Brauer Theorems in Algebra & Representations

• Brauer Theorems are a collection of foundational results in algebra and representation theory that address eigenvalue perturbations, modular character counts, and structural properties of group algebras.
• They provide explicit methods such as rank-one perturbations for eigenvalue modification and algorithmic constructions of cyclotomic crossed products via Shoda-pair theory.
• These theorems yield classification dichotomies in representation types and offer numerical bounds on group orders through character defect and centralizer size analysis.

Brauer theorems constitute a diverse and foundational collection of results in algebra and representation theory, unified by their origin in the work of Richard Brauer and his school. These theorems broadly encompass statements about the structure of algebras, modules, group representations, eigenvalues of matrices, and cohomological invariants, with many generalizations and analogues in modern mathematics. The term “Brauer theorems” may refer to results concerning eigenvalue manipulation (Brauer’s rank-one perturbation theorem), character counts and defect groups (Brauer–Nesbitt theorem), block structure and cyclotomic algebras (Brauer–Witt theorem), representation-theoretic dichotomies (Brauer–Thrall), and group-theoretic bounds involving involutions (Brauer–Fowler theorems).

1. Brauer’s Rank-One Eigenvalue Modification Theorem

The classical Brauer theorem provides an explicit method to alter a single eigenvalue of a square matrix via a rank-one perturbation, while leaving all other eigenvalues unchanged. Let $A\in\mathbb{C}^{n\times n}$ possess eigenvalues $\lambda_1,\dots,\lambda_n$, and let $x$ be an eigenvector for $\lambda_1$, with arbitrary $y\in\mathbb{C}^n$. Then the perturbed matrix $A' = A + x y^*$ has spectrum

$\spec(A') = \{\lambda_2,\ldots,\lambda_n,\,\lambda_1 + y^*x\}.$

This result extends to matrix polynomials and Laurent series, allowing the controlled shift of finite or infinite eigenvalues. The underlying mechanism generalizes to moving entire invariant subspaces and provides foundational tools for eigenvalue problems, canonical factorizations of matrix functions, Markov chains, and nonnegative inverse eigenvalue constructions (McDonald et al., 2021, Bini et al., 2015).

2. Brauer–Nesbitt and Brandt Theorems in Block Theory

In modular representation theory of finite groups, Brauer–Nesbitt and Brandt theorems characterize blocks with small defect groups by counting irreducible characters. For a block $B$ of a group algebra over a splitting field, the number $k(B)$ of ordinary irreducible characters satisfies:

• Brauer–Nesbitt: Defect group trivial $\Longleftrightarrow k(B)=1$; such blocks are Morita equivalent to the ground field.
• Brandt: Defect group of order two $\Longleftrightarrow k(B)=2$ and the number $\ell(B)=1$ of irreducible modular (Brauer) characters; these blocks are Morita equivalent to $F[X]/(X^2)$.

These statements admit generalization to arbitrary finite-dimensional algebras via the commutator codimension $k(A) = \dim_F(A/K(A))$. The Okuyama refinement links higher-radical codimension to module extensions, allowing structural characterizations beyond symmetric blocks. Morita invariance ensures that these invariants classify algebras up to equivalence (Koshitani et al., 2018).

3. Brauer–Witt and Cyclotomic Crossed Products

The Brauer–Witt theorem states that every simple component of the rational group algebra $\mathbb{Q}G$ of a finite group $G$ is Brauer-equivalent (over its center) to a cyclotomic crossed product algebra. Concretely, for each irreducible constituent, there exists a cyclotomic field extension $L/F$ and a factor set $\tau$ such that the associated crossed product $(L/F, \tau)$ realizes the same Brauer group class as the given simple algebra. While the classical proof is nonconstructive, modern approaches—such as Shoda-pair theory—provide explicit algorithms that construct the crossed product representative for each Wedderburn component, including concrete basis elements, twisting data, and Schur indices (Bakshi et al., 2022).

4. Brauer–Thrall Type Theorems in Representation Theory

Brauer–Thrall type theorems assert classification dichotomies in the representation types of algebras. For finite-dimensional $k$-algebras $A$, the derived Brauer–Thrall theorems give:

1. First Brauer–Thrall (Derived Finite ⇔ Derived Bounded): $A$ is derived finite ($D^b(A)$ has finitely many indecomposable objects up to shift and isomorphism) if and only if its global cohomological range invariant $\mathrm{gl.hr}(A)$ is finite. Precisely, such algebras are piecewise hereditary of Dynkin type.
2. Second Brauer–Thrall (Discrete vs Strong Unbounded): Every finite-dimensional algebra is either derived discrete (finitely many indecomposables of any given cohomology-dimension vector) or strongly derived unbounded (there exists a sequence of cohomological ranges each supporting infinitely many indecomposables), but not both.

These theorems use newly defined invariants—cohomological length, width, and range—to recover analogues of the classical Brauer–Thrall dichotomies, refining the classification of bounded derived categories. Piecewise hereditary and gentle algebras serve as model examples, with explicit constructions tracing the dichotomy (Zhang et al., 2013).

5. Brauer–Fowler and Structural Group Theorems

Brauer–Fowler theorems and their analogues provide bounds on the global structure of finite groups in terms of centralizer sizes, typically of involutions. Major results include:

• The order of a finite group $G$ of even order is bounded by the number of conjugacy classes of the Fitting subgroup and the fourth power of the order of the centralizer of an involution $t$:

$|G| < k(F(G)) \cdot |C_G(t)|^4.$

The exponent 4 cannot be improved to less than 3 in general.

• There exists an involution $u$ in every such group $G$ with $[G:F(G)] < |C_G(u)|^3$.
• If $t$ is an involution and $p$ divides $[G:F(G)]$, then $p \leq |C_G(t)|+1$, which is sharp in families such as $\mathrm{SL}(2,p-1)$ for Fermat primes.

These theorems not only constrain group indices, but also yield classification results (finite possibilities for almost-simple quotients given a fixed centralizer order) and quantitative lower bounds for commuting probabilities and fixed point dimensions in representations, often relying on results from the classification of finite simple groups (Guralnick et al., 2018).

6. Brauer-Type Theorems in Galois Cohomology

Extensions of the Brauer–Hasse–Noether theorem to higher-dimensional fields establish the vanishing of certain Galois cohomology groups and provide local-global exact sequences mimicking the classical structure for the Brauer group. For finite extensions $K$ of Laurent series fields or rational function fields over finite or global fields, vanishing theorems identify the top-degree cohomology that does not appear, and the derived exact sequences relate global cohomology, local completions, and the base field’s Brauer group. These results generalize the behavior of the Brauer group to contexts including surfaces and higher-dimensional arithmetic schemes (Izquierdo, 2018).

---

References

• (McDonald et al., 2021) A short and elementary proof of Brauer's theorem.
• (Bini et al., 2015) Generalization of the Brauer Theorem to Matrix Polynomials and Matrix Laurent Series.
• (Koshitani et al., 2018) On theorems of Brauer-Nesbitt and Brandt for characterizations of small block algebras.
• (Bakshi et al., 2022) A computational approach to Brauer Witt theorem using Shoda pair theory.
• (Zhang et al., 2013) Brauer-Thrall type theorems for derived module categories.
• (Guralnick et al., 2018) Variants of some of the Brauer-Fowler Theorems.
• (Izquierdo, 2018) Vanishing theorems and Brauer-Hasse-Noether exact sequences for the cohomology of higher-dimensional fields.