an:05072571
Zbl 1106.43006
Ceccherini-Silberstein, Tullio; Scarabotti, Fabio; Tolli, Filippo
Trees, wreath products and finite Gelfand pairs
EN
Adv. Math. 206, No. 2, 503-537 (2006).
00187037
2006
j
43A90 20B25 20E08 20E22 22D15 33C80 43A85
finite Gelfand pairs; wreath products; rooted trees; finite ultrametric space; Hamming scheme; Johnson scheme
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 \(V(r, s)= \Aut(T)/K(r, s)\) where \(K(r, s)\) is the stabilizer of a fixed \(T'\) in \(V(r, s)\). The authors show that \((\Aut(T), K(r, s))\) is a Gelfand pair. This generalizes known examples: the ultrametric space, the Hamming scheme, and the Johnson scheme.
Benjamin B. Wells jun. (Charlottesville)