AI 中文总结
本研究针对Katz-Long-Moody构造，明确了其生成的辫子群表示通过Hecke代数分解的条件，揭示特殊两股族外该性质与构造参数无关，还给出了通过Temperley-Lieb代数分解的精确判据。
AI 中文摘要
我们研究Katz-Long-Moody（KLM）构造，精确分类所得辫子群表示何时可通过Hecke代数分解。结果表明，除一个特殊的两股族外，该性质仅取决于自由群部分的特征值，与构造参数无关；同时给出所得表示何时可通过Temperley-Lieb代数分解的精确判据。
英文摘要
We study the Katz-Long-Moody (KLM) construction and classify exactly when the resulting braid group representations factor through Hecke algebras, for scalar braid part and semisimple free-group part, over an algebraically closed field of characteristic zero and at every convolution parameter lambda different from 1. We show that, except for one exceptional two-strand family, this property then depends only on the eigenvalues of the free-group part and is independent of lambda. Semisimplicity is a genuine hypothesis: we exhibit non-semisimple inputs, namely g = I + N with N nonzero and N^2 = 0, whose KLM quotient is nonetheless a Hecke module, realizing the permutation representation of the symmetric group. We also give an exact criterion for when the resulting representations factor through Temperley-Lieb algebras.
