Jack Polynomials And Affine Yangian

The classical Jack polynomials are a generalization of Schur functions,which are used in many fields of mathematical physics,and they are of interest to scholars.In the special case of h1=h,h2=-h-1,h3=-h+h-1,the space of classical Jack polynomials defined on two-dimensional Young diagrams is isomorphic to the MacMahon representation space of affine Yangian of g((1).According to this representation,a new kind of Jack polynomials is introduced in the first part of this thesis,which is equivalent to the classical Jack polynomials Pλα multiplied by a coefficient,using Jkλ to represent the newly introduced Jack polynomials.The properties of Jkλ are explored,and it is found that the structure constant Cλμυ in Jkλ·Jkλ=∑υCλμJkυ can be given from affine Yangian of gl(1),then we give the Boson-Fermion correspondence for Jack polynomials J kλ.In the second part of this thesis,the orthogonal basis of the vector space spanned by eiejeke0|0>is optimized to have a better expression for it.In the calculation,we find that the generator f1 of affine Yangian of gl(1)act on P2,2|0>,P3,3|0)>and the results are all zero.To satisfy this regularity,we require f1P4,4|0>=0.At the same time,we use the properties of affine Yangian of g((1)and orthogonal basis to calculate the equations,and solve a new set of orthogonal basis,and it is proved that 3-Schur functions Sπ equal Jack polynomials Jkλ in the special case of h1=h,h2=-h-1,h3=-h+h-1.