We consider the tensor square of the basic spin representations of Schur covering groups \(\widetilde{S_n}\) and \(\widetilde{S_n^{'}}\) for the symmetric group \(S_n\). It is known from work of Stembridge that the irreducible components of the tensor square of the basic spin representations for \(\widetilde{S_n}\), for n odd, are multiplicity-free and indexed by hook partitions ([3], pp. 133). In this paper, we derive similar results for \(\widetilde{S_n}\) when n is even, and for \(\widetilde{S_n^{'}}\) when n is arbitrary. We assume that \(n\ge 4,\ n\ne 6\), when discussing \(\widetilde{S_n^{'}}\).

我们考虑对称组\（S_n \）的Schur覆盖组\（\ widetilde {S_n} \）和\（\ widetilde {S_n ^ {'}} \\）的基本自旋表示的张量平方。从Stembridge的工作中知道，对于n个奇数，\（\ widetilde {S_n} \）的基本自旋表示的张量平方的张量平方的不可约成分是无重数的，并且由钩子分区索引（[3]，pp 133）。在本文中，我们得出了类似的结果\（\ widetilde {S_N} \）时Ñ是偶数，以及用于\（\ widetilde {S_N ^ {'}} \）时Ñ是任意的。在讨论时，我们假设\（n \ ge 4，\ n \ ne 6 \）\（\ widetilde {S_n ^ {'}} \）。