Major Index Statistics: Cyclic Sieving, Branching Rules, and Asymptotics

Abstract

Major index statistics have been studied for over a century in many guises and appear throughout algebraic combinatorics. We pursue major index statistics from two complementary perspectives: algebraic and asymptotic. We first prove an instance of refined cyclic sieving for the major index statistic on words with a fixed cyclic descent type. We next connect this cyclic sieving result to Schur expansions due to Kraskiewicz--Weyman, Stembridge, and Schocker related to certain reflection group branching rules and higher Lie modules. This leads to a conjectured approach to a generalization of Thrall's problem. Afterwards, we transition between the algebraic and the probabilistic by classifying the irreducible components appearing in some of these induced representations. The argument uses the underlying representation theory to prove a uniform local limit theorem, answering a conjecture of Sundaram. We then study the distribution of the major index on standard tableaux of straight shape and certain skew shapes. In particular, we classify all possible limit laws, most of them normal, providing a common generalization of results due to Canfield--Janson--Zeilberger, Chen--Wang--Wang, Diaconis, Feller, Mann--Whitney, and others. We also provide a combinatorial and constructive characterization of the irreducible representations appearing in each degree of the type A coinvariant algebra. Finally we describe a new approach to a result of Baxter--Zeilberger on the limiting joint distribution of the inversion number and major index on permutations using a generating function of Roselle, answering a $300 question of Romik and Zeilberger.