The Realization Of Some Classes Characters Of The Symmetric Group

A Gel’fand model for a finite group G is a complex representation of G which is isomorphic to the direct sum of all the irreducible representations of G. What this paper explores is a Gel’fand model of the symmetric group Sn. We also test and verify the Gel’fand model of the third symmetric S3and the fourth symmetric S4concretely. When K is the field of complex numbers,G=Sn, a finite dimensional K-subspace N of the polynomial ring K[x1,x2,…,xn] which can be obtained as the zeros of certain symmetrical operators in the Weyl algebra provides a Gel’fand model for G.G is a finite group and G×Gis the direct product of G and G. We can define a function on G by here g∈G. Let X0=1G,X1,…,Xk be all nonequivalent complex irreducible char-acters of G, then we can prove f is a character of G and can be expressed by The thesis presents a conjecture: the function f on the symmetric group be expressed f=∑H≤G aH(1H)G,here aH is an integer. And the thesis verifies the conjecture for third symmetric S3and the fourth symmetric S4.