fundamental weights

Last modified date: <%+ tp.file.last_modified_date() %>

fundamental weights

The fundamental weights \(\omega_{1}, \ldots, \omega_{n}\) are defined by the property that they form a basis of \(\mathfrak{h}_{0}\) dual to the set of coroots associated to the simple roots. That is, the fundamental weights are defined by the condition \begin{align*} 2 \frac{\left\langle\omega_{i}, \alpha_{j}\right\rangle}{\left\langle\alpha_{j}, \alpha_{j}\right\rangle}=\delta_{i, j} \end{align*} where \(\alpha_{1}, \ldots \alpha_{n}\) are the simple roots.

An element \(\lambda\) is then algebraically integral if and only if it is an integral combination of the fundamental weights.

The set of all \(\mathfrak{g}\)-integral weights is a lattice in \(\mathfrak{h}_{0}\) called the weight lattice for \(\mathfrak{g}\), denoted by \(P(\mathfrak{g})\).

🗓️ Timeline
  • 2022-05-25
    There is an isomorphism ${\mathbf{C}} { \left[ \scriptstyle {\Lambda {}^{ \vee }} \right] } ^{W_\lambda} \cong {\mathbf{C}} { \left[ \scriptstyle {\chi_1, \cdots, \chi)n} \right] } $ where \(\chi_i = \chi(\mkern 1.5mu\overline{\mkern-1.5mu\omega\mkern-1.5mu}\mkern 1.5mu_i)\) are the characters of the fundamental weights. This yields \(T_{\Lambda {}^{ \vee }}/W_{\lambda} \cong {\mathbf{A}}^n\) with coordinates \(\chi_i\).
Links to this page
  • dominant weight

    Write \(\Delta=\left\{\alpha_{1}, \ldots, \alpha_{\ell}\right\}\) for a simple system Then \(\Lambda\) is a free abelian group of rank \(\ell\), with a basis consisting of fundamental weights \begin{align*}\varpi_{1}, \ldots, \varpi_{\ell} \quad \text{ satisfying}\quad \left\langle\varpi_{i}, \alpha_{j}^{\vee}\right\rangle=\delta_{i j} .\end{align*} The subset \(\Lambda^{+}:=\mathbb{Z}^{+} \varpi_{1}+\cdots+\mathbb{Z}^{+} \varpi_{\ell}\) is called the set of dominant integral weights. From the fact that \(\left\langle\beta, \alpha^{\vee}\right\rangle=\beta\left(h_{\alpha}\right)\) when \(\beta \in \Phi\), one shows easily that \(\left\langle\lambda, \alpha^{\vee}\right\rangle=\lambda\left(h_{\alpha}\right)\) for all \(\lambda \in \Lambda\).

    There is a special weight \begin{align*} \rho \coloneqq\varpi_{1}+\cdots+\varpi_{\ell} \in \Lambda^{+} = {1\over 2}\sum_{\alpha\in \Phi^+} \alpha ,\end{align*} i.e. the sum of fundamental weights or the half-sum of positive roots. It satisfies \begin{align*} \left\langle\rho, \alpha^{\vee}\right\rangle=1 \qquad\text{ and }\qquad s_{\alpha} \rho=\rho-\alpha \qquad \forall \alpha\in \Delta .\end{align*} It is the smallest regular dominant weight fixed by no nontrivial element of \(W\), and the associated line bundle on the flag variety \(G/B\) is ample, and is in fact a square root of the canonical bundle.

  • Lie algebra

    - Tags: - #lie-theory - Refs: - #todo/add-references - Links: - Lie group - Coxeter number - adjoint representation - Weight/root theory: - Unsorted/highest weight - dominant weight - fundamental weights - weight lattice - root

  • AG_Review_Q

    fundamental weights 34 1 1969-12-19

#lie-theory #todo/add-references