Existence of elementwise invariant vectors in representations of symmetric groups (Q6614047)

Let \((\rho_{\lambda}, V_{\lambda})\) be the irreducible representation of the symmetric group \(S_{n}\) associated with a partition \(\lambda\) of \(n\). Let \(w_{\mu}\in S_{n}\) be a permutation with cycle type \(\mu\) for a partition \(\mu\) of \(n\). A vector \(v \in V_{\lambda}\) is an invariant vector for \(w \in S_{n}\) if \(\rho_{\lambda}(w)v=v\).\N\NThe main result of the paper under review is the classification of the pairs of partitions \((\lambda, \mu)\) of a given integer \(n\) such that \(w_{\mu}\) does not admit a nonzero invariant vector in \(V_{\lambda}\). They are as follows: \N\begin{itemize}\N\item[(1)] \(\lambda=(1^{n})\), \(\mu\) is any partition of \(n\) for which \(w_{\mu}\) is odd; \N\item[(2)] \(\lambda=(n-1,1)\), \(\mu=(n)\), \(n \geq 2\); \N\item[(3)] \(\lambda=(2,1^{n-2})\), \(\mu=(n)\), \(n \geq 3\) is odd; \N\item[(4)] \(\lambda=(2^{2},1^{n-4})\), \(\mu=(n-2,2)\), \(n \geq 5\) is odd; \N\item[(5)] \(\lambda=(2,2)\), \(\mu=(3,1)\); \N\item[(6)] \(\lambda=(2^{3})\), \(\mu=(3,2,1)\); \N\item[(7)] \(\lambda=(2^{4})\), \(\mu=(5,3)\); \N\item[(8)] \(\lambda=(4,4)\), \(\mu=(5,3)\); \N\item[(9)] \(\lambda=(2^{5})\), \(\mu=(5,3,2)\).\N\end{itemize}\N\NThe proof is obtained thanks to a result by \textit{J. P. Swanson} [Algebr. Comb. 1, No. 1, 3--21 (2018; Zbl 1388.05193)] and applying the Littlewood-Richardson rule [\textit{D. E. Littlewood} and \textit{A. R. Richardson}, Philos. Trans. R. Soc. Lond., Ser. A, Contain. Pap. Math. Phys. Character 233, 99--141 (1934; Zbl 0009.20203)]. Some steps in the proof involve direct calculations using the Sage Mathematical Software System.