Castelnuovo-Mumford regularity for $321$-avoiding Kazhdan-Lusztig varieties

TL;DR

This paper introduces a combinatorial method to compute the Castelnuovo-Mumford regularity of 321-avoiding Kazhdan-Lusztig varieties using K-theoretic skew excited Young diagrams, with an algorithm that provides bounds and exact values in certain cases.

Contribution

It develops a new combinatorial approach and algorithm for calculating the regularity of specific algebraic varieties, extending to mixed ladder determinantal varieties.

Findings

The regularity can be computed via K-theoretic skew excited Young diagrams.

An algorithm provides lower bounds and exact regularity in certain settings.

Specialization to mixed ladder determinantal varieties demonstrates the method's versatility.

Abstract

We prove the Castelnuovo--Mumford regularity of 321-avoiding Kazhdan--Lusztig varieties can be computed combinatorially in terms of $K$-theoretic skew excited Young diagrams. We present an algorithm which gives a lower bound for this regularity and describe a setting in which this algorithm provides precise regularity computations. This algorithm specializes to compute the regularity of all two-sided mixed ladder determinantal varieties.