---
date: 2022-06-01 20:37
modification date: Wednesday 1st June 2022 20:37:05
title: "dominant weight"
aliases: [dominant weight, dominant, dominant integral weight]
---

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

---

- Tags: 
	- #lie-theory 
- Refs:
	- #todo/add-references
- Links:
	- [fundamental weights](fundamental%20weights)
	- [weight lattice](weight%20lattice)

---

# dominant weight
The **weight space** of a representation $V$ of $\lieg$ with weight $\lambda$ is the subspace $V_{\lambda}$ given by
$$
V_{\lambda}:=\{v \in V\st \forall H \in \mathfrak{h}, \quad H \cdot v=\lambda(H) v\} .
$$
where $\lambda: \lieh\dual \to \CC$.

A **weight** is a $\lambda$ as above such that $V_\lambda \neq 0$; elements of $V_\lambda$ are **weight vectors**, which are simultaneous eigenvectors for the action $\lieh\actson V$.

If $V$ is the [adjoint representation](Unsorted/adjoint%20representation.md) of $\lieg$, then the weights are **roots** and form a [root system](Unsorted/root%20system.md).

A weight is **integral** iff
$$
\left\langle\lambda, H_{\alpha}\right\rangle=2 \frac{\langle\lambda, \alpha\rangle}{\langle\alpha, \alpha\rangle} \in \mathbf{Z}
$$
for every coroot $H_{\alpha}=2 \frac{\alpha}{\langle\alpha, \alpha\rangle}$ associated to a root $\alpha$.

Recall that a *positive root system* is a subset $\Phi^+ \subseteq \Phi$ which is closed under sums such that for $\alpha\in \Phi$, exactly one of $\pm\alpha \in \Phi^+$.

An integral weight is a **dominant weight** iff $\ip{\lambda}{\gamma} \geq 0$ for every [[positive root]] $\gamma \in \Phi^+$.

#why-care Dominant integral weights parametrize irreducible finite dimensional representations.

# From Humphreys


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]] 
$$\varpi_{1}, \ldots, \varpi_{\ell} \quad \text{ satisfying}\quad \left\langle\varpi_{i}, \alpha_{j}^{\vee}\right\rangle=\delta_{i j}
.$$ 
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** 
$$
\rho \da \varpi_{1}+\cdots+\varpi_{\ell} \in \Lambda^{+}
= {1\over 2}\sum_{\alpha\in \Phi^+} \alpha
,$$i.e. the sum of [fundamental weights](Unsorted/fundamental%20weights.md) or the half-sum of positive roots. 
It satisfies 
$$
\left\langle\rho, \alpha^{\vee}\right\rangle=1 \qquad\text{ and }\qquad s_{\alpha} \rho=\rho-\alpha \qquad \forall \alpha\in \Delta
.$$
It is the smallest [regular dominant weight](regular%20weight) fixed by no nontrivial element of $W$, and the associated line bundle on the [flag variety](Unsorted/flag%20variety.md) $G/B$ is [ample](Unsorted/ample.md), and is in fact a square root of the [canonical bundle](Unsorted/canonical%20bundle.md).

# Examples

For $\liesl_3(\CC)$:

![](attachments/Pasted%20image%2020220601204853.png)