Dot actions and Hessenberg spaces for dihedral groups
Jonathan Boretsky, Reuven Hodges, Nathan Lesnevich, Tyrrell McAllister, Joseph Pappe
math.CO
Oct 12, 2026 · v1
TL;DR
All results of the paper were formalized in Lean 4 using Mathlib, with AI assistance, and the formalization is published as a linked artifact.
Abstract
Splines on Weyl groups model the equivariant cohomology of regular semisimple Hessenberg varieties, and the natural group action on them encodes the monodromy action. The resulting representations are of great interest in algebraic combinatorics for their connections to Hecke algebra characters, LLT polynomials, and chromatic symmetric functions. We study the analogous construction for dihedral groups, which in general have no underlying geometry. Using purely algebraic methods, we show that these spline modules are free, construct explicit bases, and compute their Hilbert series, recovering the familiar properties of the geometric setting. We also determine the associated dihedral group representations completely. Unlike what is conjectured for the symmetric group case, these need not be permutation representations.
Problem
Splines on Weyl groups model the equivariant cohomology of regular semisimple Hessenberg varieties, and the group action on them encodes monodromy. The analogous spline modules for dihedral groups I_2(n) have no underlying geometry. Their algebraic structure and associated representations were not previously determined.
Approach
Purely algebraic methods are used. The authors define Schubert splines via divided differences, prove flow-up triangularity from the rank-two Bruhat order, and build flow-up bases for arbitrary Hessenberg spaces using divided differences and the dot action. They then compute the left (dot action) and right representations using equivariant freeness. All results were formalized in Lean 4 with Mathlib, with assistance from LLMs.
Results
The spline modules are free over C[x,y], with explicit bases and a palindromic Hilbert series given by an inversion-type statistic. The left and right dihedral group representations are determined completely. Unlike the conjectured symmetric group case, they need not be permutation representations.
