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The radical, socle, head, and Loewy series of a finite-dimensional module
Definition
Let be a finite-dimensional algebra and a finite-dimensional left -module.
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Its radical is equivalently the intersection of the maximal submodules, by For a finite-dimensional algebra, the module radical is exactly the action of the Jacobson radical.
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Its socle is the sum of all simple submodules, as in The socle as the sum of all simple submodules.
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Its head or top is the quotient formed using Quotient module with scalar multiplication on additive cosets.
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Its Loewy series is the descending radical filtration together with the corresponding semisimple layers .
The simple subquotients appearing in the Loewy layers are simple in the sense of Simple module: a nonzero module with no proper nonzero submodule.
Depends on
Used by
- The socle of an uncoloured modular restriction is the multiplicity-free sum over the good residues Corollary
- Modular Specht form and radical quotient Definition
- Normal and good nodes for the partition (6,4,4,3,2) Example
- The augmentation ideal and Loewy series of kCp can be written explicitly Example
- DVR cellular Jantzen filtration and exact Gram valuation Lemma
- For a finite-length module, the radical is a superfluous submodule Lemma
- Nonzero maps into tabloid quotients force dominance Lemma
- Every p-restricted partition is reached from the empty partition by cogood additions Proposition
- p-regular and p-restricted labels under transpose and sign Proposition
- Indecomposable projective kG-modules correspond to simple modules through taking the head Theorem
- James submodule theorem over every field Theorem
- Modular simple modules of the symmetric group Theorem
- Nonzero modular Specht quotient criterion Theorem
- The cogood node gives the simple socle and head of the residue induction Theorem
- The good node gives the simple socle and head of the residue restriction Theorem
- The restricted-partition component is the basic crystal in the symmetric-group convention Theorem
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)