  The first goal of this note is to extend the well-known Feigin homomorphisms
taking quantum groups to quantum polynomial algebras. More precisely, we define
generalized Feigin homomorphisms from a quantum shuffle algebra to quantum
polynomial algebras which extend the classical Feigin homomorphisms along the
embedding of the quantum group into said quantum shuffle algebra. In a recent
work of Berenstein and the author, analogous extensions of Feigin homomorphisms
from the dual Hall-Ringel algebra of a valued quiver to quantum polynomial
algebras were defined. To relate these constructions, we establish a
homomorphism, dubbed the quantum shuffle character, from the dual Hall-Ringel
algebra to the quantum shuffle algebra which relates the generalized Feigin
homomorphisms. These constructions can be compactly described by a commuting
tetrahedron of maps beginning with the quantum group and terminating in a
quantum polynomial algebra. The second goal in this project is to better
understand the dual canonical basis conjecture for skew-symmetrizable quantum
cluster algebras. In the symmetrizable types it is known that dual canonical
basis elements need not have positive multiplicative structure constants, while
this is still suspected to hold for skew-symmetrizable quantum cluster
algebras. We propose an alternate conjecture for the symmetrizable types: the
cluster monomials should correspond to irreducible characters of a KLR algebra.
Indeed, the main conjecture of this note would establish this "KLR conjecture"
for acyclic skew-symmetrizable quantum cluster algebras: that is, we conjecture
that the images of rigid representations under the quantum shuffle character
give irreducible characters for KLR algebras.
