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Let $B$ be a finite dimensional algebra over a field $K$ with dimension $n$ over $K$. Sage can give us a $K$-basis of this algebra that we denote by $e_i$ (i=1,...,n). Here is an example:

B = SymmetricGroup(4).algebra(QQ) 
U=B.basis()
display(list(U))

The structure constants $c_{l,i,j} $ of the algebra $B$ are defined by the condition: $e_i e_j = \sum\limits_{l}^{}{c_{l,i,j} e_l }$. For a list $A=[a_1,...,a_n]$ with field elements $a_i$ define the $n \times n$-matrix $P_A$ by $(P_A)_{i,j}= \sum\limits_{l}^{}{c_{l,i,j} a_l }$.

Question: How can one obtain the $P_A$ with Sage and check whether there exists a list A such that $P_A$ has non-zero determinant (meaning that $det(P_A)$ does not identically vanish to zero)?

The background of this question is that this is satisfied precisely when $B$ is a Frobenius algebra, see for example theorem 16.82 in the book "Lectures on Modules and Rings" by Lam.

Thanks again. You might be right but I want a program that works for sure so I want to avoid random things. I tried to m

Thanks again. You might be right but I want a program that works for sure so I want to avoid random things. I tried to m

Thanks again. You might be right but I want a program that works for sure so I want to avoid random things. I tried to m

Thanks again. You might be right but I want a program that works for sure so I want to avoid random things. I tried to m

I realised that my previous comment might be stupid as calculating such big determinants with variables might take too l

I realised that my previous comment might be stupid as calculating such big determinants with variables might take too l

Maybe one can view $P_A$ as a matrix with entries in the quotient field of the polynomial ring $K[a_1,a_2,...,a_n]$ and

Thank you very much. Is it possible to take as the entries of $A$ variables $a_i$ so that $det(P_A)$ is a polynomial in

Thank you very much. Is it possible to take as the entries of $A$ variables $a_i$ so that $P_A$ is a polynomial in the v

Testing whether a finite dimensional algebra is Frobenius Let $B$ be a finite dimensional algebra over a field $K$ with

Testing whether a finite dimensional algebra is Frobenius Let $B$ be a finite dimensional algebra over a field $K$ with

Testing whether a finite dimensional algebra is Frobenius Let $B$ be a finite dimensional algebra over a field $K$ with

Testing whether a finite dimensional algebra is Frobenius Let $B$ be a finite dimensional algebra over a field $K$ with

Thanks, I searched a bit and so far all solutions that work sadly delete the digraph structure and just give a graph som

Thanks I searched a bit and so far all solutions that work sadly delete the digraph structure and just give a graph some

Obtaining an unmodified latex output via Sage I use the following in Sage to obtain latex output of a digraph: G=DiGrap

Obtaining an unmodified latex output via Sage I use the following in Sage to obtain latex output of a digraph: G=DiGrap

Hmmm, thanks. Im a bit confused right now. I delete the question for now and try to pose a more precise version maybe la

Hmmm, Im a bit confused right now. I delete the question for now and try to pose a more precise version maybe later.

It means stable up to isomorphism and taking the opposite orientation. Sorry, I should have said that.

Checking whether a Dynkin graph is invariant under the canonical automorphism Let $G$ be a digraph with underlying graph

Checking whether a Dynkin graph is invariant under the canonical automorphism Let $G$ be a digraph with underlying graph

Checking whether a Dynkin graph is invariant under the canonical automorphism Let $G$ be a digraph with underlying graph

Checking whether a Dynkin graph is invariant under the canonical automorphism Let $G$ be a digraph with underlying graph

Checking whether a Dynkin graph is invariant under the canonical automorphism Let $G$ be a digraph with underlying graph