Gelfand pairs associated with the action of graph automaton groups. (English) Zbl 1522.20099
In this paper, the authors pursue a construction they introduced in [Adv. Group Theory Appl. 11, 75–112 (2021; Zbl 1495.20038)]. This construction begins with a finite graph and results in an invertible automaton. The group generated by the transformations associated with the states of the automaton is an automorphism group of the rooted regular tree whose valency is the order of the input graph. Properties of these particular automata were studied in more detail in the original paper.
In this paper, the authors show that the group \(G_n\) restriction of the action of this group to the \(n\)th level of the rooted tree, together with the stabiliser \(K_n\) in \(G_n\) of a fixed vertex in that level, are a symmetric “Gelfand pair”. That is, the algebra of bi-\(K_n\)-invariant functions on \(G_n\) is commutative under convolution. They explicitly determine the associated spherical functions.
In this paper, the authors show that the group \(G_n\) restriction of the action of this group to the \(n\)th level of the rooted tree, together with the stabiliser \(K_n\) in \(G_n\) of a fixed vertex in that level, are a symmetric “Gelfand pair”. That is, the algebra of bi-\(K_n\)-invariant functions on \(G_n\) is commutative under convolution. They explicitly determine the associated spherical functions.
Reviewer: Joy Morris (Lethbridge)
MSC:
| 20E08 | Groups acting on trees |
| 20F65 | Geometric group theory |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 43A85 | Harmonic analysis on homogeneous spaces |
| 43A90 | Harmonic analysis and spherical functions |
Keywords:
graph automaton group; Gelfand pair; rooted regular tree; 2-points homogeneous action; spherical functionCitations:
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