For simply-laced quivers, we consider the fixed-point subalgebra of the quiver Hecke algebra under the homogeneous sign map. This leads to a new family of algebras we call alternating quiver Hecke algebras. We give a basis theorem and a presentation by generators and relations which is strikingly similar to the KLR presentation for quiver Hecke algebras.
Abstract. The main result of this paper shows that, over large enough fields of characteristic different from 2, the alternating Hecke algebras are Z-graded algebras that are isomorphic to fixed-point subalgebras of the quiver Hecke algebra of the symmetric group Sn. As a special case, this shows that the group algebra of the alternating group, over large enough fields of characteristic different from 2, is a Z-graded algebra. We give a homogeneous presentation for these algebras, compute their graded dimension and show that the blocks of the quiver Hecke algebras of the alternating group are graded symmetric algebras.
Abstract. We define alternating cyclotomic Hecke algebras in higher levels as subalgebras of cyclotomic Hecke algebras under an analogue of Goldman's hash involution. We compute the rank of these algebras and construct a full set of irreducible representations in the semisimple case, generalising Mitsuhashi's results [23], [24].
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