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Let $\mathbb{Z}_m = \mathbb{Z}/m\mathbb{Z}$. Let $A$ be an $k \times n$ matrix over $\mathbb{Z}_m$. Let $f: \mathbb{Z}_m^n \to \mathbb{Z}_m^k$ be a linear map defined by $f(x) = Ax$, $x \in \mathbb{Z} …

Let $V, W$ be two vector spaces. We have $\Lambda^2(V \otimes W) \cong (\Lambda^2 V \otimes S^2 W) \oplus (S^2 V \otimes \Lambda^2 W)$. I am trying to find similar results for $\Lambda^3(V \otimes W)$ …

Let $V$ be a vector space. A reflection is a linear map $f: V \to V$ which has an eigenvalue $1$ with multiplicity $n-1$. Let $S_n$ be the symmetric group on $\{1,\ldots,n\}$. Then the reflections in …

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$. We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the …

In my research, I need to find the set of all minimal dependent sets of a given set of vectors. One method is to check every subset of the given set. But this method is very slow when the set of vecto …