Irreducible Specht modules for Iwahori-Hecke algebras of type \(B\). (Q2889346)

Let \(\mathcal B_n\) denote the Iwahori-Hecke algebra of type \(B_n\) with parameters \(Q\) and \(q\) defined over a field \(\mathbb F\). This algebra has a cellular basis in the sense of \textit{J. J. Graham} and \textit{G. I. Lehrer} [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)] with Specht modules \(S^{(\lambda,\mu)}\) indexed by the bipartitions of \(n\). This paper studies when these Specht modules are irreducible.NEWLINENEWLINE In the case when \(-Q\) is not a power of \(q\) in \(\mathbb F\), the author uses a result of \textit{R. Dipper} and \textit{G. James} [J. Algebra 146, No. 2, 454-481 (1992; Zbl 0808.20016), Theorem 4.17] that the algebra \(\mathcal B_n\) is Morita equivalent to a sum of tensor products of Hecke algebras of type \(A\) under a map sending the Specht module \(S^{(\lambda,\mu)}\) for \(\mathcal B_n\) to the tensor product \(S^\lambda\otimes S^\mu\) of Specht modules for Hecke algebras of type \(A\). This reduces to the type \(A\) case, where a complete classification of reducible Specht modules is available in all cases except \(q=-1\).NEWLINENEWLINE Section 3 of the paper gives a complete classification in the case where \(Q=-q^r\) where \(r\in\mathbb N_0\) and \(q\) is not a root of unity. Section 4 gives some partial results on the case \(q=-1\) and \(Q=\pm 1\) including a sufficient condition for \(S^{(\lambda,\mu)}\) to be reducible, stated in terms of the number of addable and removable \(1\) and \(-1\) nodes in the partitions \(\lambda\) and \(\mu\). Moreover, a reduction is given to the analogous problem for type \(A\) Hecke algebras when \(q=-1\).