Richardson tableaux and components of Springer fibers equal to Richardson varieties

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.