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Blocks partition the ordinary and Brauer irreducible characters
Statement
Every ordinary irreducible character and every irreducible Brauer character lies in exactly one block.
Facts & Assumptions
Given: The primitive central idempotents of the modular system of .
A block is the direct-product summand cut out by a primitive central idempotent (p-blocks from primitive central idempotents).
The primitive central idempotents of correspond bijectively, under reduction, to those of (Block idempotents lift uniquely from kH to OH).
Irreducible Brauer characters are attached to simple modules (Irreducible Brauer characters form a basis of the p-regular class functions).
Proof
Because the primitive central idempotents are central, pairwise orthogonal, and sum to , every module decomposes as the direct sum of the eigenspaces .
If is simple, only one summand in step 1.1 can be nonzero; otherwise would split as a nontrivial direct sum of submodules. So each simple -module, hence each irreducible Brauer character by [L1], lies in exactly one block.
By [F2], each modular block idempotent has a unique lift in . These lifted idempotents remain central, orthogonal, and sum to , and they act on -modules through . Thus step 1.1 applies to a simple -module using the lifted block idempotents: exactly one summand is nonzero. Its ordinary irreducible character consequently belongs to that unique block.
Depends on
Used by
- Generalized decomposition columns have corresponding block support Corollary
- A block with one ordinary and one Brauer character Example
- A p-section with no local block inducing to the chosen global block Example
- Brauer correspondence for a defect-D8 block of S7 in characteristic 2 Example
- Brauer correspondence for SL2(Fp) in defining characteristic Example
- FALSE: every block has one ordinary and one Brauer irreducible character False statement
- Local block projection controls p-section character support Lemma
- After block ordering, the decomposition matrix is block diagonal Proposition
- Brauer's Second Main Theorem Theorem
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
- J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
- Tudor Ciurca, Representation Theory (standard reference, not scraped)