AI 中文总结
研究对称关联方案中矩阵积分解，给出等价结构和谱准则等，分析多个标准族，如二类、\(P -\)多项式、汉明方案等，得到相关限制、定理及分类结果。
AI 中文摘要
我们研究对称关联方案中的矩阵积分解（MPFs），即等式\(A_SA_T = A_U\)，其中\(A_S\)、\(A_T\)、\(A_U\)是基本关系的无环并集，且普通矩阵积再次是一个\(0 - 1\)邻接矩阵。我们给出了MPFs的等价结构和谱准则，推导了价和秩的限制，并分析了几个标准族。对于二类方案，唯一非平凡的无环MPF来自于五边形方案。对于\(P -\)多项式方案，距离正则递推对乘积\(A_1A_i\)给出了强限制。我们还证明了\(A_SA_T = J - I\)情况下的通用五边形定理，并表明极值秩迫使\(A_U\)的所有非零特征值为\(\pm k(U)\)，从而给出二分性。最后，在汉明方案中，我们得到了秩障碍并对\(A_1A_T = A_U\)形式的MPFs进行了分类。
英文摘要
We study matrix product factorizations (MPFs) in symmetric association schemes: identities $A_SA_T=A_U$ where $A_S,A_T,A_U$ are loopless unions of basic relations and the ordinary matrix product is again a $0$-$1$ adjacency matrix. We give equivalent structural and spectral criteria for MPFs, derive valency and rank restrictions, and analyze several standard families. For $2$-class schemes, the only nontrivial loopless MPF comes from the scheme of the $5$-cycle. For $P$-polynomial schemes, the distance-regular recurrence gives strong restrictions on products $A_1A_i$. We also prove a universal pentagon theorem for the case $A_SA_T=J-I$, and show that extremal rank forces all non-zero eigenvalues of $A_U$ to be $\pm k(U)$, hence gives bipartiteness. Finally, in Hamming schemes we obtain rank obstructions and classify MPFs of the form $A_1A_T=A_U$: in $H(d,2)$, for $d\ge2$, the only non-zero loopless example is $A_1A_d=A_{d-1}$, which is trivial since $A_d$ has valency $1$; for $q>2$, no non-zero example occurs.
