Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups
Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups
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DOI：
10.1112/jlms.12852
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发表时间：
2023-03
期刊：
影响因子：
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通讯作者：
M. Fayers;A. Kleshchev;Lucia Morotti
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文献类型：
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作者：
M. Fayers;A. Kleshchev;Lucia Morotti
我们计算了具有阿贝尔缺陷的对称群双重覆盖的RoCK块的（超）分解矩阵，验证了第一作者的猜想。为此，我们利用第二作者和 Livesey 的定理，即 RoCK 块 Bρ,d$\mathcal {B}^{\rho,d}$ 是 Morita 超等价于某个颤动（超）代数与对称群 Sd$\mathfrak {S}_d$ 的花环超积。我们开发了这个花环超级产品的表示理论来计算它的嘉当不变量。然后，我们直接构造 Bρ,d$\mathcal {B}^{\rho,d}$ 的射影特征来计算其分解矩阵直至三角调整，并通过比较嘉当不变量表明这种调整是微不足道的。
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 Bρ,d$\mathcal {B}^{\rho,d}$ is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group Sd$\mathfrak {S}_d$ . We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for Bρ,d$\mathcal {B}^{\rho,d}$ to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.