Abstract

We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it.

摘要

我们介绍了一种用于生成编码图形对称性的量子置换矩阵的 Sinkhorn 型算法。我们的算法生成方矩阵，其条目是正交投影到满足一组线性关系的一维子空间上。我们用它来实验量子置换群及其量子子群的表示论。我们将其应用于给定的有限图（没有多条边）是否具有 Banica 意义上的量子对称性的问题。为此，我们运行 Sinkhorn 算法并检查生成的投影是否通勤。我们讨论了产生的数据和由此产生的未来研究的一些问题。