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Signed permutations and degree-one dot action representations for types B and C

Nathan R. T. Lesnevich

TL;DR

The paper develops a combinatorial framework for type B/C analogues of the type A dot-action representations arising from Hessenberg varieties, by studying a spline ring $\mathcal{M}_{H}$ on the signed permutation group $\mathfrak{W}_n$ with edge labels from Hessenberg root data. It proves that the degree-one piece $\mathcal{M}_{H}^1$ and the corresponding left and right dot-action representations $\mathrm{L}_{H}$ and $\mathrm{R}_{H}$ are completely determined by explicit combinatorial data $S(H)$, and provides explicit generators and bases enabling the computation of the degree-one characters. The main results yield nonnegative BC-symmetric-function expansions for the degree-one pieces, giving $h_{\lambda,\mu}$-positive expressions and linking to representations of $\mathfrak{W}_n$ via the BC Frobenius map. This work extends type A phenomena (chromatic symmetric functions and LLT polynomials) to types B and C, offering a purely combinatorial route to BC-positivity at degree one and paving the way for further exploration of BC-analogues. The methodology combines root-system combinatorics, GKM theory, and explicit spline constructions to produce a concrete, computable description of these representations.

Abstract

A spline is an assignment of polynomials to the vertices of a graph, where the difference of two polynomials along an edge must belong to the ideal labeling that edge. We consider a ring of splines $\mathcal{M}_{H}$ constructed on a graph whose vertices are the Weyl group $\mathfrak{W}_n$ of signed permutations, and whose edges and edge-ideals are defined using an order ideal $H$ of positive roots. These splines are a module over the polynomial ring in two ways, and a $\mathfrak{W}_n$-module by the dot action. These structures on $\mathcal{M}_{H}$ give rise to the graded left and right dot action representations of $\mathfrak{W}_n$. The left representation is the type B/C generalization of the type A dot action for regular semisimple Hessenberg varieties (and thus, chromatic quasisymmetric functions), and the right representation is the same for corresponding manifolds of isospectral matrices (and thus, unicellular LLT polynomials). This paper gives explicit module generators for the degree-one graded piece of $\mathcal{M}_{H}$ and computes the degree-one piece of the both dot action representations for all $H$ using the combinatorial data of $H$.

Signed permutations and degree-one dot action representations for types B and C

TL;DR

The paper develops a combinatorial framework for type B/C analogues of the type A dot-action representations arising from Hessenberg varieties, by studying a spline ring on the signed permutation group with edge labels from Hessenberg root data. It proves that the degree-one piece and the corresponding left and right dot-action representations and are completely determined by explicit combinatorial data , and provides explicit generators and bases enabling the computation of the degree-one characters. The main results yield nonnegative BC-symmetric-function expansions for the degree-one pieces, giving -positive expressions and linking to representations of via the BC Frobenius map. This work extends type A phenomena (chromatic symmetric functions and LLT polynomials) to types B and C, offering a purely combinatorial route to BC-positivity at degree one and paving the way for further exploration of BC-analogues. The methodology combines root-system combinatorics, GKM theory, and explicit spline constructions to produce a concrete, computable description of these representations.

Abstract

A spline is an assignment of polynomials to the vertices of a graph, where the difference of two polynomials along an edge must belong to the ideal labeling that edge. We consider a ring of splines constructed on a graph whose vertices are the Weyl group of signed permutations, and whose edges and edge-ideals are defined using an order ideal of positive roots. These splines are a module over the polynomial ring in two ways, and a -module by the dot action. These structures on give rise to the graded left and right dot action representations of . The left representation is the type B/C generalization of the type A dot action for regular semisimple Hessenberg varieties (and thus, chromatic quasisymmetric functions), and the right representation is the same for corresponding manifolds of isospectral matrices (and thus, unicellular LLT polynomials). This paper gives explicit module generators for the degree-one graded piece of and computes the degree-one piece of the both dot action representations for all using the combinatorial data of .

Paper Structure

This paper contains 13 sections, 30 theorems, 112 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Signed Permutations
  4. Root Systems
  5. Hessenberg spaces and inversions
  6. Splines and the Dot Action
  7. Ideals of Transpositions in Types B and C
  8. Simpler Conditions for MH1
  9. Signed Permutations with a singe H-inversion
  10. Splines in splinesMH1
  11. Generators for SplinesH1
  12. Bases of MH1 and the Left & Right Representations
  13. Type B/C-Symmetric Functions.

Key Result

Theorem 1

If $H$ is a type B or type C Hessenberg space, there exist non-negative integers $a,b \in \mathbb{N}$, integers $c,d \in \{0,1\}$, and a subset $I \subseteq [n]$, each determined by $S(H)$, such that and where Moreover, $c$ and $d$ are never simultaneously $1$, and $d$ is always $0$ in type B.

Figures (3)

  • Figure 1: The simple roots for B${}_4$ and positive root poset for B${}_3$
  • Figure 2: The simple roots for C${}_4$ and the positive root poset for C${}_3$.
  • Figure 3: The labeled graph for $H = \Delta$.

Theorems & Definitions (88)

  • Theorem 1
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 78 more