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2024-10-10 17:22:59 +0100 commented question Obtaining a finite dimensional algebra associated to Lie algebras in QPA

It might also be useful to have this algebra in Sage, but Im not very experienced with analysing algebras in Sage.

2024-10-10 17:22:52 +0100 commented question Obtaining a finite dimensional algebra associated to Lie algebras in QPA

But it might also be useful to have this algebra in Sage, but Im not very experienced with analysing algebras in Sage.

2024-10-10 17:21:50 +0100 commented question Obtaining a finite dimensional algebra associated to Lie algebras in QPA

Sage has the needed data on semisimple Lie algebras I think, for example to get a nice basis with known multiplication t

2024-10-10 16:05:17 +0100 commented question Obtaining a finite dimensional algebra associated to Lie algebras in QPA

It is about presenting it for QPA. Getting the algebra inside Sage might be not so useful as GAP and QPA have more comma

2024-10-09 14:25:25 +0100 edited question Obtaining a finite dimensional algebra associated to Lie algebras in QPA

Obtaining a finite dimensional algebra associated to Lie algebras in QPA Let g be a finite dimensional semisimple Lie al

2024-10-09 14:23:56 +0100 edited question Obtaining a finite dimensional algebra associated to Lie algebras in QPA

Obtaining a finite dimensional algebra associated to Lie algebras in QPA Let g be a finite dimensional semisimple Lie al

2024-10-09 14:23:03 +0100 asked a question Obtaining a finite dimensional algebra associated to Lie algebras in QPA

Obtaining a finite dimensional algebra associated to Lie algebras in QPA Let g be a finite dimensional semisimple Lie al

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2024-02-09 13:53:17 +0100 commented answer Obtaining mutation class of a quiver in Sage with Sage

Thank you very much again! I noted that there is an error for E7: Q = ClusterQuiver(['E',7]) Ts = Q.mutation_class() g

2024-02-09 13:51:58 +0100 commented answer Obtaining mutation class of a quiver in Sage with Sage

Thank you very much again! A last small question: Is there an easy command to filter out those quivers, which are not ac

2024-02-09 09:24:32 +0100 commented answer Obtaining mutation class of a quiver in Sage with Sage

Thank you very much. The first way works in the sage online cell, while the second outputs nothing (I dont know why). Bu

2024-02-09 07:31:19 +0100 asked a question Obtaining mutation class of a quiver in Sage with Sage

Obtaining mutation class of a quiver in Sage with Sage Given a quiver $Q$ of finite mutation class (such as a Dynkin qui

2023-10-14 11:44:34 +0100 marked best answer Generating a certain list of non-commuting polynomials with Sage

I have a set of variables $x_1,...,x_n$ and $y_1,...,y_m$ for $n,m >=1$.

Now I can build all quadratic monomials of the form $x_i y_j$ and $y_j x_i$ (but we do not have $x_i y_j= y_j x_i$ as we calcualte in the non-commutative polynomial ring). But something like $x_i x_j$ is not allowed as after an $x_i$ there must come an $y_j$ and after an $y_i$ there must come an $x_i$.

Now I want with Sage the list of all possible relations of the form $w_1 \pm w_2 \pm w_3 \cdots$ such that all $w_i$ are different quadratic relations that all start either with a $x_i$ or a $y_j$.

For example for $n=2$ and $m=1$, possible relations are (I hope I did not forget any relation) : $x_1 y_1, x_1 y_1-x_2 y_2, x_1 y_2 + x_2 y_2 , x_2 y_1,y_1 x_1, y_1 x_2, y_1 x_1 - y_1 x_2 , y_1x_1+y_1 x_2$.

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.

2023-10-13 06:56:59 +0100 commented question Generating a certain list of non-commuting polynomials with Sage

For this purpose one could also define $xi$ and $yj$ just as strings probably or formal non-commutative variables in sag

2023-10-13 06:56:48 +0100 commented question Generating a certain list of non-commuting polynomials with Sage

For this purpose one could also define $xi$ and $yj$ just as string probably or formal non-commutative variables in sage

2023-10-13 06:55:59 +0100 commented question Generating a certain list of non-commuting polynomials with Sage

For this purpose one could also define $xi$ and $yj$ just as string probably to get the needed output.

2023-10-13 06:45:43 +0100 commented question Generating a certain list of non-commuting polynomials with Sage

I omitted a formal definition because it is a bit complicated. A formal definition would be to take the connected quiver

2023-10-13 06:43:24 +0100 commented question Generating a certain list of non-commuting polynomials with Sage

I omitted a formal definition because it is a bit complicated. A formal definition would be to take the connected quiver

2023-10-13 00:15:02 +0100 edited question Generating a certain list of non-commuting polynomials with Sage

Generating a certain list of non-commuting polynomials with Sage I have a set of variables $x_1,...,x_n$ and $y_1,...,y_

2023-10-13 00:14:43 +0100 edited question Generating a certain list of non-commuting polynomials with Sage

Generating a certain list of non-commuting polynomials with Sage I have a set of variables $x_1,...,x_n$ and $y_1,...,y_

2023-10-13 00:14:20 +0100 asked a question Generating a certain list of non-commuting polynomials with Sage

Generating a certain list of non-commuting polynomials with Sage I have a set of variables $x_1,...,x_n$ and $y_1,...,y_

2023-10-12 09:51:17 +0100 marked best answer Fundamental polynomials with Sage

Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N \geq k$.Let $S$ be a subset of the set {1,2,...,k-1}. Define the fundamental polynomial in $N$ variables as $F_{k,S}(x_1,x_2,...,x_N):= \sum\limits_{1 \leq i_1 \leq i_2 \leq \cdots \leq i_k \leq N \ ; \ j \in S \implies i_j < i_{j+1}}^{}{x_{i_1} x_{i_2} \cdots x_{i_k}}$.

For example we have $F_{k,\emptyset}=h_k$, the complete symmetric function and $F_{k,S}=e_k$ for S={1,...,k-1}, the elementary symmetric function.

Question: Is there an existing command or another easy way to obtain those polynomials for a given triple $(N,k,S)$ via Sage?

I found a section about quasi-symmetric functions in Sage but it uses the language of Hopf algebras and I am not sure whether this contains the fundamental polynomials already in this form.

2023-10-12 01:47:27 +0100 edited question Fundamental polynomials with Sage

Fundamental polynomials with Sage Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N

2023-10-12 01:46:34 +0100 edited question Fundamental polynomials with Sage

Fundamental polynomials with Sage Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N

2023-10-12 01:46:04 +0100 edited question Fundamental polynomials with Sage

Fundamental polynomials with Sage Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N

2023-10-12 01:45:47 +0100 edited question Fundamental polynomials with Sage

Fundamental polynomials with Sage Let $N>0$ be a positive natural number and let $k \geq 0$ be natural number with $N