THE BRAUER HOMOMORPHISM

DAVE BENSON

The Brauer homomorphism was first used by Richard Brauer [1, 2] and developed further by Nagao [4] in order to establish a connection between the p-blocks of a finite group G and of the p-local subgroups. Since then, it has been used in a variety of other contexts within modular representation theory.

Let k be a field of characteristic p, and let G act by conjugation on the group algebra kG with fixed points (kG)^{G} = Z(kG). The primitive idempotents e_{0}, …, e_{s} in Z(kG) correspond to the blocks B_{0}, …, B_{s} of kG, so that B_{i} = e_{i}kG and kG = B_{0} ⊕ ⋯ ⊕ B_{s} as a direct sum of indecomposable two-sided ideals.

If H ≤ K are subgroups of G and M is a kGmodule, we have a relative trace map Tr_{H,K} : M^{H} → M^{K} given by m ↦ ∑_{i} g_{i}m, where g_{i} runs through a set of coset representatives of H in K. We write M_{H}^{K} for the image of Tr_{H,K}. Applying this to the conjugation action of G on kG, we say that D is a defect group for the block B_{i} if e_{i} lies in (kG)_{D}^{G} but not in (kG)_{D′}^{D} for any proper subgroup D′ < D. The defect groups of a block form a single conjugacy class of p-subgroups of G.

If D is any p-subgroup of G, still acting on G by conjugation, then we have a decomposition

(kG)^{D} = kC_{G}(D) ⊕ ∑_{D′<D}(kG)_{D′}^{D}

as a direct sum of a subring and a two-sided ideal. This is because the left-hand side has as a basis the orbit sums. The orbits of length one span kC_{G}(D), and the sums of orbits of length greater than one are transfers from proper subgroups of D. The Brauer homomorphism

Br_{D} : (kG)^{D} → kC_{G}(D)

is the projection onto the first factor in the above decomposition. Restricting to Z(kG) = (kG)^{G} ⊆ (kG)^{D}, we obtain the homomorphism

Br_{D} : Z(kG) → Z(kC_{G}(D))

originally considered by Brauer.

Brauer’s first main theorem states that the Brauer homomorphism Br_{D} establishes a one-one correspondence between blocks of G with defect group D and blocks of N_{G}(D) with defect group D. To make sense of this, first observe that every idempotent in Z(kN_{G}(D)) lies in Z(kC_{G}(D)) (because the other summand ∑_{D′<D}(kN_{G}(D))_{D′}^{D} is contained in the Jacobson radical J(kN_{G}(D))). A block idempotent e_{i} ∈ Z(kG) has defect group D if and only if e_{i} ∈ (kG)_{D}^{G} and Br_{D}(e_{i}) ≠ 0. In the commutative diagram

[figure: (kG)^{D}; ^{Br^{D}}; _{Tr_{D,G}}; kC_{G}(D); _{Tr_{D,NG(D)}}; (kG)_{D}^{G}; _{Br_{D}}; kC_{G}(D)_{D}^{N_{G}(D)}]

the top and right hand maps are surjective, and hence so is the bottom map. It therefore induces the bijection of block idempotents described in the first main theorem.

In subsequent work, the Brauer homomorphism has been generalized to arbitrary modules:

Br_{D} : M^{D} → M^{D}/ ∑_{D′<D} M_{D′}^{D}.

The concept is especially useful in case M is a Galgebra. Puig [5, 6] has developed an extensive theory of “interior G-algebras”, where G is acting via inner automorphisms, and in which the Brauer homomorphism plays a central role. His work has had extensive impact both on module theory and on block theory, especially his classification of nilpotent blocks [7]. Further discussion of the Brauer homomorphism in this context can be found in the books of Külshammer [3] and Thévenaz [8].

References

[1] R. Brauer. Zur Darstellungstheorie der Gruppen endlicher Ordnung. Math. Zeit. 63 (1956), 406–441.

[2] R. Brauer. Zur Darstellungstheorie der Gruppen endlicher Ordnung. II. Math. Zeit. 72 (1959), 25–46.

[3] B. Külshammer. Lectures on Block Theory. LMS Lecture Note Series 161, CUP 1991.

[4] H. Nagao. A proof of Brauer’s theorem on generalized decomposition numbers. Nagoya Math. J. 22 (1963), 73– 77.

[5] L. Puig. Pointed groups and construction of characters. Math. Z. 170 (1981), 265–292.

[6] L. Puig. Pointed groups and construction of modules. J. Algebra 116 (1988), 7–129.

[7] L. Puig. Nilpotent groups and their source algebras. Invent. Math. 93 (1988), 77–116.

[8] J. Thévenaz. G-Algebras and Modular Representation Theory. Oxford Math. Monographs, OUP 1995.