In this paper, we give a new realization of crystal bases for irreducible highest weight modules over Uq(G2) in terms of monomials. We also discuss the natural connection between the monomial realization and tableau realization. Introduction In 1985, the quantum groups Uq(g), which may be thought of as q-deformations of the universal enveloping algebras U(g) of Kac-Moody algebras g, were introd...

We prove a conjecture about the vertices and edges of the exchange graph of a cluster algebra A in two cases: when A is of geometric type and when A is arbitrary and its exchange matrix is nondegenerate. In the second case we also prove that the exchange graph does not depend on the coefficients of A. Both conjectures were formulated recently by Fomin and Zelevinsky. 1. Main definitions and res...

For a nonnegative matrix A and real diagonal matrix D, two known inequalities on the spectral radius, r(AD) ≥ r(AD) and r(A) r(AD) ≥ r(AD), leave open the question of what determines the order of r(AD) with respect to r(A) r(AD). This is a special case of a broad class of problems that arise repeatedly in ecological and evolutionary dynamics. Here, sufficient conditions are found on A that dete...