A correspondence in modular representation theory that relates blocks of a finite group algebra over a field of characteristic p to blocks of the group algebra of the normalizer of a p-subgroup (the defect group), using the Brauer homomorphism and thereby connecting global block decomposition to local subgroup structure.

Principle

Local-to-global defect principle: a block of the group algebra with defect group P has an associated Brauer correspondent block in the group algebra of N_G(P) obtained via the Brauer map on block idempotents; fusion, defect groups and local structure govern how blocks correspond.

Demonstration

If B is a p-block of kG with defect group P, its Brauer correspondent is a block b of kN_G(P) with defect P such that the Brauer homomorphism sends the central idempotent of B to that of b; in cases of cyclic defect groups this correspondence is concrete and greatly simplifies block analysis.

Misapplication

Applying the correspondence indiscriminately in characteristic zero or to algebras where p does not divide the group order, or assuming bijectivity or uniqueness without checking fusion-stability and hypotheses required for the existence of a Brauer correspondent.

Consequence

Permits the reduction of questions about global blocks to local subgroup data and is fundamental in formulating and testing local–global conjectures in modular representation theory, such as those relating characters, sources, and equivalences between blocks.

Reversal

Trying to reconstruct a global block uniquely from an arbitrary local block of N_G(P) without compatibility data or fusion constraints fails in general; the correspondence is not a simple global reconstruction map but a controlled local correspondence under defect conditions.

Boundary

Pertains to modular representation theory over a field of characteristic p dividing |G|; it concerns blocks with specified defect groups and uses the Brauer homomorphism, and does not apply in semisimple characteristic zero settings or outside the modular framework.

Semantic Tension

Tension appears between Brauer correspondence and broader local–global conjectures (Alperin, Broué, etc.), and between the Brauer map as an algebraic homomorphism and block-level correspondences that require additional structural hypotheses for existence or uniqueness.

Synthesis

The Brauer correspondence ties a global p-block to a local block of the normalizer of its defect group via the Brauer homomorphism: defect groups and fusion data control which local blocks correspond, allowing modular representation problems to be studied through local subgroup structure.