[Paper Review] Diameter of reduced words

Victor Reiner, Yuval Roichman|arXiv (Cornell University)|Jun 25, 2009

Advanced Combinatorial Mathematics32 references3 citations

TL;DR

This paper determines the exact diameter of the graph formed by reduced words of the longest element in symmetric and hyperoctahedral groups (types A and B), showing it equals $\frac{1}{24}(n-2)(n-1)n(3n-5)$, which matches the number of codimension-two intersection subspaces in their reflection arrangements. The result is derived using properties of supersolvable hyperplane arrangements and minimal galleries in chamber complexes.

ABSTRACT

For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to supersolvable hyperplane arrangements.

Motivation & Objective

• To determine the exact diameter of the graph of reduced words for the longest element in finite Coxeter groups of types A and B.
• To generalize the diameter problem to supersolvable hyperplane arrangements and minimal galleries in chamber complexes.
• To establish a connection between the diameter of reduced word graphs and the number of codimension-two intersection subspaces in the reflection arrangement.
• To investigate whether the diameter of the reduced word graph is bounded by the number of such subspaces, and to test tightness of known bounds.

Proposed method

• Define a graph $G(w)$ whose vertices are reduced words for an element $w$ in a Coxeter group, with edges corresponding to braid relations.
• Use the chamber complex of a central, essential hyperplane arrangement to model minimal galleries from a base chamber to its antipode.
• Construct a graph $G_2$ on minimal galleries from $c_0$ to $-c_0$, with edges corresponding to separation by codimension-two subspaces.
• Prove that for supersolvable arrangements, the graph $G_2$ is connected and its diameter is bounded below by $|L_2|$, the number of codimension-two subspaces.
• Apply the theory of reflection groups and modular flags to identify a unique minimal gallery incident to a flag, linking it to a canonical reduced word.
• Use combinatorial and geometric arguments to show that the diameter of $G(w_0)$ in types A and B equals the number of codimension-two intersections in the arrangement.

Experimental results

Research questions

• RQ1 What is the exact diameter of the graph of reduced words for the longest element in the symmetric group $\mathfrak{S}_n$?
• RQ2 Does the diameter of the reduced word graph for $w_0$ in types A and B equal the number of codimension-two intersection subspaces in the reflection arrangement?
• RQ3 Can the diameter of the reduced word graph be bounded by $|L_2(w)|$, the number of such subspaces, and is this bound tight?
• RQ4 How does the structure of supersolvable hyperplane arrangements influence the connectivity and diameter of the reduced word graph?
• RQ5 To what extent do known upper bounds on the diameter (e.g., $2|L_2(w)|$) reflect the true geometric and combinatorial structure of the graph?

Key findings

• The diameter of the reduced word graph for the longest element in type $A_{n-1}$ is exactly $\frac{1}{24}(n-2)(n-1)n(3n-5)$, matching the number of codimension-two subspaces in the reflection arrangement.
• For type $B_n$, the same diameter formula applies to the graph of reduced words for the longest element, derived via the same geometric and combinatorial framework.
• The diameter of $G(w)$ is bounded above by $2|L_2(w)|$, but examples in types $A_3$ and $B_3$ show this bound is not tight, as the diameter can be much smaller than $|L_2(w)|$.
• The paper provides counterexamples to the conjecture that the diameter is always at most $|L_2(w)|$, showing that even $|L_2(w)|$ is not a tight upper bound.
• A refined conjecture is proposed: for $\mathfrak{S}_n$, the diameter lies between $\frac{1}{2}|L_2(w)|$ and $|L_2(w)|$, and for $B_n$, between $\frac{1}{3}|L_2(w)|$ and $|L_2(w)|$.
• The construction of minimal galleries incident to a modular flag in supersolvable arrangements yields a canonical reduced word for each group element, which aligns with Armstrong’s notion of $\mathbf{w}_0$-sorted words.