Weyl Character and Multiplicity Formulas — Examples

Given: The Axiom of Choice, the realization of ${\mathfrak{s}\mathfrak{l}}_3$ with diagonal Cartan subalgebra and roots $\pm {\alpha }_1,\pm {\alpha }_2,\pm ({\alpha }_1+{\alpha }_2)$, the positive system ${\mathrm{\Phi }}^{+}=\{{\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2\}$, the Weyl vector $\rho$, the Weyl group $W=S_3$ with its elements and lengths, and the alternant $A(\rho )$.

The Axiom of Choice is assumed; it enters through the root-system and denominator suppliers below (The Axiom of Choice).

In this realization the positive roots are ${\alpha }_1,{\alpha }_2$ and ${\alpha }_1+{\alpha }_2$, the Weyl group acts by the simple reflections $s_1,s_2$ with $s_1{\alpha }_1=-{\alpha }_1$, $s_1{\alpha }_2={\alpha }_1+{\alpha }_2$, $s_2{\alpha }_2=-{\alpha }_2$, $s_2{\alpha }_1={\alpha }_1+{\alpha }_2$, so $s_1\rho ={\alpha }_2$, $s_2\rho ={\alpha }_1$, $s_1{s}_2\rho =-{\alpha }_1$, $s_2{s}_1\rho =-{\alpha }_2$ and $w_0\rho =-\rho$, while $\rho =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha ={\alpha }_1+{\alpha }_2$ (Root systems of the classical complex Lie algebras, Classical complex matrix Lie algebras, The Weyl vector rho for a chosen positive system, The root set is a reduced crystallographic root system).

The Weyl group of $A_2$ is $S_3=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,w_0\}$ with lengths $0,1,1,2,2,3$, length being the number of inversions and the least number of simple reflections in an expression (Weyl length equals inversion number).

The denominator identity states $A(\rho )=e^{\rho }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha })$, and $A(\nu )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w\nu }$ (The Weyl denominator identity, The Weyl alternation operator).

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$ with positive root $\alpha$, fundamental weight $\omega =\alpha \mathrm{/}2$, Weyl group $W=\{1,s\}$, Weyl vector $\rho =\omega$, the weight $\lambda =2\omega$, the weight $\mu =0$, and the Kostant partition function $P$.

The Axiom of Choice is assumed; it enters through the Kostant formula used in the counterexample (The Axiom of Choice).

For ${\mathfrak{s}\mathfrak{l}}_2$ the positive root satisfies $\alpha =2\omega$ and $s\omega =-\omega$, so $\rho =\alpha \mathrm{/}2=\omega$ and $\lambda +\rho =3\omega$ (Finite Weyl root system, lattice and chamber conventions, Fundamental weights, The Weyl vector rho for a chosen positive system).

$L(2\omega )$ is the irreducible three-dimensional ${\mathfrak{s}\mathfrak{l}}_2$-module whose weights are $2\omega ,0,-2\omega$, each with multiplicity one (Finite-dimensional representations of sl_2, Integral, dominant, and strictly dominant weights).

$P(\beta )$ counts the families $(n_{\alpha })$ with $n_{\alpha }\alpha =\beta$, so $P(\alpha )=P(2\omega )=1$, while $P(-4\omega )=P(3\omega )=0$ because $-4\omega$ and $3\omega =\frac{3}{2}\alpha$ are not nonnegative integral multiples of the simple root $\alpha$; the multiplicity formula of the Statement is $m_{\lambda }(\mu )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}P(w(\lambda +\rho )-(\mu +\rho ))$ (The Kostant partition function, Kostant's weight multiplicity formula).

Given: The Axiom of Choice, the realization of ${\mathfrak{s}\mathfrak{l}}_3$ with diagonal Cartan subalgebra and roots $\pm {\alpha }_1,\pm {\alpha }_2,\pm \theta$, where $\theta ={\alpha }_1+{\alpha }_2$, the Weyl group $W=S_3$ with simple reflections $s_1,s_2$, the Weyl vector $\rho$, and the adjoint module $L(\theta )$.

The Axiom of Choice is assumed; it enters through the Kostant formula and the highest-weight suppliers below (The Axiom of Choice).

In this realization the positive roots are ${\alpha }_1,{\alpha }_2,\theta$ and $\theta ={\alpha }_1+{\alpha }_2$ is the highest root; the simple reflections act by $s_1\theta ={\alpha }_2$, $s_2\theta ={\alpha }_1$, $s_1{\alpha }_1=-{\alpha }_1$, $s_2{\alpha }_2=-{\alpha }_2$, and $\rho ={\alpha }_1+{\alpha }_2=\theta$ (Root systems of the classical complex Lie algebras, Classical complex matrix Lie algebras, Height and highest root, The Weyl vector rho for a chosen positive system).

$W=S_3=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,w_0\}$ with lengths $0,1,1,2,2,3$, and the adjoint module is the irreducible module $L(\theta )$ of highest weight $\theta$; its weights are the roots together with $0$, with the root spaces ${\mathfrak{g}}_{\alpha }$ one-dimensional and $\mathfrak{h}$ of dimension $2$ (The adjoint highest weight is the highest root, Finite semisimple Cartan, root and string structure, Weyl length equals inversion number).

$P(\theta )=2$, because a family $(n_{{\alpha }_1},n_{{\alpha }_2},n_{\theta })$ with $n_{{\alpha }_1}{\alpha }_1+n_{{\alpha }_2}{\alpha }_2+n_{\theta }\theta =\theta$ is either $(0,0,1)$ or $(1,1,0)$, while $P(\nu )=0$ for $\nu \mathrm{\notin }{Q}_{+}$ (The Kostant partition function).

Kostant's formula reads $m_{\lambda }(\mu )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}P(w(\lambda +\rho )-(\mu +\rho ))$, and $\theta$ is dominant integral (Kostant's weight multiplicity formula, Integral, dominant, and strictly dominant weights).

Given: The Axiom of Choice, the realization of ${\mathfrak{s}\mathfrak{l}}_3$ with positive roots ${\alpha }_1,{\alpha }_2,\theta ={\alpha }_1+{\alpha }_2$ of equal length, the Weyl vector $\rho =\theta$, the top weight $\lambda =\theta$, the weight $\mu =0$, the adjoint module $L(\theta )$, and the multiplicities $m_{\theta }(\nu )$.

The Axiom of Choice is assumed; it enters through the Freudenthal recursion and the highest-weight suppliers below (The Axiom of Choice).

In the realization ${\alpha }_1=e_1-e_2$, ${\alpha }_2=e_2-e_3$ the positive roots are ${\alpha }_1,{\alpha }_2,\theta$ with $({\alpha }_1,{\alpha }_1)=({\alpha }_2,{\alpha }_2)=(\theta ,\theta )$, and $\rho ={\alpha }_1+{\alpha }_2=\theta$ (Root systems of the classical complex Lie algebras, Classical complex matrix Lie algebras, The Weyl vector rho for a chosen positive system).

The adjoint module is $L(\theta )$ and its weights are $0$ with multiplicity $2$, and $\pm {\alpha }_1,\pm {\alpha }_2,\pm \theta$ each with multiplicity $1$ (The adjoint highest weight is the highest root, Kostant multiplicity in the sl3 adjoint module).

Freudenthal's recursion reads $((\lambda +\rho ,\lambda +\rho )-(\mu +\rho ,\mu +\rho ))m_{\lambda }(\mu )=2{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}{\sum }_{j\ge 1}(\mu +j\alpha ,\alpha )m_{\lambda }(\mu +j\alpha )$ (Freudenthal's weight multiplicity recursion).

The extremal weights $W\lambda$ each have multiplicity one, and the recursion has the base value $m_{\lambda }(\lambda )=1$ with the indeterminate case $0=0$ occurring, among actual weights of $L(\lambda )$, only at $\mu =\lambda$ (Extremal Weyl-orbit weights, Freudenthal recursion terminates from the highest weight).

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$ with its positive root $\alpha$, fundamental weight $\omega =\alpha \mathrm{/}2$, Weyl group $W=\{1,s\}$, Weyl vector $\rho =\omega$, and the dominant integral weights $m\omega$ with $m\ge 0$.

The Axiom of Choice is assumed; it enters through the character and dimension formulas below (The Axiom of Choice).

$\alpha =2\omega$, $W=\{1,s\}$, $s\omega =-\omega$, $\rho =\omega$ and $\lambda +\rho =(m+1)\omega$; the length of $s$ is $1$ (The Weyl vector rho for a chosen positive system, Integral, dominant, and strictly dominant weights).

$A(\nu )=e^{\nu }-e^{-\nu }$ for every $\nu \in {\mathfrak{h}}^{\ast }$, and the Weyl character formula and Weyl dimension formula read $\mathrm{ch}L(\lambda )=A(\lambda +\rho )A(\rho {)}^{-1}$ and $\dim L(\lambda )={\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(\lambda +\rho ,\alpha )\mathrm{/}(\rho ,\alpha )$ (The Weyl alternation operator, The Weyl character formula, The Weyl dimension formula).

In the completed ring $\mathcal{R}$ the elements $e^{\pm \omega }$ are invertible with $e^{\omega }{e}^{-\omega }=e^0$, and $e^{\omega }-e^{-\omega }=e^{\omega }(1-e^{-\alpha })$ is invertible with inverse $e^{-\omega }(1-e^{-\alpha }{)}^{-1}$, because $1-e^{-\alpha }$ is invertible (The completed formal character ring, Geometric series are invertible in the completed character ring, The Weyl denominator identity).

For $m\ge 0$ the module $L(m\omega )$ is the finite-dimensional simple module of highest weight $m\omega$ (Highest-weight classification).

Given: The Axiom of Choice, the realization of ${\mathfrak{s}\mathfrak{l}}_3$ with simple roots ${\alpha }_1,{\alpha }_2$, positive roots ${\mathrm{\Phi }}^{+}=\{{\alpha }_1,{\alpha }_2,\theta \}$ with $\theta ={\alpha }_1+{\alpha }_2$, the fundamental weight ${\omega }_1$, the Weyl vector $\rho$ and the coroots ${\alpha }_1^{\vee },{\alpha }_2^{\vee },{\theta }^{\vee }$.

The Axiom of Choice is assumed; it enters through the dimension formula and the highest-weight suppliers below (The Axiom of Choice).

In this realization the positive roots are ${\alpha }_1,{\alpha }_2$ and $\theta ={\alpha }_1+{\alpha }_2$, whose coroots in the simply-laced system add: ${\theta }^{\vee }={\alpha }_1^{\vee }+{\alpha }_2^{\vee }$ (Root systems of the classical complex Lie algebras, Classical complex matrix Lie algebras, The root set is a reduced crystallographic root system).

The fundamental weight ${\omega }_1$ is dual to ${\alpha }_1^{\vee }$: $⟨{\omega }_1,{\alpha }_1^{\vee }⟩=1$ and $⟨{\omega }_1,{\alpha }_2^{\vee }⟩=0$, and the Weyl vector satisfies $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ and $\rho ={\alpha }_1+{\alpha }_2$; moreover $⟨\rho ,{\theta }^{\vee }⟩=2$ (Fundamental weights, Integral, dominant, and strictly dominant weights, The root set is a reduced crystallographic root system).

The Weyl dimension formula states $\dim L(\lambda )={\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(\lambda +\rho ,\alpha )\mathrm{/}(\rho ,\alpha )={\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}⟨\lambda +\rho ,{\alpha }^{\vee }⟩\mathrm{/}⟨\rho ,{\alpha }^{\vee }⟩$ (The Weyl dimension formula).

The defining representation ${\mathbb{C}}^3$ of ${\mathfrak{s}\mathfrak{l}}_3$ has weights ${\epsilon }_1,{\epsilon }_2,{\epsilon }_3$, with ${\epsilon }_1$ the highest weight ${\omega }_1$; it is irreducible, because for a nonzero $v={\sum }_i{c}_i{e}_i$ with $c_j\mathrm{\ne }0$ the matrix units $E_{ij}$ with $i\mathrm{\ne }j$ and the diagonal elements of ${\mathfrak{s}\mathfrak{l}}_3$ produce all three basis vectors $e_1,e_2,e_3$, so the submodule generated by $v$ is everything; hence $L({\omega }_1)\cong {\mathbb{C}}^3$ has dimension $3$ (Classical complex matrix Lie algebras, Highest-weight classification).