Proper permutations, Schubert geometry, and randomness (Q2166393)

The authors consider \textit{proper permutations} \(w \in S_n\), i.e., ones which satisfy \(\ell(w) - \binom{d(w) + 1}{2} \leq n\), where \(\ell(w)\) is the number of inversions and \(d(w)\) is the number of left descents of~\(w\). One of the main results of the article is that the probability that a random permutation \(w \in S_n\) is proper goes to zero in the limit when \(n \rightarrow \infty\). A very important aspect of this result is its relation to geometry: properness of \(w\) is related to the Schubert variety \(X_w\) being spherical. We say that \(X_w\) is spherical if it has a dense orbit of a Borel subgroup of some \(L_I\), a group of invertible block diagonal matrices, where blocks are determined by a set \(I\) of left descents of~\(w\). The authors conclude that the probability that for a random permutation \(w \in S_n\) the Schubert variety \(X_w\) is spherical goes to zero in the limit when \(n \rightarrow \infty\). Finally, the authors consider the notion of \(w \in S_n\) being \(I\)-spherical, introduced by \textit{R. Hodges} and \textit{A. Yong} in [J. Lie Theory, 32(2), 447--474 (2022; Zbl 1486.14070)], and show that the probability of \(w\) being \(I\)-spherical goes to zero in the limit when \(n \rightarrow \infty\). This result settles a conjecture from the article cited above.