Séminaires : Séminaire d'Algèbre

 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. The cluster algorithm due to Genz, Schumann and me is polynomial in time but it uses divisions of polynomials of several variables.
If time permits, I will report on applications of decorated graphs to analysis of the Newton polytopes of F-polynomials related to the Gross-Hacking-Keel-Kontsevich potentials. The talk is based on joint works with Volker Genz and Bea Schumann and with Yuki Kanakubo and Toshiki Nakashima.

The talk will take place in hybrid mode at the Institut Henri Poincaré