Complexity of the Zero Set of a Matrix Schubert Ideal

$T$-varieties are normal varieties equipped with an action of an algebraic torus $T$. When the action is effective, the complexity of a $T$-variety $X$ is $\dim (X)-\dim (T)$. Matrix Schubert varieties, introduced by Fulton in 1992, are $T$-varieties consisting of $n\times n$ matrices satisfying certain constraints on the ranks of their submatrices. In this paper, we focus on the complexity of certain torus-fixed affine subvarieties of matrix Schubert varieties. Concretely, given a matrix Schubert variety $\overline{X_w}$ where $w\in S_n$, we study the complexity of $Y_w$ obtained by the decomposition $\overline{X_w}=Y_w\times {\mathbb{C}}^k$ with $k$ as large as possible. Building up from results by Escobar and Mészáros and Donten-Bury, Escobar, and Portakal, we show that for a fixed $n$, the complexity of $Y_w$ with respect to this action can be any integer between $0$ and $(n-1)(n-3)$, except $1$.