Principal Block Characters in Finite Groups

Updated 31 July 2025

Principal block characters are defined by the unique p-block containing the trivial character, reflecting the interplay between local p-subgroup structures and global character invariants.
Lower bound results, including confirmations of the Héthelyi–Külshammer conjecture, ensure the number of irreducible characters meets strict arithmetic thresholds that constrain group structure.
Techniques from block theory, Deligne–Lusztig induction, and local-global conjectures underpin the sharp bounds and detailed classifications of defect groups in finite groups.

A principal block characterizes a fundamental aspect of the modular representation theory of finite groups. For a prime $p$ dividing the order of a finite group $G$ , the principal $p$ -block $B_0(G)$ is the unique block containing the trivial character and provides crucial insight into the interplay between local $p$ -subgroup structure and global character-theoretic and block-theoretic invariants. Recent research has resolved several deep conjectures and established sharp lower bounds for the number of irreducible characters in principal blocks, bridging local group theory, character field arithmetic, and block-theoretic combinatorics.

1. Definition and Structural Features of Principal p-Blocks

In the modular representation theory of finite groups, blocks arise as direct indecomposable summands of the group algebra over a field of characteristic $p$ . The principal $p$ -block $B_0(G)$ is the block containing the trivial (identity) character; equivalently, it is the unique block whose defect group contains a Sylow $p$ -subgroup of $G$ . The number of irreducible characters in $B_0(G)$ , denoted $k(B_0(G))$ , and the set of irreducible Brauer (modular) characters, $l(B_0(G))$ , are key invariants reflecting the structure and complexity of $G$ relative to the prime $p$ .

$B_0(G)$ encapsulates the essential $p$ -local information: its defect group is always a Sylow $p$ -subgroup, and constraints on its character invariants translate to often deep restrictions on the configuration of $p$ -subgroups and their normalizers.

2. Lower Bounds: Héthelyi–Külshammer Conjecture and Brauer's Problem 21

Héthelyi and Külshammer conjectured that for any $p$ -block $B$ of positive defect (i.e., with a nontrivial defect group), the inequality $k(B) \geq 2\sqrt{p-1}$ always holds. This has been explicitly confirmed for principal blocks; namely, for any finite group $G$ with a noncyclic Sylow $p$ -subgroup, the principal block $B_0(G)$ satisfies

$k(B_0(G)) \geq 2\sqrt{p-1}$

or more precisely, an arithmetic variant $k(B_0(G)) \geq 2V_p-1$ (with $V_p$ defined appropriately in certain contexts) (Hung et al., 2021).

This result simultaneously answers Brauer's Problem 21 in the context of principal blocks: for given $k$ , there are only finitely many possibilities (up to isomorphism) for the defect groups of principal blocks with $k$ irreducible characters. The principal block's character count is thus tightly bounded away from triviality unless $G$ has an exceptionally small $p$ -structure.

3. Techniques and Underlying Methods

The proof strategy combines modern and classical methods:

Block and defect group theory: Reductions to almost simple (and quasisimple) group cases.
Explicit semisimple and unipotent character construction in finite groups of Lie type, leveraging Deligne–Lusztig theory and Lusztig's e-Harish-Chandra series [Broué–Malle–Michel, Cabanes–Enguehard].
Sharp lower bounds for numbers of conjugacy classes and $p'$ -degree characters, notably from Maróti and Malle–Maróti, underpinning the count of irreducibles in terms of defect [Maróti, Malle–Maróti].
Application of the Alperin–McKay conjecture in known cases, using global-local correspondences to relate $k(B_0(G))$ to character counts in normalizers of defect groups.

Blockwise, the distribution of semisimple characters into unipotent blocks and the action of automorphism groups is a crucial aspect; formulas like $X(t)(1) = |(G^*)^F : C_{G^*F}(t)|_\ell$ (for a semisimple element $t$ ) formally encode degrees in several families.

4. Implications for the Structure of Finite Groups

The established lower bound on $k(B_0(G))$ implies that outside the case of cyclic Sylow $p$ -subgroups, the representation theory in characteristic $p$ is inevitably "rich": principal blocks cannot support too few irreducible characters. This restricts the possible structure of defect groups severely when $k(B_0(G))$ is small, as seen in explicit classifications for small $k$ (e.g., $k=4,5,6$ (Koshitani et al., 2020, Rizo et al., 2020, Hung et al., 2023)).

For instance, when $k(B_0(G)) = 6$ , one necessarily has that the Sylow 3-subgroup of $G$ is of order 9, i.e., $C_9$ or $C_3 \times C_3$ (Hung et al., 2023). Such results tie block-theoretic data to sharply delimited possibilities for the $p$ -local group structure.

The Alperin–McKay conjecture asserts that for any $p$ -block $B$ of $G$ , the number of height-zero characters in $B$ equals that in the corresponding Brauer correspondent block in the normalizer of a defect group. Known cases of the conjecture (notably for cyclic or controlled defect groups) are used to transfer lower bounds from $N_G(P)$ to $G$ and to guarantee that block-theoretic invariants reflect local subgroup structure (Hung et al., 2021).

These local-global correspondences are critical: they guarantee that even when constructing irreducible characters from Deligne–Lusztig induction or through combinatorics of Lie-type groups, the principal block contains "enough" semisimple or unipotent characters to meet the conjectural lower bounds.

6. Connections to Bounding $p'$ -Character Degrees and Conjugacy Classes

The analysis depends on sharp bounds for conjugacy class numbers and $p'$ -degree irreducible characters, for which the results of Maróti and Malle–Maróti play a central role. These results allow explicit calculations of the minimal number of complex irreducible representations under various constraints, translating directly into block-theoretic bounds for $k(B_0(G))$ .

In finite reductive groups, the combinatorics of unipotent blocks and the parametrization of characters into $e$ -Harish-Chandra series (Broué–Malle–Michel, Cabanes–Enguehard) allow precise counting of irreducibles.

7. Broader Consequences and Further Directions

The lower bound for the number of irreducibles in the principal block is not only structurally informative but also impacts the feasibility of classifying blocks of finite groups by small character count. It provides new invariants for the paper of representation growth, block equivalence, and modular local-global phenomena.

Ongoing work seeks to resolve extensions of the conjecture to arbitrary blocks, refine the bounds (both for ordinary and Brauer characters), and link block-theoretic behavior to properties of more general $p$ -local structures, potentially extending the classification of groups and blocks realized for $k(B_0) \leq 6$ (Moretó et al., 2023, Hung et al., 2023).

In conclusion, the minimal number of irreducible characters in the principal block of a finite group is not arbitrary but is instead subject to sharp arithmetic and structural lower bounds, results which rely on an overview of block theory, group theory, and the arithmetic of character fields. These findings deepen the understanding of the interplay between local $p$ -subgroup structure, modular representation theory, and global character-theoretic complexity.