On the $\mu$-admissible set in the extended affine Weyl groups of $E_6$ and $E_7$

TL;DR

This paper provides explicit examples demonstrating that in the cases of $E_6$ and $E_7$, the $5$-admissible set and the $5$-permissible set differ for non-trivial minuscule coweights, highlighting a nuanced distinction in extended affine Weyl groups.

Contribution

It offers the first explicit examples showing the discrepancy between $5$-admissible and $5$-permissible sets in the context of $E_6$ and $E_7$.

Findings

$5$-admissible and $5$-permissible sets do not always coincide.

Explicit examples are provided for $E_6$ and $E_7$ cases.

The discrepancy occurs for non-trivial minuscule coweights.

Abstract

Giving explicit examples we show that the $\mu$-admissible set and the $\mu$-permissible set do not agree for non-trivial minuscule coweights in the cases of $E_6$ and $E_7$.