Affine twisted length function

TL;DR

This paper extends Shi's characterization of affine Weyl group elements by providing a formula for the twisted affine length function using Shi vectors, enhancing understanding of affine Weyl group structures.

Contribution

It introduces a new formula for the twisted affine length function expressed through Shi coefficients, building on Shi's original characterization.

Findings

Derived a formula for the twisted affine length function

Expressed the length function in terms of Shi vectors

Enhanced understanding of affine Weyl group element properties

Abstract

Let $W_a$ be an affine Weyl group. In 1987 Jian Yi Shi gave a characterization of the elements $w \in W_a$ in terms of $\Phi^+$-tuples $(k(w,\alpha))_{\alpha \in \Phi^+}$ called the Shi vectors. Using these coefficients, a formula is provided to compute the standard length of $W_a$. In this note we express the twisted affine length function of $W_a$ in terms of the Shi coefficients.