Weyl dimensions, Casimirs, and global descent

For a dominant integral highest weight $\lambda$ of a complex simple Lie algebra, Weyl’s formula gives

$$\boxed{\displaystyle \dim V_\lambda =\prod_{\beta^\vee\in\Phi^\vee_+} \frac{\langle\lambda+\rho,\beta^\vee\rangle} {\langle\rho,\beta^\vee\rangle}}$$

Here $\rho$ is half the sum of the positive roots, and the product runs over positive coroots. The same $V_\lambda$ integrates to a continuous complex representation of the compact simply connected group. Descent to another global form is an additional question. Etingof, §28.1, Proposition 28.1; Conrad–Landesman, Lectures 27–28.

Open the highest-weight workbench · Worked examples

Inputs and mathematical category

Write

$$\lambda=\sum_{i=1}^{r}a_i\omega_i, \qquad \langle\omega_i,\alpha_j^\vee\rangle=\delta_{ij}, \qquad a_i\in\mathbb Z_{\geq0}.$$

The input is the ordered Dynkin-label vector $(a_1,\ldots,a_r)$. It does not denote a dimension, a root-coordinate vector, or a choice of real form. The workbench accepts $A_1$–$A_8$, $B_2$–$B_8$, $C_2$–$C_8$, $D_4$–$D_8$, $E_6,E_7,E_8,F_4,G_2$, with each label at most $30$. These are computational bounds; the mathematical formula has no such cutoff.

The supported category is finite-dimensional complex representations in characteristic zero. The compact real group supplies a unitary realization. Infinite-dimensional unitary representations of a noncompact real form require different classification data.

Why coroot coefficients matter

For a positive coroot

$$\beta^\vee=\sum_i m_i\alpha_i^\vee,$$

the corresponding Weyl factor is

$$\frac{\sum_i m_i(a_i+1)}{\sum_i m_i},$$

because $\rho=\sum_i\omega_i$. This turns the calculation into a product of rational numbers with integer numerators and denominators. The result is an integer, although individual factors usually are not.

For non-simply-laced systems, replacing the $m_i$ by coefficients of positive roots changes the calculation. A useful diagnostic is the defining representation: the first fundamental weights of $B_2$ and $C_2$ have dimensions $5$ and $4$, respectively. Their common Lie algebra isomorphism exchanges the conventional node labels. The different root lengths are explicit in the type-B and type-C realizations.

The implementation generates the coroot system by simple reflections, checks its cardinality, and evaluates the product using exact integers and reduced fractions. Every output includes the individual factors, so a large dimension can be inspected without constructing a matrix representation.

Cartan matrix and node numbering

The convention profile is lie-bourbaki-long-root-2-v1. Its matrix has coroots on rows:

$$A_{ij}=\langle\alpha_j,\alpha_i^\vee\rangle =\frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}.$$

Long roots have squared length $2$. The node order agrees with the Bourbaki realizations documented by SageMath. The workbench shows the actual matrix and the squared length of every simple root; these data remove any ambiguity in an arrow drawn on a Dynkin diagram. SageMath, Cartan matrices.

Type	Numbering used here
$A_r$	The chain is $1,\ldots,r$; node $1$ is the defining representation of $SU(r+1)$.
$B_r$	The chain ends at the short root $r$; weight $\omega_r$ is the spinor weight.
$C_r$	The chain ends at the long root $r$; $\omega_1$ gives the $2r$-dimensional representation of compact $Sp(r)$.
$D_r$	The chain ends at $r-2$, with branches to $r-1$ and $r$; the two branch ends are half-spin weights.
$E_6,E_7,E_8$	The horizontal chain is $1,3,4,5,6$ and continues to $7,8$ when present; node $2$ attaches to node $4$.
$F_4$	Nodes $1,2$ are long and nodes $3,4$ short.
$G_2$	Node $1$ is short and node $2$ long.

The exceptional numbering is specified in the official type-E, type-F, and type-G documentation.

Quadratic Casimir and trace index

The displayed normalization is

$$C_2(\lambda)=\frac{(\lambda,\lambda+2\rho)}{2}, \qquad T(V_\lambda)= \frac{\dim V_\lambda}{\dim\mathfrak g}\,C_2(\lambda).$$

Thus $C_2(\mathrm{adj})=h^\vee$, and the defining representation of $SU(N)$ has $T=\tfrac12$. These choices must accompany the numbers. In the normalization without the prefactor $\tfrac12$, the Casimir eigenvalue is $(\lambda,\lambda+2\rho)$; Etingof derives that version in §25.4. A different normalization of the generators rescales both quadratic quantities.

For an explicit calculation, set $D_{ii}=(\alpha_i,\alpha_i)/2$. In the row-coroot convention above,

$$(\omega_i,\omega_j)=(D A^{-1})_{ij}.$$

The matrix is symmetric even when $A$ is not. This is a practical check on a root-length convention.

For $SU(3)$, $\lambda=(p,q)$ gives

$$\dim(p,q)=\frac{(p+1)(q+1)(p+q+2)}{2}, \qquad C_2(p,q)=\frac{p^2+pq+q^2+3p+3q}{3}.$$

At $(p,q)=(1,1)$ this gives $\dim=8$, $C_2=3$, and $T=3$. At $(1,0)$ it gives $\dim=3$, $C_2=\tfrac43$, and $T=\tfrac12$. These follow by substituting the $A_2$ weight metric into the formulas above.

Center action and the adjoint global form

Let $\widetilde G$ be compact and simply connected, with simple Lie algebra. Its center is dual to the finite abelian group $P/Q$, where $P$ and $Q$ are the weight and root lattices. The class $[\lambda]\in P/Q$ is the central character of $V_\lambda$. Conrad, Centers of Simply Connected Semisimple Groups, pp. 1–4.

In our convention the simple roots are the columns of $A$ in the fundamental-weight basis. Therefore

$$\lambda\in Q \quad\Longleftrightarrow\quad A^{-1}a\in\mathbb Z^r \quad\Longleftrightarrow\quad V_\lambda\text{ descends to }\widetilde G/Z(\widetilde G).$$

The order of $[\lambda]$ is the least common multiple of the denominators of $A^{-1}a$. Dividing the order of the center by this character order gives the order of its kernel. These quantities are exact consequences of the lattice calculation.

“Descends to the adjoint global form” tests the quotient by the entire center. It does not test every intermediate quotient. For example, the vector of $\operatorname{Spin}(10)$ descends to $SO(10)$ but not to the adjoint group $PSO(10)$. Its central character has order $2$ inside a center of order $4$. The particular quotient subgroup must still be identified.

For type $A_{N-1}$ the charge of the scalar central generator is

$$t(\lambda)=\sum_{i=1}^{N-1}i\,a_i\pmod N.$$

The matter and global forms tool applies this criterion to a collection of $SU(N)$ matter representations.

Conjugation and reality type

The highest weight of the dual is $-w_0\lambda$. The workbench reports its Dynkin labels before assigning a reality type. Non-self-dual irreps have complex type. For a self-dual irrep, the Frobenius–Schur indicator is

$$\nu(V_\lambda) =\exp\!\bigl(2\pi i\langle\lambda,\rho^\vee\rangle\bigr) =(-1)^{\langle\lambda,\,2\rho^\vee\rangle}, \qquad 2\rho^\vee=\sum_{\beta^\vee>0}\beta^\vee.$$

The signs $+1$ and $-1$ indicate real and quaternionic type, respectively. This is the finite-dimensional statement of Adams, Lemma 5.3, applied to the complexification and its compact form.

Reality and descent answer different questions. The $SU(3)$ irrep $(3,0)$ has dimension $10$ and zero triality, so it descends to $PSU(3)$; its dual $(0,3)$ is different, so its type is complex.

Verification and interpretation

The scientific checks include the known classical fundamental dimensions, exceptional fundamental and adjoint dimensions, the identity

$$\dim V_{k\rho}=(k+1)^{|\Phi_+|},$$

and the closed $A_1,A_2$ formulas. For $E_8$, $\dim V_\rho=2^{120}$ is an exact integer beyond ordinary floating-point precision; the same example appears in the SageMath Weyl-character tutorial.

A dimension, index, or reality type is representation data. An anomaly, a spectrum of physical states, or an allowed Lagrangian term additionally depends on the spacetime theory and field content. Save the labels, global group, convention profile, and calculation export with any research use. Source checking and automated verification are recorded separately from the site’s pending independent expert review.