---- > [!justification] Preliminaries. > Let $\mathfrak{g}$ be a complex [[semisimple Lie algebra|semisimple]] [[Lie algebra]] with the usual semisimple data: [[Cartan subalgebra|CSA]] $\mathfrak{t}$, [[root system]] $\Phi$, [[root basis]] $\Delta$, [[Weyl group of a root system|Weyl group]] $W$. Let $X \subset \mathfrak{t}^{*}$ be the [[root and weight lattice of a root system|weight lattice]] of $\Phi$. > > Let $V$ be a finite-dimensional [[irreducible Lie algebra representation|irreducible]] [[Lie algebra representation|representation]] of $\mathfrak{g}$. By the [[classification of complex semisimple Lie algebras]], we know $V=V(\lambda)$ for some [[dominant weight in a root system|dominant]] $\lambda \in X$, where $V(\lambda)$ is the [[highest weight module|unique]] irreducible [[quotient representation|quotient]] of the [[Verma module]] $M(\lambda)$. > > Denote by $\Pi(\lambda)$ [[on factorizing a Lie algebra representation into weight spaces|the multiset of]] [[on the weights of a representation|weights]] of $V(\lambda)$, i.e., $\Pi(\lambda)=\{ \mu \in \mathfrak{t}^{*}: V(\lambda)_{\mu} \neq 0 \}$ > where $\mu$ is counted with multiplicity $\text{dim }V(\lambda)_{\mu}$. [[reading off the weights (with some information on multiplicity) of an irreducible semisimple Lie algebra representation|We know]] *which* elements $\mathfrak{t}^{*}$ belong to $V(\lambda)$: they are precisely the dominant elements $\mu$ of $X$ satisfying $\mu \leq \lambda$ along with the [[Weyl group of a root system|Weyl conjugates]] $w(\mu)$, $w\in W$. [[reading off the weights (with some information on multiplicity) of an irreducible semisimple Lie algebra representation|We also know a bit about the multiplicities]] $\text{dim }V(\lambda)_{\mu}$, for example that $\text{mult}(\lambda)=1$, $\text{mult}\big( w(\mu) \big)=\text{mult}(\mu)$, and that $\text{mult}(\mu)$ is at most the number of distinct positive root paths[^2] from $\mu$ up to $\lambda$. > > > The **Weyl character formula** is a magic formula for computing the [[character of a Lie algebra representation|character]] of $V(\lambda)$, and hence the exact information of weights and multiplicities. To state it, we first need two preparations: > > - Recall the **Weyl vector**[^3] $\rho=\frac{1}{2} \sum_{\alpha \in \Phi^{+}} \alpha \in \mathfrak{t}^{*}.$ [[the Weyl group and root bases|In the proof that]] $W$ [[group action|acts]] [[simply transitive group action|simply transitively]] on $\Delta$ and is [[subgroup generated by an element of a group|generated by]] [[reflection|reflections]] of [[root basis|simple roots]], we showed $\omega_{\alpha_{i}}(\rho)=\rho-\alpha_{i}$ for every [[root basis|simple root]] $\alpha_{i} \in \Delta$, i.e., $\rho-\alpha_{i}=\rho-\langle \rho, \check \alpha_{i} \rangle \alpha_{i}$. So $\langle \rho, \check \alpha_{i} \rangle=1$ for all $i$, meaning $\rho=\omega_{1} + \dots + \omega_{n},$ > the sum of the [[root and weight lattice of a root system|fundamental weights]] $\omega_{i} \in X$. > > - Recall from [[group|group theory]] [[symmetric group|the]] [[alternating group|sign homomorphism]] $S_{n} \to \{ \pm 1 \}$. This generalizes[^4] to any Weyl group $W$: let $w \in W$, and write $W=w_{1} \cdots w_{n}$ for $w_{i}$ simple [[reflection|reflections]]. Then set $\text{sgn}(w):=(-1)^{n}.$ > > [!theorem] Theorem. ([[Weyl character formula]]) > If $\lambda \in X$ is [[dominant weight in a root system|dominant]], we have the following equality in the ([[field of fractions|field of fractions of the]]) [[character ring of a root system|character ring]] $\mathbb{Z}[X]$: $\text{ch}\big( V(\lambda) \big)=\frac{\sum_{w \in W}\text{sgn}(w) e^{w(\lambda + \rho)}}{e^{\rho} \prod_{\alpha \in \Phi^{+}} (1-e^{-\alpha})}.$ > > > ^theorem > [!proposition] Corollary 1. (Weyl denominator formula) > Plugging $\lambda=0$ into the Weyl character formula yields the following equality in $\mathbb{Z}[X]$: $\sum_{w \in W} \text{sgn}(w)e^{w(\rho)}=e^{\rho}\prod_{\alpha \in \Phi^{+}}^{}(1- e^{-\alpha}).$ > This is because $V(0)$ is the [[trivial Lie algebra representation|trivial representation]], hence is character is just $1 \in \mathbb{Z}[X]$. So numerator=denominator in the Weyl character formula. ^proposition > [!proposition] Corollary 2. (Weyl dimension formula) (Most useful) > If $\lambda \in X$ is [[dominant weight in a root system|dominant]], then $\text{dim}\big( V(\lambda) \big)=\frac{\prod_{\alpha \in \Phi^{+}}^{} \langle \lambda+\rho, \check \alpha \rangle }{\prod_{\alpha \in \Phi^{+}}^{}\langle \rho, \check \alpha \rangle } =\prod_{\alpha \in \Phi^{+}}^{}\frac{(\lambda+\rho, \alpha)}{(\rho , \alpha)},$ where the second equality follows from the first by expanding the [[reflection|check-pairings]] and performing cancellations. > [!basicexample] Weyl character formula example. > [[special linear Lie subalgebra|Let]] $\mathfrak{g}=\mathfrak{sl}_{2}$, so $\Phi=\{ \pm \alpha \}$, $\Delta=\{ \alpha \}$. $\rho=\frac{\alpha}{2}$ and $W$ [[cyclic group|is]] $C_{2}=\{ \pm 1 \}$. So, given a [[root and weight lattice of a root system|fundamental weight]] $n \frac{\alpha}{2}$ the [[Weyl character formula]] [[classification of the irreps of sl2 over C|is]] (setting $t=e^{\alpha/2}$) $\text{Ch}\big( V( n \frac{\alpha}{2} ) \big)=\frac{t^{n+1}-t^{-(n+1)}}{t-t ^{-1}}=t^{n}+t^{n-2}+\dots+t^{-n}$ as expected. (Perform the manipulations with things like [[finite geometric series]].) > >The Weyl character formula quickly becomes hard to calculate for more involved algebras, mostly because $W$ gets really big. ^basic-example > [!basicexample] Weyl dimension formula example: $\mathfrak{sl}_{3}$ > Let $\mathfrak{g}=\mathfrak{sl}_{3}$, so $\Phi$ is [[classification of complex semisimple Lie algebras|of]] [[classification of irreducible root systems|type]] $A_{2}$: $\Phi=\{ \pm \alpha, \pm \beta, \pm(\alpha+\beta) \}$, $\Delta=\{ \alpha, \beta \}$, $\rho=\omega_{1}+\omega_{2}$. Want to calculate the terms in the [[Weyl character formula|Weyl dimension formula]]; this entails calculating $\langle \lambda+\rho, \check \gamma \rangle$ and $\langle \rho, \check \gamma \rangle$ for each [[root basis|positive root]] $\gamma \in \Phi^{+}$. Of course, $\lambda$ and $\rho$ are both linear combinations of the $\omega_{i}$, and the check-pairing is linear in its first argument. So this reduces to computing $\langle \omega_{i}, \check \gamma \rangle$ for $i=1,2$ and $\gamma=\alpha, \beta, \alpha+\beta$. > > > > > | $\langle \omega_{i}, \check \gamma \rangle$ | $\alpha$ | $\beta$ | $\alpha+\beta$ | > | ------------------------------------------- | -------- | ------- | -------------- | > | $\omega_{1}$ | 1 | 0 | 1 | > | $\omega_{2}$ | 0 | 1 | 1 | > > Where $\langle \omega_{i}, \check{(\alpha + \beta)} \rangle$ can be computed either by drawing the $A_{2}$ picture or by noting that because $A_{2}$ is [[simply laced root system|simply laced]], the check-pairing is also linear in its second argument: $(\check{\alpha + \beta}) = \check \alpha+ \check \beta$. With $\lambda=n_{1} \omega _1+ n_{2}\omega_{2}$, then, we see $\text{dim }V(\lambda)=\text{dim }V(n_{1} \omega_{1}+n_{2} \omega_{2})= \frac{(n_{1}+1)(n_{2}+1)(n_{1}+n_{2}+2)}{2}.$ > It follows that $V(\omega_{1})$ and $V(\omega_{2})$ are the two smallest nontrivial representations of $\mathfrak{g}$, which are both $3$-dimensional. An explicit calculation (using the standard [[Cartan subalgebra|CSA]] of $\mathfrak{g}$ and [[root basis]]) shows $V(\omega_{1})$ is the [[defining representation of a Lie algebra|defining representation]] and $V(\omega_{2})$ its [[dual representation|dual]]. > > We can now ask all kinds of questions, like what is the [[Weyl's theorem on complete reducibility|decomposition]] [[symmetric power|of]] $\text{Sym}^{2}V(\omega_{1})$? It is not hard to see that $2 \omega_{1}$ is a [[on the weights of a representation|highest weight]] of $\text{Sym}^{2}V(\omega_{1})$, and so $V(2 \omega _1)$ must feature in the decomposition of $\text{Sym}^{2}V(\omega_{1})$ into irreducibles.[^1] Since $\text{dim }\text{Sym}^{2}V(\omega_{1})=\frac{3(3+1)}{2}=6$, and $\text{dim }V(2 \omega_{1})=6$ by the above formula, there is no room for any other irreducibles: $\text{Sym}^{2}V(\omega_{1}) \cong V(2 \omega_{1})$. > In general: if the weights of $V$ are $\{ \lambda_{i} \}$, then - The weights of $V^{\otimes2}$ are $\{ \lambda_{i}+\lambda_{j} \}$ - The weights of $\text{sym}^{2}(V)$ are $\{ \lambda_{i}+ \lambda_{j} : i \leq j \}$ - The weights of $\Lambda^{2}(V)$ are $\{ \lambda_{i}-\lambda_{j}: i < j \}$. [^1]: This is an general fact: If $V$ is a [[Lie algebra representation|representation]] of $\mathfrak{g}$ and $v$ is a [[weight space for sl2(C)|highest weight vector]] of $V$ with highest weight $\lambda$, then $v$ [[Lie algebra subrepresentation generated by a vector|generates]] a copy of $V(\lambda)$ in $V$. However, I don't think I have written it down explicitly yet. The [[universal property]] of [[Verma module|Verma modules]] produces a [[surjection]] $M(\lambda) \twoheadrightarrow \mathcal{U}(\mathfrak{g}) \cdot v$ sending $m_{\lambda} \mapsto v$. [[characterization of quotienting a group|It factors]] through the [[highest weight module|highest weight module has a unique irreducible quotient|unique irreducible]] [[quotient representation|quotient]] $V(\lambda)$ of $M(\lambda)$ as $M(\lambda) \twoheadrightarrow V(\lambda) \twoheadrightarrow \mathcal{U}(\mathfrak{g}) \cdot v$. This is a nonzero surjection out of an irreducible representation, so it must be a [[bijection]]. hence $V(\lambda) \cong \mathcal{U}(\mathfrak{g}) \cdot v$. Let $\mathfrak{g}=\mathfrak{sp}_{4}$, [[symplectic Lie algebra|so]] $\Phi=C_{2} \cong B_{2}$, $\Delta=\{ \alpha, \beta \}$ with $\alpha$ short. We need to calculate a bunch of check-pairings of $\omega_{i}$ with positive roots. By looking at pictures, $\langle \omega_{1}, \check{\alpha + \beta} \rangle=1$. Similar calculations lead to the formula $\text{dim }V(n_{1}\omega_{1}+n_{2}\omega_{2})=\frac{(n_{1}+1)(n_{2}+1)(n_{1}+n_{2}+2)(n_{1}+2n_{2}+3)}{6}.$ This lets us read off that - $\text{dim }V(\omega_{1})=4$ - $\text{dim }V(\omega_{2})=5$ and that these are the smallest nontrival $\mathfrak{g}$-reps. So the only nontrivial 4-dimensional irrep of $\mathfrak{sp}_{4}$ is $V(\omega_{1})$. But we already know of an irreducible four-dimensional representation of $\mathfrak{sp}_{4}$: the [[defining representation of a Lie algebra|the defining representation]]. Let us draw the weights of $V(\omega_{1})$ and $\text{Sym}^{2}V(\omega_{1})$. Note that the weights of the latter are obtained as the various sums of the weights of the former. (Concentric circles indicate multiplicity) ![[Pasted image 20250524122139.png|500]] Since $\beta+2\alpha$ is the highest weight, which is the [[root basis|highest root]], this has to be the [[adjoint representation]] of $\mathfrak{g}$. Indeed, we see the weights are the roots $\Phi$ together with multiplicity-2 zero. (More examples in [[Weyl's theorem on complete reducibility]]) > [!proof]- Proof. ([[Weyl character formula]]) > No time for us in this course. See Humphreys 24.3. ---- #### [^2]: By these we mean 'distinct ways to get from $\mu$ to $\lambda$ by traveling along positive roots'. [^3]: Example: for $\Phi=A_{2}$ with the [[root basis]] $\{ \alpha, \beta \}$ in the pictures, $\rho=\alpha+\beta$ (draw picture). [^4]: Can think of as a generalization because the [[Weyl group of a root system|Weyl group]] of $A_{n-1}$ is $S_{n}$. ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```