On the sign representations for the complex reflection groups G(r, p, n)

Abstract

Abstract. We present a formula for the values of the sign representations of the complex reflection groups G(r, p, n) in terms of its image under a generalized Robinson-Schensted algorithm.

Sign Under the Domino Robinson-Schensted Maps

“…The proof involves three steps. We first verify the equation for large r by translating between Stanley's map and G r and appealing to the formula established in [15]. It is then possible to extend the result to involutions in H n for arbitrary r using a relation between consecutive maps G r described in [17], and finally to to all signed permutations by tracking the behavior of the established sign formula under M. Taşkın's plactic relations introduced in [23].…”

“…Our first goal is to verify the theorem for involutions in H n . There are two main tools, the sign formula for colored permutations under G ∞ derived from [15] and a description of the relationship between the maps G r and G r+1 obtained in [17]. When r is large relative to n, the relationship between G ∞ and G r is simple and it is a trivial task to translate one sign formula into the other.…”

“…The following is a special case of a result on the sign characters of the complex reflection groups G(r, p, n). Theorem 2.9 ( [15]). Consider w ∈ H n and let G ∞ (w) = (P, Q) be its image among same-shape standard bitableaux, where P = (P 1 , P 2 ) and Q = (Q 1 , Q 2 ).…”

“…In fact, this has already been done for the Stanley map in [15] in the more general context of complex reflection groups. For the family of maps G r , our main result relies on three domino tableaux statistics d, spin, and sign defined in Section 2.3.…”