3rd Year Candidacy Requirement: All but One: On the Complexity of Matrix Schubert Varieties

Abstract: T-varieties are affine varieties equipped with an action of a torus T. In particular, normal T-varieties are toric when an orbit of the torus action is dense. The complexity of a normal T-variety is the difference in the dimension of the T-variety and the dimension of T. In this talk, we will focus on the complexity of a specific class of T-varieties called matrix Schubert varieties. 

Introduced by Fulton, matrix Schubert varieties consist of square matrices satisfying certain constraints on the ranks of their submatrices. Every permutation w on n elements gives rise to a matrix Schubert variety $\overline{X_w}$. The cross product of n-by-n diagonal matrices $T \times T$ with nonzero complex entries is a torus that acts on $\overline{X_w}$ by $D \overline{X_w} D^{-1}$.

This talk will cover important results on the complexity of matrix Schubert varieties by Escobar-Mészáros, Donten-Bury-Escobar-Portakal, and Stelzer. Finally, we will show that for a fixed n, the complexity of a matrix Schubert variety can be between 0 and (n-1)(n-3), but not 1.

Advisors: Laura Escobar Vega and John Shareshian