# Asymptotic normality of the major index on standard tableaux

## TL;DR

This paper investigates the asymptotic distribution of the major index on standard tableaux of various shapes, providing a classification of limit laws and connecting to representation theory of complex reflection groups.

## Contribution

It introduces a cumulant-based approach to classify all possible limit laws for the major index on standard tableaux of arbitrary shapes, extending previous results.

## Key findings

- Classifies limit laws using a new auxiliary statistic, aft.
- Provides a detailed description of the distribution of irreducible representations in coinvariant algebras.
- Suggests conjectures on unimodality, log-concavity, and local limit theorems.

## Abstract

We consider the distribution of the major index on standard tableaux of arbitrary straight shape and certain skew shapes. We use cumulants to classify all possible limit laws for any sequence of such shapes in terms of a simple auxiliary statistic, aft, generalizing earlier results of Canfield--Janson--Zeilberger, Chen--Wang--Wang, and others. These results can be interpreted as giving a very precise description of the distribution of irreducible representations in different degrees of coinvariant algebras of certain complex reflection groups. We conclude with some conjectures concerning unimodality, log-concavity, and local limit theorems.