The twist for positroids

International audience There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.

Bott–Samelson Varieties and Poisson Ore Extensions

We show that associated with any $n$-dimensional Bott–Samelson variety of a complex semi-simple Lie group $G$, one has $2^n$ Poisson brackets on the polynomial algebra $A={\mathbb{C}}[z_1, \ldots , z_n]$, each an iterated Poisson Ore extension and one of them a symmetric Poisson Cauchon–Goodearl–Letzter (CGL) extension in the sense of Goodearl–Yakimov. We express the Poisson brackets in terms of root strings and structure constants of the Lie algebra of $G$. It follows that the coordinate rings of all generalized Bruhat cells have presentations as symmetric Poisson CGL extensions. The paper establishes the foundation on generalized Bruhat cells and sets the stage for their applications to integrable systems, cluster algebras, total positivity, and toric degenerations of Poisson varieties, some of which are discussed in the Introduction.

Affine cluster monomials are generalized minors

We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with$\mathbf{g}$-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type$A_{1}^{(1)}$, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.