报告题目： Products of Kirillov-Reshetikhin modules and maximal green sequences
报 告 人：Gleb Koshevoy
所在单位：Institute for Information Transmission Problems of the Russian Academy of Sciences
报告时间：20:30-22:30， Jan 14, 2025
Zoom Id: 904 645 6677，Password: 2024
会议链接：
https://zoom.us/j/9046456677?pwd=Y2ZoRUhrdWUvR0w0YmVydGY1TVNwQT09&omn=89697485456
报告摘要: We show that a $q$-character of a Kirillov-Reshetikhin module (KR-modules) for untwisted quantum affine algebras of simply laced types $A_n^{(1)}$, $D_n^{(1)}$, $E_6^{(1)}$, $E_7^{(1)}$, $E_8^{(1)}$ might be obtained from a specific cluster variable of a seed obtained 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 show an explicit construction of a cluster seed, which has cluster variables corresponding to the $q$-characters of KR-modules of such a collection. We prove that the product of KR-modules of such a collection is a simple module. We also construct cluster seeds with cluster variables corresponding to $q$-characters of KR-modules of some non-nested collections. We make a conjecture that tensor products of KR-modules for such non-nested collections are simple. We show that the cluster Donaldson-Thomas transformations for double Bruhat cells for $ADE$ types can be computed using $q$-characters of KR-modules. This is joint work with Y. Kanakubo and T. Nakashima.
报告人简介： Gleb Koshevoy is a chief researcher at the Institute for Information Transmission Problems of the Russian Academy of Sciences. His current research interests are algebraic combinatorics and combinatorics of cluster algebras. His main results include: 1) The creation of the theory of discrete convexity in 2003 (with Vladimir Danilov); 2) an almost purely combinatorial solution to the Horn problem on the spectra of a sum of Hermitian matrices in 2003 (with V. Danilov); 3) Local characterization of Kashiwara crystals for simply and doubly laced types in 2009 (with V. Danilov and A. Karzanov) 4) Affirmative answer to the Leclair-Zelevinsky conjecture on the purity of weakly separated sets in 2011 (with V. Danilov and A. Karzanov) and proposed various generalizations of this conjecture 5) Proposal of polynomial-time algorithms for subtraction-free computations of Schur functions and their various generalizations in 2014 (with S. Fomin and D. Grigoriev).
