Action of the framing operator on plethystic exponentials (BH Conjecture)

Prove that, for any level r≥1 horizontal Fock representation ρ^{(r,n0)} of the extended quantum toroidal gl(1) algebra and for any complex weights u=(u1,…,ur) and v=(v1,…,vr), the framing operator F^⟂ acts on the plethystic exponential according to the identity ρ^{(r,n0)}(F^⟂) · exp(∑_{k>0} (-1)^k/(k(1−q2^k)) · ∑_{a=1}^r v_a^k p_k(^{(a)})) = exp(−∑_{k>0} 1/(k(1−q2^k)) · ∑_{a=1}^r p_k(^{(a)}) · [(u'_a)^k + (1−q3^k) ∑_{b=a+1}^r (u'_b)^k]), where u'_a = −^{−1} u_a v_a, q3 = (q1 q2)^{−1}, the symbols p_k(^{(a)}) denote power-sum symmetric functions in the a-th alphabet, and ^ = q3^{1/2}. Establish this identity for all degrees and all levels r.

Background

The paper extends results on generalized Macdonald symmetric functions (GMP) using representation-theoretic tools from the quantum toroidal gl(1) algebra and introduces the framing operator F^⟂ as an outer automorphism element acting in level r horizontal Fock representations. This operator is diagonal on the GMP basis and plays a central role in the algebraic formulation and extension of the Garsia–Haiman–Tesler (GHT) identity to higher levels.

To complete several arguments, especially the higher-level GHT identity, the authors require a specific identity for the action of F^⟂ on a plethystic exponential built from r alphabets and the weights u and v. They formulate this as the BH Conjecture, which asserts a closed-form transformation rule converting the initial plethystic exponential with parameters v into another plethystic exponential with parameters u' determined by u and v.

The authors verify the conjecture extensively by computer for low degrees and small r, and they expect it to hold in full generality. A full proof would provide a crucial algebraic backbone for the extended GHT identity and related constructions (e.g., five-term relations and Fourier/Hopf pairings) developed in the paper.