Picky Conjecture in Finite Group Theory

• Picky Conjecture is a paradigm in finite group theory that establishes canonical bijections between nonvanishing irreducible characters of picky elements and their corresponding local subgroups.
• It refines the McKay conjecture by preserving crucial invariants such as the p- or ℓ-part of character degrees and character values, thereby deepening the local–global correspondence in representation theory.
• The conjecture leverages structural classifications, techniques like Lusztig induction, and computational tools to analyze picky elements in groups of Lie type and extend the theory to modular representations.

The Picky Conjecture is a recently formulated paradigm in the representation theory of finite groups, particularly those of Lie type, that postulates precise local–global correspondences for character values and degrees. A “picky” element (often a p-element or ℓ-element) is defined as one lying in a unique Sylow p- or ℓ-subgroup. The conjecture posits that for such elements, nonvanishing irreducible characters admit bijective correspondences between the ambient group and its pertinent local subgroup, preserving key invariants such as the p- or ℓ-part of character degrees, and often the actual character values up to sign. This refinement generalizes and sharpens the philosophy underlying the McKay conjecture, moving beyond mere counting towards a more detailed transfer of information.

1. Definition of Picky Elements and Subgroups

Let G be a finite group and p (or ℓ) a prime divisor of |G|. An element x ∈ G of order a power of p (resp. ℓ) is termed picky if it belongs to a unique Sylow p-subgroup P ≤ G. In such cases, the subgroup generated by the normalizers of Sylow p-subgroups containing x, denoted Sub_G(x), becomes pivotal: $$\text{Sub}_G(x) = \bigcap \{ H \leq G \mid \langle x \rangle \triangleleft H \}$$ In groups of Lie type, for unipotent elements (with p the defining characteristic), Sub_G(x) is typically a parabolic group and for regular unipotent elements, the Borel subgroup. For semisimple elements of prime power order (with order ℓ ≠ p), Sub_G(x) often coincides with the normalizer of a torus determined by cyclotomic data (Malle, 13 Mar 2025, Malle et al., 21 Oct 2025).

2. Canonical Character Correspondence: Statement of the Conjecture

The conjecture asserts the existence of a canonical bijection between nonvanishing irreducible characters on picky elements and those of local subgroups. For x ∈ G a picky p-element, let

$$\text{Irr}^x(G) := \{ \chi \in \text{Irr}(G) \mid \chi(x) \neq 0 \}$$

and similarly for $\text{Irr}^x(N_G(P))$, where $P$ is the unique Sylow p-subgroup containing x. The main statement is the existence of a bijection $$* : \text{Irr}^x(G) \rightarrow \text{Irr}^x(N_G(P))$$ and a sign $\epsilon_x \in \{\pm 1\}$ such that

$$\chi(x) = \epsilon_x \chi^*(x) \quad \text{and} \quad \chi(1)_p = \chi^*(1)_p$$

with $\chi(1)_p$ denoting the p-part of the degree of $\chi$ (Moretó et al., 13 Jun 2025).

A more general version exists for groups of Lie type, where the correspondence may be stated in terms of ℓ-parts and semisimple picky elements, with the local subgroup replaced by the normalizer of the appropriate Sylow ℓ-subgroup (Malle et al., 21 Oct 2025).

3. Structural Determination and Classification of Picky Elements

Extensive classification is given for picky p- and ℓ-elements in groups of Lie type. For unipotent elements, a nontrivial x ∈ G is picky if and only if certain structural conditions are met (for example, x is regular unipotent or belongs to subregular classes in twisted groups). For semisimple elements, necessary and sufficient criteria are established: x is picky if its centralizer in G matches that of a uniquely specified Sylow d-torus, typically expressible via cyclotomic polynomial factors (Malle, 13 Mar 2025).

The structure of Sub_G(x) and the occurrence of picky elements are completely determined in many families. For classical groups, arbitrary p or ℓ, and exceptional groups, explicit lists of picky elements (especially for ℓ = 2, 3) and their centralizers are provided, often supported by computational methods using GAP for low rank or exceptional cases (Malle et al., 21 Oct 2025).

Group Type	Characteristic	Picky Element Structure
Lie type (classical)	p (defining)	Regular unipotents in Borel or certain parabolic subgroups
Lie type (semisimple)	ℓ ≠ p	Elements in maximal tori, subject to cyclotomic order constraints
Twisted/reductive	p, ℓ	Via centralizers or normalizers, with modifiers for disconnected groups

4. Bijections, Character Invariants, and Connection to McKay

The conjecture is a strong local–global principle, subsuming the McKay conjecture as a special case. The classical McKay conjecture claims

$|\text{Irr}_{p'}(G)| = |\text{Irr}_{p'}(N_G(P))|$

where $\text{Irr}_{p'}(G)$ are irreducible characters of p'-degree. The Picky Conjecture refines this by constructing explicit bijections between nonvanishing character sets, preserving not only cardinality but also p- (or ℓ-) parts of degrees and values on the picky element. For p-solvable groups (p odd), uniqueness of sign is established (Moretó et al., 13 Jun 2025). Technical tools such as relative Glauberman correspondences and character triple isomorphisms are fundamental (Moretó et al., 13 Jun 2025, Malle, 13 Mar 2025):

$$\chi_C = e\chi^* + p\Delta + \Xi, \qquad e \equiv \pm1 \ (\mathrm{mod}\ p)$$

often yielding the desired correspondence of character values.

For quasi-simple groups of Lie type and non-defining primes, the strong form of the Picky Conjecture (preservation up to sign) is proven, utilizing Lusztig induction/restriction, Jordan decomposition, and deep combinatorics of d-split Levi subgroups and relative Weyl groups (Malle et al., 21 Oct 2025).

5. Generalizations and Algebraic Group Extensions

In the context of algebraic groups over algebraically closed fields (characteristic p), the subnormaliser of a unipotent element x is the subgroup generated by all Borel subgroups containing x. For regular unipotent x, this reduces to the (unique) Borel subgroup. In disconnected extensions (e.g. via graph automorphisms), the subnormaliser becomes the centralizer of the automorphism (Malle, 13 Mar 2025). These structural results are essential for extending the theory beyond finite groups to algebraic group settings.

6. Implications, Applications, and Future Directions

The Picky Conjecture, when verified (as for p-solvable groups with p odd, most classical and exceptional groups of Lie type, and certain twisted or covering groups (Moretó et al., 13 Jun 2025, Malle et al., 21 Oct 2025)), offers a sharpened lens on how local subgroup structure controls character theory globally.

• It provides a pathway for testing and refining broader representation-theoretic conjectures.
• The explicit preservation of character values and degrees supports more robust forms of the McKay philosophy, suggesting extensions to other invariants (fields of values, modular representations).
• The techniques developed (relative Glauberman correspondence, Lusztig theory, computational confirmation) are foundational for ongoing work in computational and algebraic representation theory.

A plausible implication is that these fine-grained correspondences will influence future approaches to local–global problems, modular representation theory, and the paper of special elements in algebraic groups. The classification of picky elements, especially for primes 2 and 3, is now essentially complete in groups of Lie type, setting a standard for testing similar conjectural correspondences elsewhere.

7. Relation to Analytic and Functional Analysis Contexts

In certain analytic settings (e.g. extremal problems for entire functions, Paley problems for plurisubharmonic functions), “picky” inequalities also arise, in which an integral condition forces a sharp pointwise bound, sometimes featuring Beta and Gamma functions (Khabibullin, 2010). While these analytic “picky” conjectures are structurally distinct from group-theoretic ones, both share the feature that carefully chosen local constraints dictate highly selective global behavior.

Summary Table: Core Forms of the Picky Conjecture

Context	Bijection	Values/Invariant Preservation
Finite group, picky p-element	χ ↔ χ* (local subgroup)	χ(x) = εₓ χ(x); χ(1)_p = χ(1)_p
Lie type, semisimple picky ℓ-element	Ω: ^x(G) → ^x(N_G(P))	ℓ-part of degree, (possibly sign of value)
Analytic (function theory)	S ↔ h, q (aux functions)	Pointwise bounds from integral estimates

The Picky Conjecture thus stands as a central organizing principle in contemporary representation theory, synthesizing local subgroup analysis, explicit character-theoretic invariants, and broader algebraic and analytic perspectives.