Brauer's Problem 41 in Modern Algebra

• Brauer's Problem 41 is a central question in modern algebra, linking local properties like defect groups and period phenomena to global invariants such as irreducible characters and division algebra indices.
• Researchers employ techniques such as Cartan matrix analysis, perfect isometries, and index reduction formulas to address conjectures in block theory and representation theory.
• The problem’s applications span computational methods, combinatorial enumeration, and arithmetic geometry, influencing studies on the Brauer group and Brauer–Manin obstructions.

Brauer's Problem 41 is a central open question and organizing principle in modern algebra, representation theory, and related areas of arithmetic geometry. Its impact is felt across block theory, the structure of division algebras, group cohomology, and the theory of central simple algebras. Although the phrase “Brauer’s Problem 41” has been used for a variety of distinct but deeply related conjectures and topics since its original formulation in the 1963 Séminaire Bourbaki, they share the overarching theme of connecting local structure to global invariants—most notably, linking defect groups or period-exponent phenomena to explicit invariants like the number of irreducible characters or the structure of the Brauer group.

1. Problem Statement and Historical Formulation

The archetypal form of Brauer's Problem 41 asks for precise restrictions or relationships between "local" properties (such as defect groups in modular representation theory, or period in the theory of central simple algebras) and "global" invariants (such as the number of irreducible characters or the index of a division algebra). In block theory, this is most notably encapsulated by Brauer’s k(B)-conjecture: for a prime $p$, a $p$-block $B$ of a finite group $G$ with defect group $D$ satisfies

$k(B) \leq |D|,$

where $k(B)$ is the number of irreducible ordinary characters in $B$. Equality is expected to occur if and only if $D$ is abelian.

In the context of central simple algebras and the Brauer group, the problem is closely tied to the period–index problem: given a Brauer class $[A]$ over a field $k$, what is the relationship between its period (order in $\operatorname{Br}(k)$) and its index (the degree of the corresponding division algebra)?

Both questions are unified by the drive to determine how much the local structure controls (or restricts) the global arithmetic, representation-theoretic, or cohomological invariants.

2. The k(B)-Conjecture: Results and Reduction Theorems

A substantial body of work addresses Brauer's Problem 41 via the $k(B)$-conjecture. The conjecture is verified in many important cases:

• For blocks with abelian or minimal nonabelian defect groups under various structure constraints, the bound $k(B) \leq |D|$ is established using Cartan matrix invariants, extensions, and coprime action arguments (1012.4444, Sambale, 2014, Sambale, 2014).
• In blocks where the defect group D is a central extension of a metacyclic group or contains a central cyclic subgroup of index at most 9, refined analyses of the Cartan matrix, relying on properties such as determinant and quadratic forms, yield the conjectured bound (see inequalities such as $k(B) \leq \sum_{i\leq j} q_{ij} c_{ij}$ for appropriate $q$, $c_{ij}$) (1012.4444).
• For $2$-blocks of defect at most $4$ and $3$-blocks of defect at most $3$, explicit enumeration confirms the conjecture—and for some blocks with minimal nonmetacyclic defect groups, the precise invariants $k(B)$, $k_0(B)$, $k_1(B)$, $l(B)$ are computed.
• The reduction theorem for Feit's conjecture shows that if all simple subquotients of a finite group $G$ satisfy the inductive Feit condition (a local-to-global control property for character conductors), Feit’s conjecture—and thus Brauer’s Problem 41—holds for $G$ as well (Boltje et al., 25 Jul 2025).

In the context of $\pi$-blocks of $\pi$-separable groups, a generalization using a k(GV)-type argument proves that $k(B) \leq |D|$ holds for all such blocks, with rigorously established equalities and counterexamples to certain generalizations (e.g., Olsson's conjecture for $\pi$-blocks) (Sambale, 2017).

For finite quasi-simple groups, including blocks of Lie type groups and covering groups, the conjecture is almost completely resolved: it is shown that no minimal counterexample to the $k(B)$-inequality exists in these families for $p \ge 5$ or abelian defects, and explicit combinatorial formulas are obtained for unipotent blocks (Malle, 2017).

3. Block Theory: Local–Global Correspondences and Extensions

Modern block theory not only investigates scalar invariants but also structural correspondences between global and local blocks:

• Perfect isometry conjectures extend the philosophy of “local determination” even to generalized character sets. Notably, for double covers of symmetric and alternating groups (“spin groups”), perfect isometries between blocks and their Brauer correspondents have been constructed, confirming far-reaching versions of the local-global principle. These isometries preserve not only scalar products but also intricate Galois and sign data, relying on advanced combinatorics of Schur Q-functions and character-theoretic restriction (Boltje et al., 25 Jul 2025).
• For blocks with a unique simple module (i.e., $l(b) = 1$ for a Brauer correspondent $b$), explicit character-theoretic conditions connect the absence of nontrivial $p$-rational, height-zero irreducible characters with the simplicity of $b$. Such results further clarify the precise mechanism by which local subgroup structure (e.g., the normal $p$-complement in $N_G(P)$) is reflected in global representation-theoretic invariants (Navarro et al., 2018).

The reduction approach, culminating in inductive and perfect isometry conditions, systematically reduces the full verification of Brauer’s k(B)-conjecture to the analysis of simple group factors and their coverings.

4. The Period–Index Problem and Cohomological Generalizations

In the context of division algebras and the Brauer group, Problem 41 is intimately related to period–index phenomena:

• For a field $k$, the period of a Brauer class $[A]$ is its order $n$, the index is the degree of the unique division algebra representing $[A]$. The core question is whether the index divides a bounded power of the period, and, more precisely, what is the minimal “symbol length” in the decomposition of Brauer group elements, especially over iterated Laurent series fields and function fields (Colliot-Thélène, 2023, Chapman, 2019).
• Index reduction formulas, such as

$$\operatorname{Index}_{k(t)}(A \otimes_k (K/k, t)) = [K : k] \cdot \operatorname{Index}_K(A_K),$$

play a key role in showing that, even for central simple algebras of small period, the index can be arbitrarily large after suitable base change, and control can be lost under transcendental extensions (Colliot-Thélène, 2023, Chipchakov, 2015).

• The precise symbol length for the $p$-torsion Brauer group of fields of iterated Laurent series $F = k((a_1))\cdots((a_n))$ is sharply determined: if $\operatorname{char}(k) = p$ and $k$ is algebraically closed, then $\operatorname{Sym}_p(F) = n-1$; if $\operatorname{char}(k) \neq p$, then the classic bound $\left\lfloor \frac{n}{2} \right\rfloor$ holds (Chapman, 2019).

Cohomological and motivic generalizations (using, e.g., Bloch’s cycle complex in place of $G_m$) allow the extension of classical Brauer group concepts to derived and arithmetic contexts, yielding generalized Gersten sequences and lifting Artin’s theorems to new settings (Sakagaito, 2015).

5. Arithmetic and Geometric Applications

The arithmetic importance of Problem 41 is manifest in various branches of number theory and arithmetic geometry:

• The computation and structure of the Brauer group for the rationals ($\mathbb{Q}$), their completions, and function fields play a central role in the classification of division algebras and the application of cohomological invariants in number theory (Chen, 2019).
• The Brauer–Manin obstruction, which uses the global Brauer group to explain failures of the Hasse principle and weak approximation on algebraic varieties, is closely analyzed via explicit calculation of Brauer groups on diagonal quartic and K3 surfaces. For instance, nonconstant odd-torsion Brauer elements can obstruct weak approximation without violating the Hasse principle (Ieronymou et al., 2013, Colliot-Thélène, 2023).
• In the geometric and motivic context, relative units–Picard complexes and their derived category properties yield five-term exact sequences governing the Brauer groups of fibered products and products of schemes, extending the reach of classical period–index results (Gonzalez-Aviles, 2016).

6. Combinatorial and Enumerative Dimensions

While Problem 41 is most prominent in representation and number theory, it also includes enumerative and combinatorial problems:

• In permutation and partition theory, connections are drawn between classes of pattern-avoiding permutations (e.g., avoiding the dashed patterns 32-41 and 41-32) and indecomposable set partitions. Such enumerative coincidences are related via bijections and ultimately tied to Bell numbers, highlighting the interplay between combinatorial invariants and classical partition problems in Brauer’s framework (Callan, 2014).

7. Algorithmic and Computational Aspects

Brauer’s Problem 41 has motivated significant advances in algorithmic mathematics:

• Determining Ramsey numbers—famously $R(3,10)$—has been sharpened to $R(3,10) \leq 41$ using substantial computational searches, sophisticated “neighbourhood gluing” techniques, and efficient processing of maximal independent sets. The residual uncertainty (whether $R(3,10) = 40$ or $41$) is now an active front in the computational branch of Problem 41 (Angeltveit, 2023).

Summary Table: Major Problem 41 Facets and Key Results

Area	Core Problem	Key Tools/Results
Block theory, $k(B)$-conjecture	$k(B)\leq|D|$	Cartan matrices, reduction theorems, perfect isometry (1012.4444, Malle, 2017, Boltje et al., 25 Jul 2025)
Period–index problem	$\operatorname{ind}(A)?$ vs $\operatorname{per}(A)$	Index-reduction, symbol length, cohomology (Colliot-Thélène, 2023, Chipchakov, 2015)
Local-global in Brauer groups	$Br(k)$ from $Br(k_v)$	Artin–Wedderburn, valuation theory, cohomology, class field theory (Chen, 2019)
Geometric/arithmetic applications	Rational points, obstructions	Brauer–Manin obstruction, motivic cohomology (Ieronymou et al., 2013, Sakagaito, 2015, Gonzalez-Aviles, 2016)
Combinatorial/algorithmic	Enumeration, Ramsey bounds	Pattern bijections, computational gluing (Callan, 2014, Angeltveit, 2023)

Conclusion and Future Directions

Brauer's Problem 41 stands as a nexus connecting group theory, representation theory, number theory, algebraic geometry, and combinatorics. Modern advances include partial or near-complete solutions for the $k(B)$-conjecture in vast families of blocks, deep connections between period and index in the Brauer group, and the unifying power of cohomological and representation-theoretic methods to relate local and global quantities.

Ongoing research focuses on closing the remaining exceptional cases (e.g., explicit minimal counterexamples in block theory, sharp period–index coincidences over function fields, full resolve of certain enumerative combinatorics), developing new computational methods for intractable search spaces, and extending the framework to even more general settings (motivic cohomology, derived categories, and quantum symmetries).

In all its incarnations, Brauer’s Problem 41 remains a compelling test of local-global principles, offering both a proving ground for new mathematical techniques and a source of enduring challenges.