Katz-Long-Moody Hermite 型的一个闭式签名公式

中文摘要 AI 辅助

Katz-Long-Moody 构造将一个自由群与辫群的半直积（由 Artin 作用定义）的表示以及一个非零参数关联为该同一群的一个新表示。在纯辫群上，它对应于 Haraoka 对 KZ 型方程的乘法中间卷积。对于酉输入和单位圆上除 1 以外的卷积参数，该构造赋予商表示一个典范的非退化不变 Hermite 型。我们给出了其签名的一个闭式公式，该公式以输入局部单值本征角、其有序乘积的本征角以及卷积参数表示。该公式考虑了在取商之前形式的核，以及共振参数处的签名变化。它精确确定了诱导形式何时是正定的，从而回答了伴随论文中提出的正定性问题。正定性意味着输出表示的可酉化性；当该表示为不可约时，其逆命题也成立。证明使用了初等线性代数：一个行列式恒等式、显式的分块主元公式，以及酉矩阵的 Cayley 变换之和的惯性公式。作为应用，我们将秩一构造与 Haraoka 对 Pochhammer 系统的不变形式进行了显式比较，恢复了 Beukers-Heckman 交错准则的 Gauss 情形，并确定了 Hecke 和 Temperley-Lieb 特殊化的正定参数区间。

英文摘要

The Katz-Long-Moody construction associates to a representation of the semidirect product of a free group and a braid group, defined by the Artin action, and a nonzero parameter a new representation of the same group. On the pure braid group it corresponds to Haraoka's multiplicative middle convolution for KZ-type equations. For unitary input and a convolution parameter on the unit circle other than one, the construction equips the quotient representation with a canonical non-degenerate invariant Hermitian form. We give a closed formula for its signature in terms of the eigenangles of the input local monodromies, the eigenangles of their ordered product, and the convolution parameter. The formula accounts for the kernel of the form before passage to the quotient and for signature changes at resonant parameters. It determines precisely when the induced form is definite, answering the definiteness problem posed in the companion paper. Definiteness implies unitarizability of the output representation; the converse holds when that representation is irreducible. The proof uses elementary linear algebra: a determinant identity, explicit block-pivot formulas, and an inertia formula for sums of Cayley transforms of unitary matrices. As applications, we compare the rank-one construction explicitly with Haraoka's invariant form for Pochhammer systems, recover the Gauss case of the Beukers-Heckman interlacing criterion, and determine the definite parameter intervals for the Hecke and Temperley-Lieb specializations.

发表机构

• Chiba University（千叶大学）

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