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arxiv: 2604.24573 · v1 · submitted 2026-04-27 · 🧮 math.CO

Commutation classes of reduced words and higher Bruhat orders for affine permutations

Sara Billey , Herman Chau , Kevin Liu This is my paper

Pith reviewed 2026-05-08 02:26 UTC · model grok-4.3

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keywords bruhathighergrouporderorderssymmetricaffineclasses
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The pith

The authors define second higher Bruhat orders as posets on commutation classes of reduced words for any affine permutation and prove direct analogs of classical results for all higher orders in the symmetric group.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In the symmetric group, reduced words for the longest element can be grouped into commutation classes, and higher Bruhat orders put a structure on those classes where moves correspond to braid relations. This paper takes that idea and builds similar orders for affine permutations, which are like permutations that repeat in a periodic way and live in an infinite group. They show that for any interval from the identity to some affine permutation, you can define these higher orders and they behave like posets, meaning you can compare classes in a consistent way. For the ordinary symmetric group they also recover the full set of higher orders as in the classical theory. The work stays within pure combinatorics but connects to the study of Coxeter groups and their representations.

Core claim

Paralleling the classical case, we show that the second higher Bruhat order is a poset on commutation classes of reduced words for any affine permutation.

Load-bearing premise

That the notions of commutation classes, braid relations, and covering relations extend directly from the finite symmetric group to the affine symmetric group without requiring extra conditions or modifications to the definitions.

Figures

Figures reproduced from arXiv: 2604.24573 by Herman Chau, Kevin Liu, Sara Billey.

Figure 1
Figure 1. Figure 1: Rpwq (left) and Gpwq (right) for the affine permutation w “ p1, 7, 2, 0q P Sr 4. ones using relations among the generators of Sr n. Two reduced words are commutation equivalent if they differ by a sequence of commutation relations. If a reduced word for w has ipi ` 1qi (resp. pi ` 1qipi ` 1q) in consecutive entries, then a braid is the operation that replaces these three elements with pi ` 1qipi ` 1q (resp… view at source ↗
Figure 2
Figure 2. Figure 2: The Hasse diagram of Pwp6, 3q for w “ p6, 4, 5, 2, 3, 1q P S6. To give some further intuition on the relations, consider the reflection orders for w in Definition 2.1. One can show that for any quasi-inversion rx, y, zs, if Pprx, y, zsq X Inv2pwq “ trx, ys,rx, zsu (resp. trx, zs,ry, zsu), then rx, ys (resp. ry, zs) must appear before rx, zs. These aligns with the relations in Pwpn, 2q. See view at source ↗
Figure 3
Figure 3. Figure 3: The Hasse diagrams for Bwp6, 3q (left), Pwp6, 4q (middle), and Cwp6, 4q (right) for the permutation w “ p6, 4, 5, 2, 3, 1q from view at source ↗
Figure 4
Figure 4. Figure 4: The graph GR for w “ p6, 4, 5, 2, 3, 1q and R “ tr1, 2, 5, 6s,r1, 3, 5, 6su P Cwp6, 4q view at source ↗
read the original abstract

The higher Bruhat orders are partial orders that generalize the weak order on the symmetric group $S_n$, and the second higher Bruhat order is a poset on commutation classes of reduced words for the longest element in $S_n$, where covering relations correspond to braid relations. Constructing analogs in other settings is an area of recent interest, and we present an analog that generalizes any interval $[id,w]$ in the weak order of both the symmetric group and the affine symmetric group. Paralleling the classical case, we show that the second higher Bruhat order is a poset on commutation classes of reduced words for any affine permutation. For the symmetric group, we also establish results for all higher Bruhat orders that are direct analogs of those in the classical case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard theory of Coxeter groups, reduced words, and weak orders; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • standard math Reduced words and commutation classes are well-defined in the affine symmetric group via the standard Coxeter presentation.
    The construction presupposes the usual length function and braid relations extend to the affine case.

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Reference graph

Works this paper leans on

16 extracted references · 1 canonical work pages

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