The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(∞) and of the Lie superalgebra osp(1|∞). A complete solution to the problem is presented, in which the Fock...

A combinatorial construction of a Gelfand model for the symmetric group and its Iwahori-Hecke algebra is presented.

The Gelfand representation of $${\mathcal {S}}_n$$ is the multiplicity-free direct sum irreducible representations . In this paper, we use a result Adin, Postnikov, and Roichman to find generating function for character. order function, investigate descents so-called $$\lambda $$ -unimodal involutions.