»ã±¨±êÌâ (Title)£ºThe modified Macdonald polynomials and mu-Mahonian statistics£¨ÓÅ»¯µÄMacdonald¶àÏîʽºÍmu-Mahonianͳ¼ÆÁ¿£©

»ã±¨ÈË (Speaker)£º½ùÓî ½ÌÊÚ£¨ÏÃÃÅ´óѧ£©

»ã±¨¹¦·ò (Time)£º2025Äê11ÔÂ6ÈÕ£¨ÖÜËÄ£©15:00¡ª16:00

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»ã±¨ÌáÒª£ºThe modified Macdonald polynomials indexed by partitions are the basis of the symmetric functions in infinitely many variables with coefficients in the field of rational functions of two variables. The combinatorial investigation of modified Macdonald polynomials has been greatly promoted by the celebrated breakthrough on the connections between them and mu-Mahonian statistics on fillings of Young diagrams due to Haglund, Haiman and Loehr (2005).

Recently, Corteel, Haglund, Mandelshtam, Mason and Williams (2021) discovered a compact formula for the modified Macdonald polynomials and made a conjecture on an equivalent form of them. This was subsequently affirmed by Ayyer, Mandelshtam and Martin (2023) and they proposed a stronger conjecture on a refined equivalence. Our main result confirms their conjecture. That is, we establish the equidistribution between the pairs (inv, maj) and (quinv, maj) on any row-equivalency class of a given filling of a Young diagram. In particular if the Young diagram is rectangular, the triples (inv, quinv, maj) and (quinv, inv, maj) have the same distribution over the row-equivalence class. This talk is based on joint work with Xiaowei Lin.