论文标题
基质乘法张量的各向同性组
The isotropy group of the matrix multiplication tensor
论文作者
论文摘要
By an {\em isotropy group} of a tensor $t\in V_1 \otimes V_2\otimes V_3=\widetilde V$ we mean the group of all invertible linear transformations of $\widetilde V$ that leave $t$ invariant and are compatible (in an obvious sense) with the structure of tensor product on~$\widetilde V$.我们考虑$ t $是矩形矩阵乘法图的结构张量的情况。 De Groote，Strassen和Brockett-Dobkin在1970年代研究了该张量的各向同性组。在目前的工作中，我们扩大，更精确，以张量空间的小组动作语言揭露，并赋予证据以前已知的结果。这对于研究接收对称性的快速基质乘法的算法是必不可少的。后者似乎是构建快速算法的新方法。
By an {\em isotropy group} of a tensor $t\in V_1 \otimes V_2\otimes V_3=\widetilde V$ we mean the group of all invertible linear transformations of $\widetilde V$ that leave $t$ invariant and are compatible (in an obvious sense) with the structure of tensor product on~$\widetilde V$. We consider the case where $t$ is the structure tensor of multiplication map of rectangular matrices. The isotropy group of this tensor was studied in 1970s by de Groote, Strassen, and Brockett-Dobkin. In the present work we enlarge, make more precise, expose in the language of group actions on tensor spaces, and endow with proofs the results previously known. This is necessary for studying the algorithms of fast matrix multiplication admitting symmetries. The latter seems to be a promising new way for constructing fast algorithms.
