Simple Lie Algebra Roots & Weights

Given a simple Lie algebra of type $X_N$ and a highest weight $\lambda$, this demo produces the Cartan matrix, Dynkin diagram, and root information in terms of fundamental weights for this algebra. In addition, we find all of the weights of the irreducible $X_N$-module with highest weight $\lambda$. When working in ranks 1, 2, or 3, we also provide a plot of the weights of our irreducible module. The small grey dots in these plots represent part of the fundamental weight lattice (i.e., integral linear combinations of fundamental weights).

Notes:

• The default choice for highest weight is the highest long root of $X_N$ (i.e., the highest weight for the adjoint representation).
• Our highest weight is inputted as a list: $[k_1,k_2,\dots,k_N]$. If "in terms of weights" is selected, this is $\lambda = k_1\lambda_1+\cdots+k_N\lambda_N$ where $\lambda_1,\dots,\lambda_N$ denote the fundamental weights. If "in terms of roots" is selected, this is $\lambda = k_1\alpha_1+\cdots+k_N\alpha_N$ where $\alpha_1,\dots,\alpha_N$ denote the simple roots.

  Note to self: Finish commenting code. Clean up output (e.g., line breaks).
  
  

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