Tonal partition algebras: fundamental and geometrical aspects of representation theory
Tonal partition algebras: fundamental and geometrical aspects of representation theory
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DOI：
10.1080/00927872.2023.2239357
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发表时间：
2019-12
影响因子：
0.7
通讯作者：
C. Ahmed;Paul Martin;V. Mazorchuk
中科院分区：
文献类型：
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作者：
C. Ahmed;Paul Martin;V. Mazorchuk
对l，n∈N，定义了Z[δ]上的调分划代数Pnl。我们为Pnl在Z[δ]上构造模{Δμ <$}μ <$，从而在任何包含Z[δ]的整环（如C[δ]）上构造模，它传递到分式域上的不可约模的完备集。我们证明了Pnl在那里是半单的。也就是说，我们为调划分代数构造了一个Brauer意义下的模系统。利用Δ-模的“几何”指标集，给出了C上分解矩阵（δ∈C×）的上单三角阶.我们建立了Δ-模的几个重要性质。这些性质包括关于n的塔性质，在绿色和Cox-Martin-Parker-Xi的意义上;关于自然对合反自同构的逆变形式;最高权范畴性质;和分支规则。
Abstract For l,n∈N we define tonal partition algebra Pnl over Z[δ]. We construct modules {Δμ¯}μ¯ for Pnl over Z[δ], and hence over any integral domain containing Z[δ] (such as C[δ]), that pass to a complete set of irreducible modules over the field of fractions. We show that Pnl is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer. Using a “geometrical” index set for the Δ-modules, we give an order with respect to which the decomposition matrix over C (with δ∈C×) is upper-unitriangular. We establish several crucial properties of the Δ-modules. These include a tower property, with respect to n, in the sense of Green and Cox-Martin-Parker-Xi; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.