AI 中文总结
该研究解决了矩阵中心化子的固定源实现问题，明确与给定矩阵中心化子代数导出等价的代数结构，分类了基础倾斜模并确定其性质，还刻画了相关偏序集与突变图的结构。
AI 中文摘要
设$A$为任意域上某矩阵的中心化子代数。我们通过证明与$A$导出等价的矩阵中心化子代数恰好是$A$上倾斜模的相反自同态代数，解决了矩阵中心化子的固定源实现问题。我们对基础倾斜模进行分类并确定其相反自同态代数。倾斜偏序集是对称群上右弱序的乘积，每个素块对应一个因子，其阶等于该素块中不同指数的数量。结合中心，该偏序集可恢复所有素块的这些数量的多重集，尽管它无法将这些数量与局部中心因子进行规范匹配。对于每个素块，目标代数通过置换指数间的连续间隙得到，其同构类由间隙字的稳定子确定。因此，带标记突变图模去目标代数同构的商是Schreier多重图。我们还刻画了弱序定向何时降至其非环边。
英文摘要
Let $A$ be the centralizer algebra of a matrix over an arbitrary field. We solve the fixed-source realization problem for matrix centralizers by proving that the matrix centralizer algebras derived equivalent to $A$ are precisely the opposite endomorphism algebras of tilting modules over $A$. We classify the basic tilting modules and determine their opposite endomorphism algebras. The tilting poset is a product of right weak orders on symmetric groups, with one factor for each primary block and degree equal to the number of distinct exponents in that block. Together with the center, this poset recovers the multiset of these numbers across all primary blocks, although it does not canonically match them with the local center factors. For each primary block, the target algebras are obtained by permuting the successive gaps between exponents, and their isomorphism classes are determined by the stabilizer of the gap word. Consequently, the quotient of the labeled mutation graph by target-algebra isomorphism is a Schreier multigraph. We also characterize when the weak-order orientation descends to its nonloop edges.
Comments28 pages
