# Wreath Product Generalizations of the Triple $(S_{2n},H_{n},\phi)$ and Their Spherical Functions

## TL;DR

This paper generalizes the Gelfand triple involving symmetric and hyperoctahedral groups to wreath products, exploring their spherical functions and connections to symmetric functions, revealing new algebraic structures and relationships.

## Contribution

It introduces a wreath product generalization of the Gelfand triple and analyzes the associated spherical functions and their link to symmetric functions.

## Key findings

- The generalized triplet remains a Gelfand triple.
- Spherical functions relate to multi-partition symmetric functions.
- Provides a framework for studying wreath product symmetries.

## Abstract

The symmetric group $S_{2n}$ and the hyperoctaheadral group $H_{n}$ is a Gelfand triple for an arbitrary linear representation $\phi$ of $H_{n}$. Their $\phi$-spherical functions can be caught as transition matrix between suitable symmetric functions and the power sums. We generalize this triplet in the term of wreath product. It is shown that our triplet are always to be a Gelfand triple. Furthermore we study the relation between their spherical functions and multi-partition version of the ring of symmetric functions.