Title: Products of Kirillov-Reshetikhin modules and maximal green sequences
Speaker: Gleb Koshevoy
Institution: Institute for Information Transmission Problems of the Russian Academy of Sciences
Date & Time: Jan 14, 2025, 20:30–22:30
Location: Zoom ID: 904 645 6677, Password: 2024
Meeting Link: https://zoom.us/j/9046456677?pwd=Y2ZoRUhrdWUvR0w0YmVydGY1TVNwQT09&omn=89697485456
We show that a
q-character of a Kirillov-Reshetikhin module (KR-modules) for untwisted quantum affine algebras of simply laced types
An(1),
Dn(1),
E6(1),
E7(1),
E8(1) can be obtained from a specific cluster variable of a seed derived by applying a maximal green sequence to the initial (infinite) quiver of the Hernandez-Leclerc cluster algebra. For a collection of KR-modules with nested supports, we provide an explicit construction of a cluster seed, where cluster variables correspond to the
q-characters of KR-modules in such a collection. We prove that the product of KR-modules in this collection is a simple module. We also construct cluster seeds with cluster variables corresponding to
q-characters of KR-modules from some non-nested collections and conjecture that tensor products of KR-modules for such non-nested collections are simple. Additionally, we show that cluster Donaldson-Thomas transformations for double Bruhat cells of
ADE types can be computed using
q-characters of KR-modules. This is joint work with Y. Kanakubo and T. Nakashima.
Gleb Koshevoy is a chief researcher at the Institute for Information Transmission Problems of the Russian Academy of Sciences. His current research focuses on algebraic combinatorics and the combinatorics of cluster algebras. His key contributions include:
Co-founding the theory of discrete convexity in 2003 (with Vladimir Danilov);
Providing an almost purely combinatorial solution to the Horn problem on the spectra of Hermitian matrix sums in 2003 (with V. Danilov);
Locally characterizing Kashiwara crystals for simply and doubly laced types in 2009 (with V. Danilov and A. Karzanov);
Affirmatively resolving the Leclair-Zelevinsky conjecture on the purity of weakly separated sets in 2011 (with V. Danilov and A. Karzanov), and proposing generalizations of this conjecture;
Developing polynomial-time algorithms for subtraction-free computations of Schur functions and their generalizations in 2014 (with S. Fomin and D. Grigoriev).
