Computations of $\tilde{A}_2$ Bruhat intervals via shadows

Abstract

We develop an explicit geometric and algebraic description of shadows in the affine Coxeter complex of type $\tilde{A}_2$, by introducing a coordinate system based on a decomposition of the complex into tunnels and channels. We also provide an algorithm for converting arbitrary reduced words into coordinates. Using this framework, we identify geometric symmetries of shadows and show that they are governed by the underlying channel structure. This allows us to derive explicit, piecewise formulas for the cardinality of shadows in $\tilde{A}_2$, depending on the parity of the coordinates. Furthermore, we establish a simple criterion for shadow membership via a counting function that detects admissible positions within channels. These results provide a concrete and computationally effective description of shadows in affine type $\tilde{A}_2$, bridging the gap between combinatorial definitions and geometric realisations.