English

This interactive tool is a companion application for the article Weyl Groups and the Modified Kostant Game. It lets you explore the explicit bijection between the modified Kostant game on type $A_n$ and Standard Young Tableaux directly in your browser, with no installation required.

The game is played on an $A_n$ Dynkin diagram where a chosen modified vertex $k$ is connected to a special "always-happy" source node permanently fixed at 1. Nodes evaluate their state based on their neighbors and can be Happy (light green), Excited (dark green), or Sad (blue). At each step, a Sad node can be reflected, updating its chip count according to the formula $c_i \to \sum c_j - c_i$.

Simultaneously, the sequence of valid reflections automatically populates a Standard Young Tableau of rectangular shape $k \times (n-k+1)$ following the filling rule $i - k = c - r$. The game ends when the tableau is complete and no sad nodes remain.

Features: define the size of the graph ($n$) and the modified vertex ($k$), step through the reflections manually to study specific paths, or use the Auto-Play feature with adjustable speed to watch the Young Tableau construct itself in real-time.

You are invited to experiment with different dimensions, help the sad nodes, and watch the bijection unfold!