Picky Conjecture (Moretó–Rizo)

Establish, for every finite group G, prime ℓ, Sylow ℓ-subgroup P ≤ G, and picky ℓ-element x ∈ P (i.e., an ℓ-element lying in a unique Sylow ℓ-subgroup of G), a bijection f: Irr^x(G) → Irr^x(N_G(P)) between irreducible characters that do not vanish at x such that (i) the ℓ-part of the degrees is preserved, χ(1)_ℓ = f(χ)(1)_ℓ for all χ ∈ Irr^x(G), and (ii) the fields generated by the values at x coincide, Q(χ(x)) = Q(f(χ)(x)).

Background

The Picky Conjecture, proposed by Moretó and Rizo, introduces a new global-to-local character correspondence that extends the McKay correspondence. An ℓ-element x is called “picky” if it lies in a unique Sylow ℓ-subgroup P of G; write Irr^x(G) for the set of irreducible characters of G that do not vanish at x.

The conjecture predicts a bijection between Irr^x(G) and Irr^x(N_G(P)) that simultaneously preserves the ℓ-part of character degrees and matches the fields generated by the values at x. As observed in the paper, whenever G has a picky ℓ-element, the Picky Conjecture implies the McKay Conjecture for G at the prime ℓ. The authors prove the conjecture for quasi-simple groups of Lie type in non-defining characteristic, but the general conjecture remains open.