A Bijective Proof of a Factorization Formula for Specialized Macdonald Polynomials

Let μ and ν = (ν1, . . . , νk) be partitions such that μ is obtained from ν by adding m parts of size r. Descouens and Morita proved algebraically that the modified Macdonald polynomials H̃μ(X; q, t) satisfy the identity H̃μ = H̃νH̃(rm) when the parameter t is specialized to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ νk and r ∈ {1, 2}. This note gives a bijective proof of the formula for all r ≤ νk. AMS Subject Classification Numbers: 05A19, 05E05, 05A05.