Journal of Algebraic Combinatorics ( IF 0.6 ) Pub Date : 2020-09-13 , DOI: 10.1007/s10801-020-00972-1 Kazuya Aokage
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 \(\widetilde{S_n^{'}}\).
中文翻译:
对称群的Schur覆盖群的基本自旋表示的张量平方
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\(n \ ge 4,\ n \ ne 6 \)\(\ widetilde {S_n ^ {'}} \)。