Equivariant multiplicities of simply-laced type flag minors

Abstract: The study of remarkable bases of (quantum) coordinate rings has been an area of intensive research since the early 90's. For instance, the multiplicative properties of these bases (in particular the dual canonical basis) was one of the main motivations for the introduction of cluster algebras by Fomin and Zelevinsky around 2000. In recent work, Baumann-Kamnitzer-Knutson introduced an algebra morphism  $\overline{D}$ from the coordinate algebra $\mathbb{C}[N]$ of a maximal unipotent subgroup $N$ to the function field of a maximal torus. It is related to the geometry of Mirkovic-Vilonen cycles via the notion of equivariant multiplicity. This morphism turns out to be useful for comparing good bases of the coordinate algebra $\mathbb{C}[N]$. We will focus on comparing the values taken by $\overline{D}$ on several distinguished elements of the Mirkovic-Vilonen basis and the dual canonical basis. For the latter one, we will use Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster structure of the cluster structure of $\mathbb{C}[N]$ via quiver Hecke algebras as well as recent results by Kashiwara-Kim. This will lead us to an explicit description of the images under $\overline{D}$ of the flag minors of $\mathbb{C}[N]$ as well as remarkable identities between them.

K-theory and homologyquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Paris algebra seminar

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