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Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties

Laura Escobar, Alex Fink, Jenna Rajchgot, Alexander Woo

Abstract

We study a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan-Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme-theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan-Lusztig ideals that arise are exactly those where the opposite cell is $123$-avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley-Reisner ideals of subword complexes) and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.

Gröbner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties

Abstract

We study a class of combinatorially-defined polynomial ideals which are generated by minors of a generic symmetric matrix. Included within this class are the symmetric determinantal ideals, the symmetric ladder determinantal ideals, and the symmetric Schubert determinantal ideals of A. Fink, J. Rajchgot, and S. Sullivant. Each ideal in our class is a type C analog of a Kazhdan-Lusztig ideal of A. Woo and A. Yong; that is, it is the scheme-theoretic defining ideal of the intersection of a type C Schubert variety with a type C opposite Schubert cell, appropriately coordinatized. The Kazhdan-Lusztig ideals that arise are exactly those where the opposite cell is -avoiding. Our main results include Gröbner bases for these ideals, prime decompositions of their initial ideals (which are Stanley-Reisner ideals of subword complexes) and combinatorial formulas for their multigraded Hilbert series in terms of pipe dreams.

Paper Structure

This paper contains 26 sections, 41 theorems, 107 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Gröbner bases and initial ideals
  4. Torus actions, multigradings, and $K$-polynomials
  5. Simplicial complexes and Stanley-Reisner ideals
  6. Kazhdan-Lusztig varieties
  7. Schubert cells and varieties
  8. Permutations and left-right weak order
  9. Opposite Schubert cells as spaces of matrices
  10. Rank conditions on type C Kazhdan-Lusztig varieties
  11. Small patches
  12. Small patches and symmetric matrices
  13. Equations for type C Kazhdan-Lusztig varieties
  14. Torus action of type C Kazhdan-Lusztig varieties and the weights for $v\ge v_\square$
  15. Type C subword complexes and vertex decomposition
  16. ...and 11 more sections

Key Result

Theorem 1

Let $v,w\in C_n$ and $v$ be $123$-avoiding. The determinants defining the ideal of $\mathcal{N}_{v,w}$ form a Gröbner basis with respect to any diagonal term order.

Figures (11)

  • Figure 1: On the left we have the diagram of $v=462513>v_\square$ and on the right its associated skew partition.
  • Figure 2: The simplicial complex $\Delta_{\overline{vc_0},w}$ for $v=321654$ and $w=635241$.
  • Figure 3: The simplicial complex $\Delta_{\overline{v},w} = \operatorname{cone}_{z_{33}}\Delta_{\overline{vc_0},w} \cup \Delta_{\overline{vc_0}, wc_0}$ for $v=321654$ and $w=632541$.
  • Figure 4: The entries of $M_{\overline{v}}$ in relevant rows and columns.
  • Figure 5: The pipe dream on the left is reduced and contains the pipe dream on the right, which is not reduced. These are both pipe dreams for $1432$.
  • ...and 6 more figures

Theorems & Definitions (92)

  • Theorem
  • Theorem 3.1
  • Example 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • proof
  • Corollary 3.6
  • proof
  • ...and 82 more