Christopher Koch *Butler University*

**Faculty Sponsor(s): **Amber Russell *Butler University*

A Young Diagram is an arrangement of boxes. If there are n boxes in the Diagram, then you can fill those boxes with n numbers. This is called a Young Tableau. Utilizing two types of Young Tableau labelings, we can create a permutation, which tracks how entries change between the labelings. A permutation is a rearrangement of the numbers of the Young Tableau. We call these two forms of labelings the Standard and Upright Tableaux. The Standard Tableau increases consecutively going across the right and down, whereas the Upright Tableau increases consecutively going up and to the right.

The Young Tableaux have applications in Lie Theory and the main goal of this project is to explore this connection. In particular, we have related the permutations found from the Tableaux to the longest element of a Lie theoretic Weyl Group. This is a report of work begun at the Math Research Camp at Butler University.

Mathematics & Computer Science