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algebras Definition: For an exceptional complex simple Lie algebra $\mathfrak g$ CITE{WikiExceptional},- let $V_{\mathfrak g}(\lambda)$ be the irreducible representation of $\mathfrak g$- with highest weight $\lambda$; when $\mathfrak g$ is fixed, $V(\lambda)$ means $V_{\mathfrak- g}(\lambda)$. This table stores $\dim V_{\mathfrak g}(\lambda)$ for nonzero dominant- integral weights $\lambda$.+ $V_{\mathfrak g}(\lambda)$ is the irreducible representation of highest weight $\lambda$.+ This table stores $\dim V_{\mathfrak g}(\lambda)$ for nonzero dominant integral+ weights $\lambda$. Parameters: algebra:
bourbaki-numbering: Weights are written as Dynkin labels in Bourbaki node numbering, the numbering used by Sage's root-system data CITE{SageRootSystem}. For example,- in this convention $E_8$ has $\dim V(1,0,0,0,0,0,0,0)=3875$ and $\dim V(0,0,0,0,0,0,0,1)=248$.+ in this convention the $E_8$ row for $(1,0,0,0,0,0,0,0)$ has dimension $3875$,+ while $(0,0,0,0,0,0,0,1)$ has dimension $248$. trivial-representation: The zero highest weight, whose representation has dimension $1$, is omitted.
classical-types: The classical simple Lie algebras are not included. Formulas:- formula-weyl-dimension: Weyl's dimension formula is $\dim V(\lambda)=\prod_{\alpha\in\Phi^+}+ formula-weyl-dimension: Weyl's dimension formula is $\dim V_{\mathfrak g}(\lambda)=\prod_{\alpha\in\Phi^+} \frac{\langle\lambda+\rho,\alpha^\vee\rangle} {\langle\rho,\alpha^\vee\rangle}$, where $\Phi^+$ is the set of positive roots and $\rho$ is the half-sum of the
rigour: exact complete: 'no'- complete-note: it holds every nontrivial irreducible representation with $\dim V(\lambda)\leq10^6$- for $G_2$, $\dim V(\lambda)\leq10^7$ for $F_4$ and $E_6$, $\dim V(\lambda)\leq10^8$- for $E_7$, and $\dim V(\lambda)\leq10^{12}$ for $E_8$+ complete-note: it holds every nontrivial irreducible representation with $\dim V_{\mathfrak+ g}(\lambda)\leq10^6$ for $G_2$, $\dim V_{\mathfrak g}(\lambda)\leq10^7$ for $F_4$+ and $E_6$, $\dim V_{\mathfrak g}(\lambda)\leq10^8$ for $E_7$, and $\dim V_{\mathfrak+ g}(\lambda)\leq10^{12}$ for $E_8$ rigour details: 'All entries are exact integers computed from Weyl''s dimension formula CITE{formula-weyl-dimension}. The generator reads the positive roots and
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