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A splitting p-modular system for a finite group is a p-modular system whose fraction and residue fields split the needed group algebras
Definition
Let be a finite group. A splitting -modular system for is a -modular system such that both and are splitting fields for every subgroup .
Thus the characteristic- fraction field and the characteristic- residue field satisfy the splitting-field condition of A splitting field for a finite group: every irreducible representation has scalar endomorphism ring simultaneously for the determinate family of subgroups of .
Depends on
Used by
- Modular Specht modules need not be simple, and form heads can vanish Counterexample
- An OG-lattice is a finite free module over the valuation ring with G-action, and reduction modulo the maximal ideal produces a kG-module Definition
- Block bimodule for the double group Definition
- Generalized decomposition numbers Definition
- Lifted modular trace on p-regular elements Definition
- Modular Specht form and radical quotient Definition
- p-blocks from primitive central idempotents Definition
- Teichmuller lift in a splitting p-modular system Definition
- Decomposition matrices of S3 at p=2 and p=3 Example
- A projective simple in a symmetric block forces a matrix block Lemma
- Block idempotents lift uniquely from kH to OH Lemma
- Prime-to-p roots lift uniquely in a complete DVR Lemma
- p-regular and p-restricted labels under transpose and sign Proposition
- Triangularity does not compute every modular decomposition number Remark
- Brauer reciprocity Theorem
- Brauer-Nesbitt determines semisimplifications Theorem
- Defect zero blocks are simple algebras Theorem
- Dominance unitriangularity of the symmetric-group decomposition matrix Theorem
- Modular block central characters correspond to blocks Theorem
- Modular simple modules of the symmetric group Theorem
- Nakayama block classification for symmetric groups Theorem
- Nonzero modular Specht quotient criterion Theorem
Dependency tree · two levels
7 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)