Sage for QPA

Obtaining admissible relations for acyclic tree quivers with Sage for QPA Let $Q$ be a finite acyclic quiver which is a

2021-07-11 18:20:13 +0200 asked a question Obtaining admissible relations for acyclic tree quivers with Sage for QPA

Obtaining admissible relations for acyclic tree quivers with Sage for QPA Let $Q$ be a finite acyclic quiver which is a

2021-06-11 18:04:31 +0200 commented question Finding generalised braid relations for finite Coxeter groups with Sage

Another small question: In the link the example is only for Weyl groups but there are some other Coxeter graphs not corr

2021-06-11 16:54:10 +0200 commented question Finding generalised braid relations for finite Coxeter groups with Sage

Thank you for the comment. This looks like exactly like what is needed but instead of [1, 2, 1], [2, 1, 2] the output sh

2021-06-11 16:53:53 +0200 commented question Finding generalised braid relations for finite Coxeter groups with Sage

Thank you for the comment. This looks like exactly like what is needed but instead of [1, 2, 1], [2, 1, 2] the output sh

2021-06-11 16:53:32 +0200 commented question Finding generalised braid relations for finite Coxeter groups with Sage

Thank you for the comment. This looks like exactly like what is needed but instead of [1, 2, 1], [2, 1, 2] the output sh

2021-06-11 16:51:49 +0200 commented answer Obtaining the Solomon-Orlik algebra in QPA with the help of Sage

Thanks again!

2021-06-11 16:51:44 +0200 marked best answer Obtaining the Solomon-Orlik algebra in QPA with the help of Sage

The Solomon-Orlik algebra (see Sage documentation: Orlik-Solomon algebra) associates to a finite matroid a finite dimensional algebra. It is implemented in Sage to obtain generators and relations of those algebras but there seems to be no easy way to translate them into the GAP-package QPA where many more functions are available to study those algebras.

Let M be a finite matroid which we can assume for simplicity given as a finite set X of vectors with set of circuits C, which are simply the minimal linearly dependent subsets of X.

Recall that the exterior algebra on a ground set of variables x1,...,xn (assume them to be ordered x1 < ... < xn) is given as the quotient of the free algebra on x1,...,xn modulo the relations xi^2 and xi xj +xj xi for i < j.

In QPA the exterior algebra for n=4 is for example obtained as follows:

Q := Quiver(1,[[1,1,"x1"],[1,1,"x2"],[1,1,"x3"],[1,1,"x4"]]);
KQ := PathAlgebra(Rationals,Q);
AssignGeneratorVariables(KQ);
rel := [x1^2,x2^2,x3^2,x4^2,x1*x2+x2*x1,x1*x3+x3*x1,x1*x4+x4*x1,x2*x3+x3*x2,x2*x4+x4*x2,x3*x4+x4*x3];
A := KQ/rel;

Now the Solomon-Orlik algebra (see Sage documentation: Orlik-Solomon algebra) of a matroid M is given by adding the relations $$\sum_{r=1}^{t}{(-1)^{r-1}{x_{j_1} ... \hat{x_{j_r}} .... x_{j_t}}$$ for every element (x1< ... < xt) in the circuits C.

For example when M is the matroid given by the vectors x1=t=(1,-1,0), x2=u=(0,1,-1), x3=v=(-1,0,1), x4=w=(1,1,1). then we have one circut given by x1 x2 x3 and thus one additional relation given by x2 x3 - x1 x3 +x1x2 and the algebra in QPA can be obtained as follows:

Q := Quiver(1,[[1,1,"x1"],[1,1,"x2"],[1,1,"x3"],[1,1,"x4"]]);
KQ := PathAlgebra(Rationals,Q);
AssignGeneratorVariables(KQ);
rel := [x1^2,x2^2,x3^2,x4^2,x1*x2+x2*x1,x1*x3+x3*x1,x1*x4+x4*x1,x2*x3+x3*x2,x2*x4+x4*x2,x3*x4+x4*x3,x2*x3-x1*x3+x1*x2];
A := KQ/rel;

Question: Is there a direct way to directly obtain the quiver and relations for QPA in the format as in the examples using Sage for a given matroid?

2021-06-11 14:36:19 +0200 edited question Finding generalised braid relations for finite Coxeter groups with Sage

Finding generalised braid relations for finite Coxeter groups with Sage Let $S={x1,...,xn }$ be a finite set and $A=(a_{

2021-06-11 14:22:59 +0200 edited question Finding generalised braid relations for finite Coxeter groups with Sage

Finding generalised braid relations for finite Coxeter groups with Sage Let $S={x1,...,xn }$ be a finite set and $A=(a_{

2021-06-11 14:22:00 +0200 edited question Finding generalised braid relations for finite Coxeter groups with Sage

Finding generalised braid relations for finite Coxeter groups with Sage Let $S={x1,...,xn }$ be a finite set and $A=(a_{

2021-06-11 14:21:06 +0200 asked a question Finding generalised braid relations for finite Coxeter groups with Sage

Finding generalised braid relations for finite Coxeter groups with Sage Let $S={x1,...,xn }$ be a finite set and $A=(a_{

2021-06-11 09:18:30 +0200 commented answer Obtaining the Solomon-Orlik algebra in QPA with the help of Sage

Thank you very much! This works. I am not so experienced in Sage: Is there a quick way to make Sage automatically write

2021-06-10 06:16:55 +0200 received badge  Nice Question (source)
2021-06-09 14:00:12 +0200 asked a question Finding a diagonal matrix for a given matrix

Finding a diagonal matrix for a given matrix Let $A$ be a real $n \times n$ matrix. Then there exist orthogonal matrices

2021-06-07 18:37:22 +0200 edited question Obtaining the Solomon-Orlik algebra in QPA with the help of Sage

Obtaining the Solomon-Orlik algebra in QPA with the help of Sage The Solomon-Orlik algebra (see https://doc.sagemath.org

2021-06-07 18:37:07 +0200 edited question Obtaining the Solomon-Orlik algebra in QPA with the help of Sage

Obtaining the Solomon-Orlik algebra in QPA with the help of Sage The Solomon-Orlik algebra (see https://doc.sagemath.org

2021-06-07 18:35:55 +0200 asked a question Obtaining the Solomon-Orlik algebra in QPA with the help of Sage

Obtaining the Solomon-Orlik algebra in QPA with the help of Sage The Solomon-Orlik algebra (see https://doc.sagemath.org

2021-06-03 23:55:27 +0200 asked a question Obtaining a poset of matrices in Sage

Obtaining a poset of matrices in Sage Let $R=(r_1,...,r_m)$ and $S=(S_1,...,s_n)$ be a sequence of nonnegative integers

2021-06-03 21:04:36 +0200 commented answer Finding a permutation matrix associated to a non-singular matrix

Thank you for the answer. But in my question we need A=LPU instead of A=PLU.

2021-06-03 18:42:03 +0200 asked a question Finding a permutation matrix associated to a non-singular matrix

Finding a permutation matrix associated to a non-singular matrix Let $A$ be a non-singular complex matrix. Then there ex

2021-06-03 18:38:57 +0200 marked best answer Finding centraliser algebras of a finite set of matrices

Let $S$ be a finite set of $n \times n$-matrices over a field $K$ (lets say finite or the real or complex field).

Is it possible to obtain the $K$-algebra (or at least its vector space dimension) of $n \times n$-matrices X in Sage with $XY=YX$ for all $Y \in S$?

(I can only think of a way for doing this for finite field with very small $n$ by looking at all elements,but maybe there is a better technique in Sage)

2021-05-31 11:49:47 +0200 received badge  Nice Question (source)
2021-05-30 13:49:54 +0200 edited question Finding centraliser algebras of a finite set of matrices

Finding centraliser algebras of a finite set of matrices Let $S$ be a finite set of $n \times n$-matrices over a field $

2021-05-30 13:48:31 +0200 asked a question Finding centraliser algebras of a finite set of matrices

Finding centraliser algebras of a finite set of matrices Let $S$ be a finite set of $n \times n$-matrices over a field $

2021-05-30 13:46:27 +0200 marked best answer Obtaining the lattice of equivalence relations

Is there an easy method to obtain the lattice of all equivalence relations $L_n$ of a set with $n$ elements in Sage?

2021-05-24 16:57:50 +0200 received badge  Nice Question (source)
2021-05-24 13:18:33 +0200 asked a question Obtaining the lattice of equivalence relations

Obtaining the lattice of equivalence relations Is there an easy method to obtain the lattice of all equivalence relation

2021-04-29 19:48:32 +0200 received badge  Good Question (source)
2021-04-25 13:35:26 +0200 marked best answer Obtaining the immanent associated to a partition

For a partition $\lambda$ let $y_{\lambda}$ be the corresponding irreducible representation of the symmetric group $S_n$. Let $p_{\lambda}=\sum\limits_{\pi \in S_n}^{}{y_\lambda( \pi) x_{1 \pi(1)} ... x_{n \pi(n)}}$ be the immanent corresponding to $\lambda$. (For the sign representation we will just get the determinant for example). This is a polynomial in the $n^2$ variables $x_{i,j}$ over $\mathbb{Z}$.

My question is how can I obtain the immanent given a parition $\lambda$ using Sage?

My first problem is already that we need the polynomial ring in the $n^2$ variables $x_{i,j}$ and I am not sure how to define this in Sage depending on $n$.

2021-04-23 14:30:23 +0200 received badge  Nice Question (source)
2021-04-22 17:55:28 +0200 edited question Obtaining the immanent associated to a partition

Obtaining the immanent associated to a partition For a partition $\lambda$ let $y_{\lambda}$ be the corresponding irredu

2021-04-22 17:52:13 +0200 asked a question Obtaining the immanent associated to a partition

Obtaining the immanent associated to a partition For a partition $\lambda$ let $x_{\lambda}$ be the corresponding irredu

2021-03-30 19:42:14 +0200 edited question Obtaining directed graphs associated to matrices

Obtaining directed graphs associated to matrices Let $M$ be an $n \times n$-matrix with entries only 0 or 1 and all diag

2021-03-30 19:41:51 +0200 commented question Obtaining directed graphs associated to matrices

Thank you for your comment. I forgot that the graph is defined via the matrix $R-U$ instead of $U$. I hope it it correct

2021-03-30 19:41:25 +0200 edited question Obtaining directed graphs associated to matrices

Obtaining directed graphs associated to matrices Let $M$ be an $n \times n$-matrix with entries only 0 or 1 and all diag

2021-03-30 17:56:35 +0200 edited question Obtaining directed graphs associated to matrices

Obtaining directed graphs associated to matrices Let $M$ be an $n \times n$-matrix with entries only 0 or 1 and all diag

2021-03-30 17:53:51 +0200 edited question Obtaining directed graphs associated to matrices

Obtaining directed graphs associated to matrices Let $M$ be an $n \times n$-matrix with entries only 0 or 1 and all diag

2021-03-30 17:53:04 +0200 edited question Obtaining directed graphs associated to matrices

Obtaining directed graphs associated to matrices Let $M$ be an $n \times n$-matrix with entries only 0 or 1. Let $R$ be

2021-03-30 17:51:39 +0200 edited question Obtaining directed graphs associated to matrices

Obtaining directed graphs associated to matrices Let $M$ be an $n \times n$-matrix with entries only 0 or 1. Let $R$ be

2021-03-30 17:51:14 +0200 asked a question Obtaining directed graphs associated to matrices

Obtaining directed graphs associated to matrices Let $M$ be an $n \times n$-matrix with entries only 0 or 1. Let $R$ be

2021-03-26 21:45:39 +0200 marked best answer 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

2021-03-21 12:29:03 +0200 marked best answer Obtaining incidence algebras for GAP via Sage

Gap (via its package QPA) can obtain the incidence algebras of a given connected poset P as a quiver algebra KQ/I (note that any incidence algebra is isomorphic to the quiver algebra where Q is the Hasse quiver and the relations I are generated by all commutativity relations w1-w2 where the paths w1 and w2 start and end at the same points).

However, sadly GAP is very slow with this and obtaining the quiver algebra for a poset with 40 or more points can take days or even weeks.

I wonder whether there is a way to use Sage to obtain two lists of quiver and relations from a given poset and then use those lists to input them in GAP to directly obtain the quiver algebra in GAP (so that GAP has to do no computations for the relations, which seems to be the main problem although I'm not really sure why it takes so long).

Solving this problem would be very important to deal with large posets in GAP (and one really needs the incidence algebra as a quiver algebra in QPA to do homological algebra with it as Sage has no such functions).

Here is an example, namely the strong Bruhat order of the symmetric group $S_3$ with quiver and relations.

First here it is in Sage:

Y = posets.SymmetricGroupBruhatOrderPoset(3)
display(Y)

Now quiver and relations for GAP should look as follows:

Quiver(
    ["x123", "x132", "x213", "x231", "x312", "x321"],
    [["x123", "x132", "x123_x132"],
     ["x123", "x213", "x123_x213"],
     ["x132", "x231", "x132_x231"],
     ["x132", "x312", "x132_x312"],
     ["x213", "x231", "x213_x231"],
     ["x213", "x312", "x213_x312"],
     ["x231", "x321", "x231_x321"],
     ["x312", "x321", "x312_x321"]])

[ x123_x132*x132_x231-x123_x213*x213_x231, 
  x123_x132*x132_x312-x123_x213*x213_x312,
  x132_x231*x231_x321-x132_x312*x312_x321,
  x213_x231*x231_x321-x213_x312*x312_x321 ]

Here the first entry is the quiver specified by the points

["x123", "x132", "x213", "x231", "x312", "x321"]

and by the arrows

[["x123", "x132", "x123_x132"],
 ["x123", "x213", "x123_x213"],
 ["x132", "x231", "x132_x231"],
 ["x132", "x312", "x132_x312"],
 ["x213", "x231", "x213_x231"],
 ["x213", "x312", "x213_x312"],
 ["x231", "x321", "x231_x321"],
 ["x312", "x321", "x312_x321"]]

that specify the Hasse diagram of the poset. Here we see that a point is called for example x132 , so we put an x before each name of the point (here the point is named 132 in Sage).

An arrow is named for example x132_x312 and the list entry ["x132","x312","x132_x312"] specifies that the arrow x132_x312 starts at the point x132 and ends at the point x312.

The relations are then of the form for example x123_x132*x132_x231-x123_x213*x213_x231.

Note that we do not need really all relations of the form w1-w2 in general as some of those relations might be implied by some other.

So to make things faster it might be a good idea to find "minimal" relations, but I am not quite sure how to do that.

As a test, it would be interesting if one can make things fast enough to obtain for example the strong Bruhat order of $S_5$ (which has 120 points) as a quiver algebra in GAP within 2 or 3 hours. At the moment this takes more than a month with GAP.