Sign-twisted generating functions of the odd length for Weyl groups of type $D$

TL;DR

This work establishes explicit closed product formulas for the sign-twisted generating functions of the odd length $L(w)$ over parabolic quotients of the Weyl groups of type $D$. The authors develop a cohesive toolkit—shifting, compressing, chessboard-element analysis, and odd-sandwich structures—to reduce general parabolic subsets $I$ to canonical compressed shapes and derive exact product expressions for $\mathcal{D}_n^I(x)$. They verify Brenti–Carnevale conjectures on evaluating these generating functions and resolve Stembridge’s cyclotomic-polynomial question by giving a necessary and sufficient condition for $\mathcal{D}_n^I(x)$ to be a product of cyclotomic polynomials, including a precise non-cyclotomic factor in the exceptional cases. The results connect type $D$ sign-twisted generating functions to explicit product structures, generalizing known type $A$ and $B$ cases and providing concrete combinatorial descriptions with potential applications in representation zeta functions and geometry of related varieties.

Abstract

The odd length in Weyl groups is a new statistic analogous to the classical Coxeter length, and features combinatorial and parity conditions. We establish explicit closed product formulas for the sign-twisted generating functions of the odd length for parabolic quotients of Weyl groups of type $D$. As a consequence, we verify three conjectures of Brenti and Carnevale on evaluating closed forms for these generating functions. We then give an equivalent condition for the sign-twisted generating functions to be expressible as products of cyclotomic polynomials, settling a conjecture of Stembridge.

Theorems & Definitions (71)

• Theorem 1.1: BC17D, Theorem 2.8
• Theorem 1.2: BC17D, Theorem 2.10
• Conjecture 1.3: BC17D, Conjectures 6.1 and 6.2
• Conjecture 1.4: BC17D, Conjecture 6.5
• Conjecture 1.5: Ste19, Conjecture 7.4(a)
• Lemma 2.1
• proof
• Lemma 3.1: BC17D, Proposition 3.4
• Lemma 3.2
• proof