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Given a square $n\times n$ matrix $M$, let $M_i^j$ denote the $(n-1)\times(n-1)$ matrix obtained from M by omitting the $i$-th row and $j$-th column of $M$. The Desnanot–Jacobi Identity states $$\det(...

Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as: $$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...

Denote the falling factorials by $(x)_k=x(x-1)\cdots(x-k+1)$. The Vandermonde determinant is given by $\det\left[x_i^{j-1}\right]_1^n=\prod_{i<j}(x_j-x_i)$. It is well-known that in as much as ...

Assume that you are explicitly given the representation matrices of a group. How does one go about finding that common basis which will find the irreducible components of all of them simultaneously? ...

Update: see below Let $M$ be an $n\times n$ matrix that's constructed as follows. Construct the right-most column of $M$ as $[\alpha_1(x_1),\cdots,\alpha_n(x_n)]^T$ for some class of fixed functions $...