Gr\"obner bases, symmetric matrices, and type C Kazhdan-Lusztig varieties

TL;DR

This paper develops Gr"obner bases and prime decompositions for a class of symmetric matrix ideals related to type C Kazhdan-Lusztig varieties, linking algebraic, combinatorial, and geometric aspects.

Contribution

It introduces a new class of symmetric determinantal ideals as type C analogs of Kazhdan-Lusztig ideals, with explicit Gr"obner bases and combinatorial formulas.

Findings

Gr"obner bases for the ideals are constructed.

Prime decompositions of initial ideals are provided.

Multigraded Hilbert series are expressed via pipe dreams.

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 Gr\"obner 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.