The Deletion Order and Coxeter Groups

TL;DR

Let $W$ be a finitely generated Coxeter group with simple reflections $S$ and Bruhat order $<_B$. The deletion order $<_\Delta^W$ refines $<_B$, and its associated normal form ${\rm NF}_\Delta(w)$ coincides with the reverse lexicographic normal form ${\rm NF}_{\mathrm{RLex}}(w)$. This enables a Prim-like labeling of the Cayley graph and a constructive way to compute normal forms and labelings. The paper also characterizes when $<_\Delta^W$ is Artinian (finite, affine, or compact hyperbolic) and proves duality $L(w) + L(\omega_0 w) = |W|+1$ for types $A_n$ and $B_n$, clarifying when duality holds or fails across types.

Abstract

The deletion order of a finitely generated Coxeter group W is a total order on the elements which, as is proved, is a refinement of the Bruhat order. This order is applied in [8] to construct Elnitsky tilings for any finite Coxeter group. Employing the deletion order, a corresponding normal form of an element w of W is defined which is shown to be the same as the normal form of w using right to left lexicographic ordering. Further results on the deletion order are obtained relating to the property of being Artinian and, when W is finite, its interplay with the longest element of W.

Figures (3)

• Figure 1: The Cayley Graph of $\mathrm{Sym}({3})$ with the labelling $L$ produced by Algorithm \ref{['Algorithm min min']} (left) and the resulting minimal spanning tree (right).
• Figure 2: The Cayley Graph of $\mathrm{Sym}({4})$ with the labelling $L$ produced by Algorithm \ref{['Algorithm min min']}.
• Figure 3: The minimal spanning tree of $\mathrm{Sym}({4})$ with the labelling $L$ produced by Algorithm \ref{['Algorithm min min']}.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Example 2.1
• Definition 2.2: The Deletion Order
• Remark 2.3
• Example 2.4
• Example 2.5
• Lemma 3.1
• proof
• Lemma 3.2