Richardson tableaux and components of Springer fibers equal to Richardson varieties

TL;DR

This paper introduces Richardson tableaux, characterizes them combinatorially and geometrically, and links them to Springer fibers, Richardson varieties, and totally nonnegative Springer fibers, revealing elegant enumeration formulas and smoothness properties.

Contribution

It defines Richardson tableaux and characterizes them both combinatorially and geometrically, connecting Springer fibers, Richardson varieties, and totally nonnegative parts.

Findings

Richardson tableaux are characterized combinatorially via evacuation and reading words.

Each Richardson tableau corresponds to a smooth component of a Springer fiber.

The enumeration of Richardson tableaux relates to binomial coefficients and Motzkin numbers.

Abstract

Motivated by the study of Springer fibers and their totally nonnegative counterparts, we define a new subset of standard tableaux called Richardson tableaux. We characterize Richardson tableaux combinatorially using evacuation as well as in terms of a pair of associated reading words. We also characterize Richardson tableaux geometrically, proving that a tableau is Richardson if and only if the corresponding component of a Springer fiber is a Richardson variety, which in turn holds if and only if its positive part is a top-dimensional cell of the totally nonnegative Springer fiber studied by Lusztig (2021). We prove that each such component is smooth by leveraging a combinatorial description of the corresponding pair of reading words, generalizing a result of Graham-Zierau (2011). Another application is that the cohomology classes of these components can be computed in the Schubert basis using Schubert calculus. Finally, we show that the enumeration of Richardson tableaux is surprisingly elegant: the number of Richardson tableaux of fixed partition shape is a product of binomial coefficients, and the number of Richardson tableaux of size $n$ is the $n$th Motzkin number. As a result, we obtain a novel refinement for the Motzkin numbers, as well as a formula for the number of top-dimensional cells in the totally nonnegative Springer fiber.