Reflection Automorphisms

• Reflection automorphisms are automorphisms that act compatibly with distinguished reflective symmetries in algebraic, combinatorial, and categorical structures.
• They are constructible via operations such as conjugation, diagram automorphisms, and quiver mutations, which facilitate the classification of symmetry groups.
• These automorphisms play a pivotal role in invariant theory, Yang–Baxter equations, and quantum group studies, bridging combinatorial and topological frameworks.

A reflection automorphism is an automorphism of an algebraic, combinatorial, or categorical structure associated with an underlying system of reflections, such as a Coxeter group, a reflection monoid, or a solution of the Yang–Baxter or reflection equations, which either conjugates, permutes, or otherwise acts compatibly with the set of distinguished involutive symmetries ("reflections") in the structure. Reflection automorphisms play a fundamental role in the classification of automorphism groups of reflection groups and related objects, in invariant theory, representation theory, the theory of monoids and semigroups, and the combinatorics of Coxeter groups and clusters.

1. Reflection Automorphisms in Coxeter Systems and Weyl Groups

Let $(W,S)$ be a Coxeter system, where $S$ is the set of simple reflections generating $W$. An automorphism of $W$ is called:

• a diagram automorphism if it permutes $S$ according to a graph automorphism of the Coxeter–Dynkin diagram,
• an inner automorphism if it is conjugation by some $g \in W$,
• an inner by diagram automorphism if it is generated by all inner and diagram automorphisms.

An automorphism of $W$ is said to be reflection-preserving if it permutes the set of all reflections. For irreducible Weyl groups, every reflection-preserving automorphism is inner by diagram (Duan et al., 2017).

These automorphisms are explicitly constructible via finite sequences of cluster algebra quiver mutations that preserve the underlying Dynkin diagram. In particular, mutating twice at a vertex $i$ replaces the generator $s_i$ by its $g$-conjugate for a suitable word $S$0, and any inner by diagram automorphism is realized by such a sequence.

Automorphism groups of complex reflection groups, notably $S$1 and their quotients $S$2, are classified by explicit field, scalar, and diagram automorphisms, where those preserving the set of reflections are easily characterized in terms of their action on cycle type and phase, with exceptional diagram automorphisms (as in $S$3, $S$4, etc.) arising only in very specific low-rank cases (Marberg, 2010, Caselli et al., 2012).

2. Reflection Automorphisms of Reflection Monoids

A reflection monoid $S$5 is associated to a group $S$6 acting by linear automorphisms on a vector space $S$7 and a $S$8-invariant Boolean system of subspaces $S$9. The Boolean reflection monoids $W$0, for Weyl group $W$1 and the Boolean system $W$2 of coordinate subspaces, provide monoidal generalizations of Coxeter/Weyl groups. Their automorphism theory mirrors that of their unit group (Duan et al., 2017):

• Every inner automorphism of $W$3 extends uniquely to $W$4;
• Inner by diagram automorphisms of the monoid correspond to sequences of compatible double-mutations on mutable vertices of appropriately frozen quivers, generalizing the theory of quiver mutations for cluster algebras;
• The group of inner automorphisms of $W$5 is isomorphic to $W$6.

These automorphisms act on the monoid generators $W$7, where the $W$8 correspond to simple reflections and $W$9 is the generator associated to a frozen vertex. Explicit presentations encode the relations necessary to generalize the classical behavior of symmetric inverse semigroups and signed partial permutation monoids.

3. Reflection Automorphisms in Invariant Theory

In invariant theory, a reflection automorphism is closely tied to the structure of the group acting on a variety $W$0:

• For a finite group $W$1 acting on $W$2, a $W$3-reflection is an element with fixed-point locus of codimension $W$4. Reflection groups are those generated by 1-reflections (codimension-one fixed loci).
• The existence of minimal separating sets in the invariant ring implies $W$5 must be generated by reflections; more generally, the minimal number $W$6 for which $W$7 is generated by $W$8-reflections is governed by the Cohen–Macaulay defect of associated separating algebras (Reimers, 2013).

Thus, reflection automorphisms connect the combinatorial and homological properties of group actions with the geometry of separating varieties and the algebraic properties of the invariant ring.

4. Reflection Automorphisms in Set-Theoretic Yang–Baxter Theory

For a set-theoretic solution $W$9 of the Yang–Baxter equation, with $S$0, a reflection is a map $S$1 satisfying the reflection equation: $S$2 which imposes compatibility constraints formalized by twist and quasi-commutation conditions.

For bijective, non-degenerate solutions, the set of reflection automorphisms is identified with the centralizer of the left multiplication group in the automorphism group of the associated rack. This provides a powerful unifying structure for classifying and constructing reflections and their automorphisms in the context of braces, racks, and matched products (Albano et al., 2024).

5. Reflection Automorphisms in Quantum Groups and Reflection Algebras

In the theory of quantum groups, reflection automorphisms are induced by automorphisms of quantum symmetric pairs, affine Lie algebras, and associated reflection equations:

• Generalized Satake diagram automorphisms $S$3 and their quantum analogues induce automorphisms of quantized enveloping algebras $S$4 and their right coideal subalgebras $S$5, acting by automorphisms of the second kind and parametrized by diagram data $S$6 (Regelskis et al., 2016).
• These automorphisms correspond to symmetries of trigonometric and rational solutions of the reflection equation, and their orbits classify all solutions with prescribed sparsity and parameter structure.
• In the context of the AdS/CFT correspondence and boundary Yangians, the reflection automorphism is a Hopf algebra involution which implements $S$7 at the level of central charges, preserving or inducing correspondences among boundary and bulk representations (MacKay et al., 2011).

This automorphism structure guarantees the coideal property of subalgebras relevant for boundary integrability and the uniqueness of physical $S$8-matrices under symmetry constraints.

6. Combinatorial and Geometric Realizations: Bruhat Graphs and Braid Groups

In combinatorial algebra, reflection automorphisms manifest as automorphisms induced by reflection elements:

• In Bruhat graphs of Coxeter groups, automorphisms generated by left, right, and "middle" multiplication by reflections correspond to involutive, structure-preserving automorphisms—often coinciding with "special matchings" in the context of Bruhat intervals (Gaetz et al., 2022).
• The full automorphism group of the undirected Bruhat graph is, in classical type $S$9 and right-angled Coxeter groups, generated by these reflection-based operations, and their orbits have a conjectural structure as single intervals.
• In the category of braid groups, the reflection automorphism $g \in W$0 acts on the pure braid group $g \in W$1 by reversing strand order: $g \in W$2, preserving all the defining relations and, in the induced action on the homotopy groups $g \in W$3 (via the Moore complex of the $g \in W$4-group structure), acting trivially for $g \in W$5 (Alekseev et al., 2021).

These reflection automorphisms ensure that deep topological invariants and graph-theoretic or poset-theoretic symmetries are compatible with the combinatorial structure induced by reflections.

7. Schematic Table: Reflection Automorphism Classes in Key Settings

Structure	Reflection Automorphism Type	Reference and Description
Coxeter/Weyl group	Inner by diagram	Sequences of conjugation and diagram sym. (Duan et al., 2017)
Boolean reflection monoid	Inner/inner by diagram	Conjugation by units, diagram perm. (Duan et al., 2017)
Complex reflection groups	Conjugation, field, diagram	Explicit automorphisms, field scalar, etc. (Marberg, 2010, Caselli et al., 2012)
Set-theoretic YBE solution	Centralizer of rack-mult.	Satisfies reflection equation; rack structure (Albano et al., 2024)
Quantum group/coideal subalg.	Diagram automorphisms	Satake diagram, quantum symmetric pairs (Regelskis et al., 2016)
Bruhat graphs	Multiplication by reflection	Graph automorphisms and special matchings (Gaetz et al., 2022)
Pure braid group	Strand reflection automorphism	Reversal of strand order (Alekseev et al., 2021)

This structural uniformity reflects (modulo precise technical conditions) the remarkable persistence of reflection-induced automorphism classes across a wide range of algebraic, combinatorial, and topological settings.