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Trees, wreath products and finite Gelfand pairs. (English) Zbl 1106.43006
Let $$G$$ be a finite group and $$K$$ a subgroup of $$G$$. Let $$L(X)$$ denote the complex-valued functions on $$X$$. The pair $$(G, K)$$ is a Gelfand pair if the algebra $$L(K/G\setminus K)$$ of bi-$$K$$-invariant functions is commutative.
Let $$T$$ be a finite rooted tree of depth $$m$$ and let $$r= \{r_1,r_2,\dots, r_m\}$$ be an $$m$$-tuple of integers $$\geq 2$$. $$T$$ is of type $$r$$ when each vertex at distance $$k$$ from the root has exactly $$r_{k+1}$$ sons, for $$k= 0,1,2,\dots, m-1$$. If $$s$$ is another $$m$$-tuple with $$1\leq s_k\leq r_k$$ then $$V(r, s)$$ denotes the variety of subtrees of $$T$$ of type $$s$$. Then $$V(r, s)= \operatorname{Aut}(T)/K(r, s)$$ where $$K(r, s)$$ is the stabilizer of a fixed $$T'$$ in $$V(r, s)$$. The authors show that $$(\operatorname{Aut}(T), K(r, s))$$ is a Gelfand pair. This generalizes known examples: the ultrametric space, the Hamming scheme, and the Johnson scheme.

MSC:
 43A90 Harmonic analysis and spherical functions 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures 20E08 Groups acting on trees 20E22 Extensions, wreath products, and other compositions of groups 22D15 Group algebras of locally compact groups 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 43A85 Harmonic analysis on homogeneous spaces
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