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Splicing skew shaped positroids

Eugene Gorsky, Soyeon Kim, Tonie Scroggin, José Simental

Abstract

Skew shaped positroids (or skew shaped positroid varieties) are certain Richardson varieties in the flag variety that admit a realization as explicit subvarieties of the Grassmannian $\mathrm{Gr}(k,n)$. They are parametrized by a pair of Young diagrams $μ\subseteq λ$ fitting inside a $k \times (n-k)$-rectangle. For every $a = 1, \dots, n-k$, we define an explicit open set $U_a$ inside the skew shaped positroid $S^{\circ}_{λ/μ}$, and show that $U_a$ is isomorphic to the product of two smaller skew shaped positroids. Moreover, $U_a$ admits a natural cluster structure and the aforementioned isomorphism is quasi-cluster in the sense of Fraser. Our methods depend on realizing the skew shaped positroid as an explicit braid variety, and generalize the work of the first and third authors for open positroid cells in the Grassmannian.

Splicing skew shaped positroids

Abstract

Skew shaped positroids (or skew shaped positroid varieties) are certain Richardson varieties in the flag variety that admit a realization as explicit subvarieties of the Grassmannian . They are parametrized by a pair of Young diagrams fitting inside a -rectangle. For every , we define an explicit open set inside the skew shaped positroid , and show that is isomorphic to the product of two smaller skew shaped positroids. Moreover, admits a natural cluster structure and the aforementioned isomorphism is quasi-cluster in the sense of Fraser. Our methods depend on realizing the skew shaped positroid as an explicit braid variety, and generalize the work of the first and third authors for open positroid cells in the Grassmannian.

Paper Structure

This paper contains 28 sections, 41 theorems, 177 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Braid varieties
  4. Background
  5. Flags and framed flags
  6. Relative position
  7. Transversality
  8. Braid varieties
  9. Skew shaped positroid varieties
  10. Positroid varieties
  11. Skew shaped positroids
  12. Lattice path labeling of a skew Young diagram
  13. Grassmann necklace for a skew shaped positroid
  14. Skew shaped positroid as a braid variety
  15. Skew shaped positroid as a braid variety on $n$ strands
  16. ...and 13 more sections

Key Result

Theorem 1.1.2

Let $n > 0$ and $0 < k < n$. Let $\mu \subseteq \lambda$ be partitions fitting inside a $k \times (n-k)$-rectangle. Choose $1 \leq a \leq n-k$, and let $U_a$ be the principal open set in $S_{\lambda/\mu}^\circ$ defined by the non-vanishing of the cluster variables in the $a$-th column of $\lambda/\m Furthermore, $\Phi_a$ is a quasi-cluster equivalence.

Figures (9)

  • Figure 1: On the left is the diagram for $\lambda/\mu$ with corresponding matrix $(v_1,\dots,v_{12})$, where $\lambda=(7,7,5,3,1)$ and $\mu=(3,1)$. Performing the cut at $a=6$ is equivalent to cutting the diagram along the boundary of the 5-th and 6-th column from the right, as indicated by the blue dotted line. The resulting skew diagrams under the map $\Phi_6$ are shown on the right, corresponding to the diagrams for $\lambda^{6,L}/\mu^{6,L}$ with matrix $(w_1,\dots,w_7)$ and $\lambda^{6,R}/\mu^{6,R}$ with matrix $(u_1,\dots,u_{10})$, respectively.
  • Figure 2: The permutation $w_\lambda$ for $\lambda = (7,7,5,3,1)$ considered inside the rectangle $(7^5)$. The factors $C_{\lambda, j}$ correspond to the columns of the complement of $\lambda$ inside the rectangle.
  • Figure 3: The length-additive decomposition $w_\mu = {\color{blue} w_{\lambda/\mu}}{\color{red} w_{\lambda}}$.
  • Figure 4: The wiring diagram associated to the bounded affine permutation $f_{\lambda/\mu}$. For horizontal arrows, we have to add $n$ to the value of the target. Note that $5$ is a fixed point of $f_{\lambda/\mu}$ for the right-hand figure.
  • Figure 5: Braid $\underline{w_{0}}\beta_{\lambda/\mu}$ for $\lambda=(7,7,5,3,1)$, and $\mu=(3,3,2)$ with subspaces $V(a,i)$ and $W_i,W^{\mathrm{op}}_i$ filled in.
  • ...and 4 more figures

Theorems & Definitions (140)

  • Remark 1.1.1
  • Theorem 1.1.2
  • Example 1.1.3
  • Corollary 1.1.4
  • Remark 1.1.5
  • Example 1.2.1
  • Remark 1.2.2
  • Theorem 1.2.3
  • Definition 2.4.1
  • Lemma 2.4.2
  • ...and 130 more