Standard Young tableaux are fundamental in combinatorics, representation theory, and geometry. In this talk, we will define these objects and discuss some of their remarkable properties. In particular, we will discuss the connection between the major index generating functions for standard Young tableaux and the $S_n$-representation theory of the coinvariant algebras. The major index statistic was originally defined 100 years ago for permutations and has been very successfully extended to tableaux.
In recent joint work with Matjaz Konvalinka and Joshua Swanson, we study the probability distribution of major index on standard tableaux of fixed partition shape chosen uniformly in terms of the corresponding generating function. We give an explicit hook length style formula for all of the cumulants of these distributions using recent work of Chen--Wang--Wang and Hwang--Zacharovas. The cumulant formula allows us to classify all possible limit laws for any sequence of shapes in terms of a simple auxiliary statistic, aft. We show that any such sequence of distributions with aft approaching infinity is asymptotically normal. This leads to a series of questions concerning locations of zero coefficients, unimodality, and asymptotic estimates for the major index generating functions over all standard tableaux of a fixed shape. We give conjectured answers concerning unimodality and asymptotic estimates.
Sara Billey is a Professor and John Rainwater Faculty Fellow at the University of Washington. Her PhD is from UCSD, and she was at MIT before coming to UW. She is a Fellow of the American Math Society.