Bases and Symmetric Structure Of Alternating Cyclotomic Hecke Algebras

Abstract

Let \(n\in \mathbb {Z}^{{\ge }2}, \ell \in \mathbb {Z}^{{\ge }1}\). In this paper we use (some slightly modified versions of) the distinguished bases \(\{\mathcal {B}_{\mathfrak {s}\mathfrak {t}}\}\) and \(\{\check{\mathcal {B}}_{\mathfrak {s}\mathfrak {t}}\}\) of the cyclotomic Hecke algebra \(\mathscr {H}_{\ell ,n}\) of type \(G(\ell ,1,n)\) introduced by Mathas and the first named author (Hu and Mathas, A. Math. Ann. 364, 1189–1254, 2016) to study the alternating cyclotomic Hecke algebra \(\mathscr {H}_{\ell ,n}^{\#}\). We construct an explicit integral basis for the alternating cyclotomic Hecke algebra \(\mathscr {H}_{\ell ,n}^{\#}\) of arbitrary higher levels. We also present an explicit seminormal basis for the semisimple alternating cyclotomic Hecke algebra. We show that the alternating cyclotomic Hecke algebra is a symmetric algebra over an infinite field.