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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
Let be a splitting -modular system for a finite group . An -lattice is a left -module that is finite free as an -module.
Its reduction modulo the maximal ideal of is
Because the -action on is -linear, it descends to a -action on , so is a -module in the sense of An -linear action of on a left -module, and a -module over .
Depends on
Used by
- An ordinary irreducible representation can have reducible reduction modulo p Counterexample
- Reducing a standard integral lattice for S3 modulo 3 produces a reducible kS3-module Example
- Every reduction modulo p of an ordinary irreducible lattice stays irreducible False statement
- Reducing an OG-lattice modulo the maximal ideal gives a finite-dimensional kG-module Lemma
Dependency tree · two levels
9 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)