arXiv 2007.115352020-07math.RT

抛物型Hecke子代数的自由度与迹图

The freeness and trace conjectures for parabolic Hecke subalgebras

Eirini Chavli, Maria Chlouveraki

备注：: 24 pages
发表信息：: Journal of Pure and Applied Algebra 226 (2022) 107050
DOI：: 10.1016/j.jpaa.2022.107050

摘要：

关于与复反射群$W$相关联的一般Hecke代数$\mathcal{H}(W)$的结构的两个最主猜测表明，$\mathcal{H}(W)$是其定义环上秩为$|W|$的自由模，并且$\mathcal{H}(W)$允许一个规范对称迹.第一个猜想最近已经成为一个定理，而第二个猜想，已知为实反射群，只证明了一些特殊的非实复反射群（所有秩$2$，但一个）。关于与$W$的抛物子群$W'$相关联的抛物Hecke子代数$\mathcal{H}(W')$的结构的两个最主猜测表明，$\mathcal{H}(W)$是秩$|W|/|W'|$的自由左、右$\mathcal{H}(W')$ -模，并且$\mathcal{H}(W')$的正则对称迹是$\mathcal{H}(W)$的正则对称迹对$\mathcal{H}(W')$的限制.到目前为止，这两个定理只对实反射群成立。我们证明他们的所有复杂的反射群的秩$2$的BMM对称跟踪猜想是已知的。

英文摘要：

The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace. The first conjecture has recently become a theorem, while the second conjecture, known to hold for real reflection groups, has only been proved for some exceptional non-real complex reflection groups (all of rank $2$ but one). The two most fundamental conjectures on the structure of the parabolic Hecke subalgebra $\mathcal{H}(W')$ associated with a parabolic subgroup $W'$ of $W$ state that $\mathcal{H}(W)$ is a free left and right $\mathcal{H}(W')$-module of rank $|W|/|W'|$, and that the canonical symmetrising trace of $\mathcal{H}(W')$ is the restriction of the canonical symmetrising trace of $\mathcal{H}(W)$ to $\mathcal{H}(W')$. Until now, these two conjectures have only be known to be true for real reflection groups. We prove them for all complex reflection groups of rank $2$ for which the BMM symmetrising trace conjecture is known to hold.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文