Affine Weyl group

Okay, so imagine you have a group of friends and you like to play dress-up. One friend might wear a hat, another might wear a cape, and so on. But sometimes you run out of costumes, so you have to start using your imagination and get creative.

In math, there's a similar idea where you have a group of transformations that work together to transform geometric shapes. This is called an affine Weyl group. The transformations can be things like translations (moving something up, down, left, or right) and reflections (flipping something over a line).

But sometimes you want to work with more complicated shapes that don't fit neatly into traditional transformations. That's where the creativity comes in. You can start combining transformations in different ways, like doing a translation followed by a reflection, or a reflection followed by a rotation. This is kind of like your friends using their imagination to create new costumes by combining different pieces of clothing they already have.

The affine Weyl group lets you do all of these transformations and combinations in a structured way, so you can still keep track of what you're doing and know how it's going to change your geometric shapes. It's kind of like having a dress-up box with different compartments for hats, capes, shoes, and so on. You can mix and match them in different ways, but you still know where everything is and what it's supposed to do.