矩阵乘法指数的同时三重乘积性质及群论结果
The Simultaneous Triple Product Property and Group-theoretic Results for the Exponent of Matrix Multiplication
AI总结：
该研究通过群论方法探讨矩阵乘法指数，利用同时三重乘积性质得到指数界，并展示特定群结构可使指数低于2.84甚至接近2.02。
摘要：
我们描述了从群论中某些基本方法研究矩阵乘法代数复杂性所得到的某些特殊结论，这些方法由H.Cohn、C.Umans等人于2003年和2005年发展而来。这里的复杂性度量是矩阵乘法指数，一个介于2和3之间的实数，曾被推测为2。更具体地说，如果一个有限群通过其正则代数同时“实现”多个独立的矩阵乘法，并且具有一个满足所谓同时三重乘积性质（STPP）的索引子集族，那么这些多个乘法的复杂度不会超过该代数的秩（复杂度）。这导致了指数与群的大小及其STPP三元组大小以及其不同不可约表示的维度之间的界。在这一点上，阿贝尔群与对称群的 wreath 积显得尤为重要，我们给出了这样一个例子，显示指数小于2.84，并且根据它所实现的多个矩阵乘法的数量，可能小至2.02。
英文摘要：
We describe certain special consequences of certain elementary methods from group theory for studying the algebraic complexity of matrix multiplication, as developed by H. Cohn, C. Umans et. al. in 2003 and 2005. The measure of complexity here is the exponent of matrix multiplication, a real parameter between 2 and 3, which has been conjectured to be 2. More specifically, a finite group may simultaneously "realize" several independent matrix multiplications via its regular algebra if it has a family of triples of "index" subsets which satisfy the so-called simultaneous triple product property (STPP), in which case the complexity of these several multiplications does not exceed the rank (complexity) of the algebra. This leads to bounds for the exponent in terms of the size of the group and the sizes of its STPP triples, as well as the dimensions of its distinct irreducible representations. Wreath products of Abelian with symmetric groups appear especially important, in this regard, and we give an example of such a group which shows that the exponent is less than 2.84, and could be possibly be as small as 2.02 depending on the number of simultaneous matrix multiplications it realizes.
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