Torsion proliferation
We prove that, in every classical type and every characteristic, a random Schubert variety almost surely has a stratum where the p-Kazhdan–Lusztig and Kazhdan–Lusztig stalk polynomials differ.
Data
- Title: Torsion proliferation
- Authors: Joseph Baine and Daniel Tubbenhauer
- Status: Preprint, 2026. Comments welcome.
- Code and data: GitHub
- ArXiv link: coming soon
- MSC 2020: Primary: 14M15, 20C08; Secondary: 05A05, 60C05.
Abstract
We prove that, in every classical type and every characteristic, a random Schubert variety almost surely witnesses a difference between the p-Kazhdan–Lusztig and Kazhdan–Lusztig bases. More precisely, for the classical Weyl groups of ranks tending to infinity, the proportion of elements whose entire lower Bruhat interval is unchanged tends to zero exponentially fast.
What is the point?
Kazhdan–Lusztig theory has two natural integral shadows. One is the usual KL basis, or intersection cohomology in characteristic zero. The other is the p-canonical basis, or parity sheaves in characteristic p. These two worlds often agree in small examples, but the agreement is not typical. The theorem says that, in the classical towers, almost every sufficiently large element contains a local witness where the two theories differ.
The proof is intentionally soft. One starts with a finite bad type A interval. A consecutive copy of this interval inside a larger permutation, or a positive consecutive copy inside a signed permutation, transports the same local discrepancy upstairs. Finally, a random large permutation almost surely contains the fixed consecutive pattern.
The first picture
Available finite-type computations at \(p=2\). The colors indicate the Coxeter family.
Computational snapshot
The three ingredients
For every prime \(p\), there is a finite type A interval where the pKL and KL local stalk polynomials differ.
Consecutive pattern embeddings preserve the relevant length drop, so the local singularity is transported to larger rank.
A fixed consecutive pattern appears with probability tending to one. Avoidance is exponentially rare.
A few extra words
The result is not a sharp finite-rank prediction. The first rank where the p-canonical basis differs visibly from the ordinary KL basis depends strongly on the prime and the type. The point is asymptotic: once one finite bad seed exists, copies of it become overwhelmingly likely in large rank.
The accompanying computations are meant to show what already happens in small rank. Besides the percentage of changed elements, the data record correction size, graded corrections, length distributions, first witnesses, and several structural summaries. Most of this is too much for the paper, but it belongs on the GitHub page.
The moral is cheerfully bleak: the unchanged cases survive in small ranks, but in the classical towers the bad behavior becomes generic.