Cauchon diagrams for quantized enveloping algebras

TL;DR

This paper provides an algorithmic method to describe Cauchon diagrams associated with quantized enveloping algebras, linking combinatorial objects to algebraic prime ideals, and confirms their count matches the Weyl group size.

Contribution

It introduces an explicit algorithmic description of Cauchon diagrams for good orderings of positive roots in quantized enveloping algebras, connecting combinatorics with algebraic structures.

Findings

Number of Cauchon diagrams equals the Weyl group order.

Algorithmic description based on Lusztig's admissible planes.

Explicit Cauchon diagrams for each type and reduced decomposition.

Abstract

Let $\mathfrak{g}$ be a finite dimensional complex simple Lie algebra, $\mathbb{K}$ a commutative field and $q$ a nonzero element of $\mathbb{K}$ which is not a root of unity. To each reduced decomposition of the longest element $w_0$ of the Weyl group $W$ corresponds a PBW basis of the quantised enveloping algebra $\mathcal{U}_q^+(\mathfrak{g})$, and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of $w_0$, this theory constructs a bijection between the set of prime ideals in $\mathcal{U}_q^+(\mathfrak{g})$ that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of $w_0$ corresponds to a good ordering (in the sense of Lusztig \cite{Lu2}) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes \cite{Lu2}: each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitely describe the set of Cauchon diagrams for explicit reduced decomposition of $w_0$ in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of $W$. In \cite{CM}, we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of $w_0$. So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of $w_0$ corresponding to a good ordering.