A bijective proof of a factorization formula for Macdonald polynomials   at roots of unity
  We give a combinatorial proof of the factorization formula of modified Macdonald polynomials when the parameter t is specialized at a primitive root of unity. Our proof is restricted to the special case of partitions with 2 columns. We mainly use the combinatorial interpretation of Haglund, Haiman and Loehr giving the expansion of the modified Macdonald polynomials on the monomial basis. 
Francois Descouens, Hideaki Morita, Yasuhide Numata