Monomial–polyhedral realization conjecture for adapted sequences

Establish, for every symmetrizable Kac–Moody algebra g and every adapted sequence ι, that the image Im(Ψι) of the Nakashima–Zelevinsky crystal embedding Ψι:B(∞)↪Z∞ equals the set of integer points a∈Z∞ satisfying ϕ(a)≥0 for every tropicalization Trop(M) of every monomial M in the monomial realizations Ms,k,ι of the fundamental crystals B(Λk) for the Langlands dual algebra gL, with s∈Z≥1 and k∈I.

Background

The paper studies two realizations of crystal bases associated with a symmetrizable Kac–Moody algebra: polyhedral realizations, in which Im(Ψι) is described by integer points satisfying linear inequalities, and monomial realizations, in which fundamental crystal bases are represented by Laurent monomials. For an adapted sequence ι, the paper applies a tropicalization map to the monomials Ms,k,ι for the Langlands dual algebra gL, converting them into linear forms on the coordinates of Z∞.

The conjecture asserts that the inequalities obtained from all such tropicalized monomials define exactly the polyhedral realization Im(Ψι). The paper proves the conjecture for finite-dimensional simple Lie algebras of types An, Bn, Cn, and Dn, rank-2 Kac–Moody algebras, and the listed classical affine Lie algebras, but the general statement remains unresolved beyond these cases. Theorem 4.2 gives a sufficient condition for the conjecture, while Theorem 4.3 verifies it in the specified families.