Séminaires : Séminaire d'Algèbre

Admissible chains of $\bold i$-boxes are important combinatorial tools in the monoidal categorification of cluster algebras via representations of quantum affine algebras, since they provide some seeds of the cluster algebra. For a given sequence $\bold i$ with indices ranging over the interval [a,b], we define a subinterval [x,y] of [a,b] as an $\bold i$-box if the color of $\bold i$ at x matches the color at y. Two $\bold i$-boxes are said to commute if the extension of one of the $\bold i$-boxes by one step to the left and one step to the right properly contains the other $\bold i$-box. A maximal commuting family of $\bold i$-boxes yields a seed in the category of finite-dimensional modules over the quantum affine algebra, and any such family can be constructed from an admissible chain.  In this talk, I will introduce the notion of $\bold i$-boxes and present recent results on the exchange matrices of a maximal commuting family of $\bold i$-boxes. This is a joint work with Masaki Kashiwara.

This talk will take place on Zoom only.