论文标题
在$ \ text {sl} _2（\ mathbb {z}）$的对称表示
On symmetric representations of $\text{SL}_2(\mathbb{Z})$
论文作者
论文摘要
我们介绍了$ \ text {sl} _2（\ mathbb {z}）$的对称和对称表示的概念。 $ \ text {sl} _2（\ mathbb {z}）$的线性表示是对称的，并且具有一致的内核。相反，也可以从有限维对称的$ \ text {sl} _2（\ Mathbb {z}）$中重建模块化数据。通过调查$ \ mathbb {z}/2 \ mathbb {z} $ - 某些Weil表示的对称性，我们证明所有有限二维的一致性表示$ \ text {sl} _2 _2 _2 _2 _2（\ mathbb {z}）$都可以模样。我们还提供了$ \ text {sl} _2（\ mathbb {z}）$的$ \ text {sl} _2的不可弥补的不符合表示表示的示例。
We introduce the notions of symmetric and symmetrizable representations of $\text{SL}_2(\mathbb{Z})$. The linear representations of $\text{SL}_2(\mathbb{Z})$ arising from modular tensor categories are symmetric and have congruence kernel. Conversely, one may also reconstruct modular data from finite-dimensional symmetric, congruence representations of $\text{SL}_2(\mathbb{Z})$. By investigating a $\mathbb{Z}/2\mathbb{Z}$-symmetry of some Weil representations at prime power levels, we prove that all finite-dimensional congruence representations of $\text{SL}_2(\mathbb{Z})$ are symmetrizable. We also provide examples of unsymmetrizable noncongruence representations of $\text{SL}_2(\mathbb{Z})$ that are subrepresentations of a symmetric one.
