Polyhedral parametrization of canonical bases
Gleb Koshevoy (Moscou)
Parametrizations of the canonical bases, string basis and theta basis, can be obtained by the tropicalization of the Berenstein-Kazhdan decoration function and the Gross-Hacking-Keel-Kontsevich potential respectively. For a classical Lie algebra and a reduced decomposition $\mathbf i$, the decorated graphs are constructed algorithmically, vertices of such graphs are labeled by monomials which constitute the set of monomials of the Berenstein-Kazhdan potential. Due to this algorithm we obtain a characterization of $\mathbf i$-trails introduced by Berenstein and Zelevinsky. Our algorithm uses multiplication and summations only, its complexity is linear in time of writing the monomials of the potential. For $SL_n$, there is an algorithm due to Gleizer and Postnikov which gets all monomials of the Berenstein-Kazhdan potential using combinatorics of wiring diagrams. For this case, our algorithm uses simpler combinatorics and is faster than the Gleizer-Postnikov algorithm. For computing GHKK potential there is an algorithm using cluster mutations due to Genz, Schumann and me, which is polynomial in time but it uses divisions of polynomials of several variables.
The talk is based on joint works with Yuki Kanakubo and Toshiki Nakashima and with Volker Genz and Bea Schumann.