Symmetric functions and Jucys-Murphy elements

  This page gives new data for the class expansion of the Hall-Littlewood polynomial taken at the Jucys-Murphy elements of the symmetric group. Our results have been published there.

  We consider the Hall-Littlewood polynomial P_k(x₁,x₂,...,x_n;z), its specialization at the Jucys-Murphy elements J₁,J₂,...,J_n of the symmetric group of n letters, and the development of this specialization in terms of the Farahat-Higman classes, i.e. we write

  We give the generating functions of the coefficients, i.e.
 

  For z = 0 ( respectively z = 1 ), our tables correspond to the case of complete symmetric functions h_k (respectively power sums p_k).

  Tables giving the function &Phi_&rho(t) for weight of &rho from 3 to 14

  Tables for bigger values may be given upon request.

  Our results are in Maple format. They should be read as follows.

• The letter z stands for the parameter of the Hall-Littlewood polynomial.
• The letter q stands for exp(t).
• The table gives first the partition &rho, then &Phi_&rho(t).

Example :
[3]
-1/3*q+1/3*q^2+1/6/q*z+1/6*q*z-1/6/q^2*z-1/3/q+1/3/q^2-1/6*q^2*z

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