[Paper Review] Cyclic symmetry loci in Grasssmannians

Chris Fraser|arXiv (Cornell University)|Oct 12, 2020

Algebraic structures and combinatorial models45 references4 citations

TL;DR

This paper studies cyclic symmetry loci in Grassmannians, describing fixed points under cyclic group actions, providing a cell decomposition of totally nonnegative $ $-fixed points, and constructing efficient total positivity tests. It conjectures a generalized cluster algebra structure on these loci when the orbifold order is sufficiently large, generalizing Postnikov's results to the symmetric setting.

ABSTRACT

The Grassmannian admits an action by a finite cyclic group via the cyclic shift map. We give a simple description of the points fixed by each element of this cyclic group, extending Karp's description of the points fixed by the cyclic shift itself. We give a cell decomposition of the set of totally nonnegative points in each cyclic symmetry locus and describe efficient total positivity tests, extending results of Postnikov to the cyclically symmetric setting. We describe a conjectural generalized cluster structure on cyclic symmetry loci provided the order of the orbifold point is sufficiently large. The generalized exchange relations we find should be a Higher Teichmüller analogue of the relations Chekhov and Shapiro used to study Teichmüller theory of orbifolds.

Motivation & Objective

• To systematically describe the fixed point loci of the cyclic shift action on the Grassmannian $\mathrm{Gr}(k,n)$.
• To extend Postnikov's cell decomposition and total positivity theory to the setting of cyclically symmetric points.
• To construct efficient total positivity tests for $\ell$-fixed points in $\mathrm{Gr}(k,n)$.
• To conjecture a generalized cluster algebra structure on the component of $\ell$-fixed TNN points, especially when the orbifold order $p = n/\gcd(\ell,n)$ is at least $k$.

Proposed method

• Characterize each cyclic symmetry locus $\mathrm{Gr}(k,n)^{\rho^\ell}$ as a disjoint union of products of smaller Grassmannians via linear coordinate changes.
• Show that the $\ell$-fixed TNN locus $\mathrm{Gr}(k,n)^{\rho^\ell}_{\geq 0}$ is homeomorphic to a closed ball using techniques from Galashin-Karp-Lam.
• Develop a cyclically symmetric version of bridge decompositions in plabic graphs to design efficient total positivity tests for all cells in the $\ell$-fixed TNN locus.
• Define a conjectural atlas of generalized cluster charts on the distinguished component $\mathcal{D}_{n}(k,\ell)$, with one generalized exchange relation and the rest binomial.
• Specialize the generalized cluster algebra $\mathcal{A}_{\rm cyc}(k,\ell)$ by setting indeterminate coefficients $z_s$ to $q$-binomial coefficients at $p$th roots of unity to recover the coordinate ring $\mathbb{C}[\mathcal{D}]$.
• Use the dual poset of an order ideal in the affine symmetric group $\widetilde{S}_\ell$ to describe the cell closure order in $\mathrm{Gr}(k,n)^{\rho^\ell}_{\geq 0}$.

Experimental results

Research questions

• RQ1 How can the fixed point loci of the cyclic shift automorphism $\rho^\ell$ on $\mathrm{Gr}(k,n)$ be described algebraically and geometrically?
• RQ2 What is the cell decomposition of the totally nonnegative $\ell$-fixed locus $\mathrm{Gr}(k,n)^{\rho^\ell}_{\geq 0}$, and how is its closure order related to the affine symmetric group?
• RQ3 Can efficient total positivity tests be constructed for $\ell$-fixed points, and if so, how do they generalize Postnikov’s methods?
• RQ4 Does a generalized cluster algebra structure exist on the $\ell$-fixed TNN locus, and how does it relate to higher Teichmüller theory?

Key findings

• The $\ell$-fixed locus $\mathrm{Gr}(k,n)^{\rho^\ell}$ is isomorphic to a disjoint union of products of smaller Grassmannians, and is a Richardson variety in $\mathrm{Gr}(k,n)$.
• The totally nonnegative $\ell$-fixed locus $\mathrm{Gr}(k,n)^{\rho^\ell}_{\geq 0}$ is homeomorphic to a closed ball, extending results from Galashin-Karp-Lam.
• A cell decomposition of $\mathrm{Gr}(k,n)^{\rho^\ell}_{\geq 0}$ is given, with the closure order dual to an order ideal in the Bruhat order on $\widetilde{S}_\ell$.
• Efficient total positivity tests for $\ell$-fixed points are constructed using cyclically symmetric bridge decompositions of plabic graphs, valid for all cells, not just the top cell.
• A conjectural generalized cluster algebra structure is proposed on the component $\mathcal{D}_{n}(k,\ell)$ containing $\ell$-fixed TNN points, with one generalized exchange relation and the rest binomial.
• The coordinate ring $\mathbb{C}[\mathcal{D}]$ is shown to coincide with the upper generalized cluster algebra for convenient parameters, and the structure arises as a specialization of $\mathcal{A}_{\rm cyc}(k,\ell)$ by setting $z_s \mapsto \binom{k}{s}_q$ at a $p$th root of unity.