A tightness property of relatively smooth permutations

TL;DR

This paper explores a combinatorial property related to the smoothness of Schubert varieties in type A, linking geometric smoothness conditions with permutation properties and their implications for Kazhdan--Lusztig polynomial identities.

Contribution

It introduces a new combinatorial consequence of the smooth locus containment condition for Schubert varieties, connecting geometric and algebraic aspects in type A.

Findings

Identifies a combinatorial criterion for smooth locus containment

Links geometric properties of Schubert varieties to permutation characteristics

Supports interpretation of Kazhdan--Lusztig polynomial identities

Abstract

It is well known that many geometric properties of Schubert varieties of type $A$ can be interpreted combinatorially. Given two permutations $w,x\in S_n$ we give a combinatorial consequence of the property that the smooth locus of the Schubert variety $X_w$ contains the Schubert cell $Y_x$. This is a necessary ingredient for the interpretation of recent representation-theoretic results of the author with M\'inguez in terms of identities of Kazhdan--Lusztig polynomials.