Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

Abstract

We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block $\mathcal B^{ρ,d}$ is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group $\mathfrak S_d$. We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for $\mathcal B^{ρ,d}$ to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.

Theorems & Definitions (18)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof