Folding homomorphism conjecture for twisted q-characters

Prove that for any simply-laced complex simple Lie algebra g with a nontrivial Dynkin diagram automorphism σ and for every dominant monomial m in the Frenkel–Reshetikhin Y-variables, the folding homomorphism φ^σ_g satisfies φ^σ_g(χ_q(L(m))) = χ_q^σ(L^σ(φ^σ_g(m))), equivalently giving a dimension-preserving bijection between simple isomorphism classes in the untwisted category C_g and the twisted category C_g^σ.

Background

For twisted quantum loop algebras U_q(Lgσ), Hernandez defined twisted q-characters and a folding homomorphism relating untwisted and twisted Grothendieck rings. This map is known to match q-characters for Kirillov–Reshetikhin modules, and it has been proved for all simples in type A and for simples in certain core subcategories.

The current paper proves the assertion for doubly twisted algebras of types A and D with σ of order 2, lending strong evidence to the general claim. The conjecture seeks to extend this equality to all simply-laced types and all simples, thereby computing every simple twisted q-character from its untwisted counterpart.

References

Conjecture. The equalities eq:folding hold for any dominant monomial $m$.

eq:folding:

$\Phi^\sigma_\fg[L(m)] = [L^\sigma(\phi_\fg^\sigma(m))], \quad \text{or equivalently,} \quad \phi^\sigma_\fg(\chi_q(L(m)))= \chi_q^\sigma(L^\sigma(\phi^\sigma_\fg(m))) $

— Freezing operators in representation theory of quantum loop algebras  (2601.00687 - Fujita et al., 2 Jan 2026) in Section 3.4 (Folding homomorphism)