Nilpotency indices for quantum Schubert cell algebras

TL;DR

This paper investigates the nilpotency indices of quantum adjoint actions in quantum Schubert cell algebras, providing a classification and explicit calculations for these indices based on combinatorial and algebraic structures.

Contribution

It introduces a new framework for understanding nilpotency indices in quantum Schubert cell algebras using Bruhat order and element classification, with explicit computations.

Findings

Nilpotency indices are classified via an equivalence relation on triples involving Weyl group elements.

Each equivalence class contains a representative with specific algebraic properties.

Explicit nilpotency indices are computed for key representatives.

Abstract

We study quantum analogs of $\operatorname{ad}$-nilpotency and Engel identities in quantum Schubert cell algebras ${\mathcal U}_q^+[w]$. For each pair of Lusztig root vectors, $X_\mu$ and $X_\lambda$, in ${\mathcal U}_q^+[w]$, where $w$ belongs to a finite Weyl group $W$ and $\mu$ precedes $\lambda$ with respect to a convex order on the roots in $\Delta_+ \cap w(\Delta_-)$, we find the smallest natural number $k$, called the nilpotency index, so that $\left(\operatorname{ad}_q X_\mu \right)^k$ sends $X_\lambda$ to $0$, where $\operatorname{ad}_q X_\mu$ is the $q$-adjoint map. We start by observing that every pair of Lusztig root vectors can be naturally associated to a triple $(u, i, j)$, where $u \in W$ and $i$ and $j$ are indices such that $\ell(s_i u s_j) = \ell(u) - 2$. In light of this, we define an equivalence relation, based upon the weak left and weak right Bruhat orders, on the set of such triples. We show this equivalence relation respects nilpotency indices, and that each equivalence class contains an element of the form $(v, r, s)$, where $v$ is either (1) a bigrassmannian element satisfying a certain orthogonality condition, or (2) the longest element of some subgroup of $W$ generated by two simple reflections. For each such $(v, r, s)$, we compute the associated nilpotency index.