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The Picky Conjecture for finite groups

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Let GG be a finite group, let ℓ\ell be a prime, let P∈Syl⁡ℓ(G)P\in\operatorname{Syl}_\ell(G), and let x∈Px\in P be a picky ℓ\ell-element, meaning that xx lies in a unique Sylow ℓ\ell-subgroup of GG. For any finite group HH and y∈Hy\in H, write Irr⁡y(H)\operatorname{Irr}^y(H) for the set of complex irreducible characters of HH that take a nonzero value at yy.

Picky Conjecture. There exists a bijection

f:Irr⁡x(G)→Irr⁡x(N⁡G(P))f:\operatorname{Irr}^x(G)\to\operatorname{Irr}^x(\operatorname{N}_G(P))

such that, for every χ∈Irr⁡x(G)\chi\in\operatorname{Irr}^x(G),

χ(1)ℓ=f(χ)(1)ℓ\chi(1)_\ell=f(\chi)(1)_\ell

and

Q(χ(x))=Q(f(χ)(x)).\mathbb{Q}(\chi(x))=\mathbb{Q}(f(\chi)(x)).

We say that the Picky Conjecture holds for (G,ℓ,x)(G,\ell,x) when these degree and character-field conditions hold.

The conjecture proposes a character correspondence extending the McKay conjecture. The source paper proves it for all quasi-simple groups of Lie type in non-defining characteristic; its general status beyond those cases is not resolved here.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The Picky Conjecture for finite groups

    Let pp be a prime, let GG be a finite group, and let P∈Syl⁡p(G)P\in\operatorname{Syl}_p(G). For x∈Px\in P, call xx picky if it lies in a unique Sylow pp-subgroup of GG. Define

    Irr⁡x(G)={χ∈Irr⁡(G)∣χ(x)≠0}.\operatorname{Irr}^x(G)=\{\chi\in\operatorname{Irr}(G)\mid \chi(x)\ne 0\}.

    Picky Conjecture. If x∈Px\in P is picky, then there exists a bijection

    Γ:Irr⁡x(G)⟶Irr⁡x(NG(P))\Gamma:\operatorname{Irr}^x(G)\longrightarrow\operatorname{Irr}^x(\mathbf{N}_G(P))

    satisfying (I)(I) Γ(χ)(1)p=χ(1)p\Gamma(\chi)(1)_p=\chi(1)_p for every χ∈Irr⁡x(G)\chi\in\operatorname{Irr}^x(G), and (II)(II) Q(Γ(χ)(x))=Q(χ(x))\mathbb{Q}(\Gamma(\chi)(x))=\mathbb{Q}(\chi(x)) for every χ∈Irr⁡x(G)\chi\in\operatorname{Irr}^x(G). This character-theoretic local-global conjecture is related to the McKay conjecture and was formulated in the cited source. It has been proved by Cabanes and Späth.

    source: Juan Martínez Madrid, “The Picky and Subnormalizer Conjectures for symmetric groups”, arXiv:2508.05180 (2026).

References

Primary source

Gunter Malle and A. A. Schaeffer Fry, “The Picky Conjecture for groups of Lie type”, arXiv:2510.18397 (2025).

Additional references

2 papers in this index state this conjecture (2025). The statement above is taken from the most recent of them; the others are arXiv:2508.05180.

character-correspondencescharacter-theoryfinite-groupsmckay-conjecturerepresentation-theory
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