Preliminaries on classical type combinatorics

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 23 February 2012

Preliminaries on classical type combinatorics

The Lie algebras $𝔤={\mathrm{𝔤𝔩}}_r$ and ${\mathrm{𝔰𝔩}}_r$ are given by $${\mathrm{𝔤𝔩}}_r=\mathrm{End}\left(V\right)and{\mathrm{𝔰𝔩}}_r=\{x\in {\mathrm{𝔤𝔩}}_r | \mathrm{tr}\left(x\right)=0\},$$ with bracket $[x,y]=xy-yx.$ Then $${\mathrm{𝔤𝔩}}_{r}has\; basis\left\{E_{ij} \right| 1\le i,j\le r\},$$ where $E_{ij}$ is the matrix with 1 in the $\left(i,j\right)$ entry and 0 elsewhere. A Cartan subalgebra of ${\mathrm{𝔤𝔩}}_r$ is $${𝔥}_{\mathrm{𝔤𝔩}}=\{x\in {\mathrm{𝔤𝔩}}_r | x \; is\; diagonal\}with\; basis\{E_{11},E_{22},...,E_{rr}\},$$ and the dual basis $\{{\epsilon }_1,...,{\epsilon }_r\}$ of ${𝔥}_{\mathrm{𝔤𝔩}}^{*}$ is specified by $${\epsilon }_i:{𝔥}_{\mathrm{𝔤𝔩}}\to ℂgiven\; by{\epsilon }_i\left(E_{jj}\right)={\delta }_{ij}.$$ The form $$⟨,⟩:𝔤\times 𝔤\to ℂgiven\; by⟨x,y⟩={\mathrm{tr}}_V\left(xy\right)(Pctc\; 1.1)$$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to $𝔥$ is a nondegenerate form $⟨,⟩:𝔥\times 𝔥\to ℂ$ on $𝔥.$ $$\begin{array}{l}\{E_{11},...,E_{rr}\} \; is\; an\; orthonormal\; basis\; with\; respect\; to\; ⟨,⟩:𝔥\times 𝔥\to ℂ,and\\ \{{\epsilon }_1,...,{\epsilon }_r\} \; is\; an\; orthonormal\; basis\; with\; respect\; to\; ⟨,⟩:{𝔥}^{*}\times {𝔥}^{*}\to ℂ,\end{array}$$ the form on ${𝔥}^{*}$ induced by the form on $𝔥$ and the vector space isomorphism $\nu :{𝔥}_{\mathrm{𝔤𝔩}}\to {𝔥}_{\mathrm{𝔤𝔩}}^{*}$ given by $\nu \left(h\right)=⟨h,\cdot ⟩.$ A Cartan subalgebra of ${\mathrm{𝔰𝔩}}_r$ is $$𝔥={(E_{11}+\cdots +E_{rr})}^{\perp }=\{x\in {𝔥}_{\mathrm{𝔤𝔩}} | ⟨x,E_{11}+\cdots +E_{rr}⟩=0\},$$ the orthogonal subspace to $ℂ(E_{11}+\cdots +E_{rr}).$ The dominant integral weights for ${\mathrm{𝔤𝔩}}_r,$ $$P^{+}=\{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_r{\epsilon }_r | {\lambda }_i\in \mathbb{Z}, {\lambda }_1\ge \cdots \ge {\lambda }_r\}$$ index the irreducible finite dimensional representations $L\left(\lambda \right)$ of ${\mathrm{𝔤𝔩}}_r$ and the irreducible finite dimensional representations of ${\mathrm{𝔰𝔩}}_r$ are $$L\left(\stackrel{\_}{\lambda }\right)={\mathrm{Res}}_{{\mathrm{𝔰𝔩}}_r}^{{\mathrm{𝔤𝔩}}_r}\left(L\left(\lambda \right)\right),where\begin{array}{ccl}{𝔥}_{\mathrm{𝔤𝔩}}^{*}& \to & {𝔥}_{\mathrm{𝔰𝔩}}^{*}={({\epsilon }_1+\cdots +{\epsilon }_r)}^{\perp }\\ \lambda & \mapsto & \stackrel{\_}{\lambda }\end{array}$$ is the orthogonal projection.

The matrix units $\left\{E_{ij} \right| 1\le i,j\le r\}$ form a basis of ${\mathrm{𝔤𝔩}}_r$ for which the dual basis with respect to the form in (Pctc 1.1) is $\left\{E_{ji} \right| 1\le i,j\le r\}$ so that $$\begin{array}{cl}{\gamma }_{\mathrm{𝔤𝔩}}=\sum _{1\le i,j\le r}{E}_{ij}\otimes E_{ji}=\sum _{\begin{array}{c}1\le i,j\le r\\ i\ne j\end{array}}{E}_{ij}\otimes E_{ji}+\sum _{i=1}{E}_{ii}\otimes E_{ii},and& (Pctc\; 1.2)\\ {\gamma }_{\mathrm{𝔰𝔩}}={\gamma }_{\mathrm{𝔤𝔩}}-E_{+}\otimes E_{+},where{E}_{+}=E_{11}+\cdots +E_{rr}.& (Pctc\; 1.3)\end{array}$$ If the Casimir for ${\mathrm{𝔤𝔩}}_r,$ $${\kappa }_{\mathrm{𝔤𝔩}}=\sum _{1\le i,j\le r}{E}_{ij}{E}_{ji},acts\; on\; L\left(\lambda \right) \; by\; the\; constant\; {\kappa }_{\mathrm{𝔤𝔩}}\left(\lambda \right)$$ then the Casimir for ${\mathrm{𝔰𝔩}}_r$ $$\begin{array}{cl}{\kappa }_{\mathrm{𝔰𝔩}}={\kappa }_{\mathrm{𝔤𝔩}}-E_{+}{E}_{+}acts\; on\; L\left(\stackrel{\_}{\lambda }\right) \; by\; the\; constant\; {\kappa }_{\mathrm{𝔤𝔩}}\left(\lambda \right)-\frac{1}{r}{\left|\lambda \right|}^2,& (Pctc\; 1.4)\end{array}$$ where $\left|\lambda \right|={\lambda }_1+\cdots +{\lambda }_r.$

The Lie algebras $𝔤={\mathrm{𝔰𝔬}}_{2r+1}, {\mathrm{𝔰𝔭}}_{2r},$ and ${\mathrm{𝔰𝔬}}_{2r}$ are given by $$𝔤=\{x\in \mathrm{𝔤𝔩}\left(V\right) | (x{v}_1,v_2)+(v_1,x{v}_2)=0 \; for\; all\; v_1,v_2\in V\},$$ where $(,):V\times V\to ℂ$ is a nondegenerate bilinear form such that $$\begin{array}{lll}(v_1,v_2)=\epsilon (v_2,v_1),& where\epsilon =\{\begin{array}{ll}1,& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1},\\ -1,& if\; 𝔤={\mathrm{𝔰𝔭}}_{2r},\\ 1,& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r}.\end{array}\phantom{\}}& (Pctc\; 1.5)\end{array}$$

Choose $$a\; basis\; \left\{v_i\right|i\in \hat{V}\} \; of\; V,where\hat{V}=\{\begin{array}{ll}\{-r,...,-1,0,1,...,r\},& if 𝔤={\mathrm{𝔰𝔬}}_{2r+1},\\ \{-r,...,-1,1,...,r\},& if 𝔤={\mathrm{𝔰𝔭}}_{2r},\\ \{-r,...,-1,1,...,r\},& if 𝔤={\mathrm{𝔰𝔬}}_{2r},\end{array}\phantom{\}}$$ so that the matrix of the bilinear form $(,):V\times V\to ℂ$ is $$J=\left(\begin{array}{cccccc}& & & & & 1\\ & 0& & & ⋰& \\ & & & 1& & \\ & & \epsilon & & & \\ & ⋰& & & 0& \\ \epsilon & & & & & \end{array}\right)and𝔤=\{x\in {\mathrm{𝔤𝔩}}_N | x^{t}J+Jx=0\},$$ where $N=\dim \left(V\right).$ Then, as in Molev [Mo, (7.9)] and [Bou, Ch. 8 §13 2.I, 3.I, 4.I], $$\begin{array}{ll}𝔤=\mathrm{span}\left\{F_{ij} \right| i,j\in \hat{V}\}where{F}_{ij}=E_{ij}-{\theta }_{ij}{E}_{-j,-i},& (Pctc\; 1.6)\end{array}$$ where $E_{ij}$ is the matrix with 1 in the $(i,j)-$entry and 0 elsewhere, and $${\theta }_{ij}=\{\begin{array}{ll}1,& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1},\\ \mathrm{sgn}\left(i\right)\cdot \mathrm{sgn}\left(j\right),& if\; 𝔤={\mathrm{𝔰𝔭}}_{2r},\\ 1,& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r}.\end{array}\phantom{\}}$$

A Cartan subalgebra of $𝔤$ is $$\begin{array}{ll}𝔥=\mathrm{span}\left\{F_{ii} \right| i\in \hat{V}\}with\; basis\{F_{11},F_{22},...,F_{rr}\}.& (Pctc\; 1.7)\end{array}$$ The dual basis $\{{\epsilon }_1,...,{\epsilon }_r\}$ of ${𝔥}^{*}$ is specified by $${\epsilon }_i:𝔥\to ℂgiven\; by{\epsilon }_i\left(F_{jj}\right)={\delta }_{ij}.$$ The form $$\begin{array}{ll}⟨,⟩:𝔤\times 𝔤\to ℂgiven\; by⟨x,y⟩=\frac{1}{2}{\mathrm{tr}}_V\left(xy\right)& (Pctc\; 1.8)\end{array}$$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to $𝔥$ is a nondegenerate form $⟨,⟩:𝔥\times 𝔥\to ℂ$ on $𝔥.$ the form on ${𝔥}^{*}$ induced by the form on $𝔥$ and the vector space isomorphism $\nu :𝔥\to {𝔥}^{*}$ given by $\nu \left(h\right)=⟨h,\cdot ⟩.$

With $F_{ij}$ as in (Pctc 1.6), $𝔤$ has basis $$\begin{array}{ll}\left\{F_{i,i} \right| 0<i\in \hat{V}\}\cup \left\{F_{\pm i,\pm j} \right| 0<i<j\in \hat{V}\}\cup \left\{F_{0,\pm i} \right| 0<i\in \hat{V}\}& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1},\\ \{F_{i,i},F_{-i,i},F_{i,-i} | 0<i\in \hat{V}\}\cup \left\{F_{\pm i,\pm j} \right| 0<i<j\in \hat{V}\}& if\; 𝔤={\mathrm{𝔰𝔭}}_{2r},\\ \left\{F_{i,i} \right| 0<i\in \hat{V}\}\cup \left\{F_{\pm i,\pm j} \right| 0<i<j\in \hat{V}\}& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r}.\end{array}$$ With respect to the nondegenerate ad-invariant symmetric bilinear form $⟨,⟩:𝔤\otimes 𝔤\to ℂ$ given in (Pctc 1.8), $⟨x,y⟩=\frac{1}{2}{\mathrm{tr}}_V\left(xy\right),$ the dual basis with respect to $⟨,⟩$ is $$D_{ij}^{*}=F_{ji}ifi\ne -j,and{F}_{i,-i}^{*}=\frac{1}{2}{F}_{-i,i}.$$ Since the sets $$\begin{array}{ll}\left\{F_{-i,-i} \right| 0<i\in \hat{V}\}\cup \left\{F_{\pm i,\pm j} \right| 0<j<i\in \hat{V}\}\cup \left\{F_{\pm i,0} \right| 0<i\in \hat{V}\}& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1},\\ \{F_{-i,-i},F_{-i,i},F_{i,-i} | 0<i\in \hat{V}\}\cup \left\{F_{\pm i,\pm j} \right| 0<j<i\in \hat{V}\}& if\; 𝔤={\mathrm{𝔰𝔭}}_{2r},\\ \left\{F_{-i,-i} \right| 0<i\in \hat{V}\}\cup \left\{F_{\pm i,\pm j} \right| 0<j<i\in \hat{V}\}& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r},\end{array}$$ form alternate bases, and $F_{i,-i}=0$ when $𝔤={\mathrm{𝔰𝔬}}_{2r+1}$ or $𝔤={\mathrm{𝔰𝔬}}_{2r},$ $$\begin{array}{ll}2\gamma =\sum _{i,j\in \hat{V}}{F}_{ij}\otimes F_{ji}^{*}+\sum _{i\in \hat{V}}{F}_{i,-i}\otimes F_{i,-i}^{*}=\sum _{i,j\in \hat{V}}{F}_{ij}\otimes F_{ji}.& (Pctc\; 1.9)\end{array}$$

To compute the value in (1.17) Where does this reference? choose positive roots $$\begin{array}{ll}{R}^{+}=\{\begin{array}{ll}\{{\epsilon }_i-{\epsilon }_j | 1\le i<j\le r\},& for\; {\mathrm{𝔰𝔩}}_r,\\ \{{\epsilon }_i\pm {\epsilon }_j | 1\le i<j\le r\}\cup \left\{{\epsilon }_i \right| 1\le i\le r\},& for\; {\mathrm{𝔰𝔬}}_{2r+1},\\ \{{\epsilon }_i\pm {\epsilon }_j | 1\le i<j\le r\}\cup \left\{2{\epsilon }_i \right| 1\le i\le r\},& for\; {\mathrm{𝔰𝔭}}_{2r},\\ \{{\epsilon }_i\pm {\epsilon }_j | 1\le i<j\le r\},& for\; {\mathrm{𝔰𝔬}}_{2r}.\end{array}\phantom{\}}& (Pctc\; 1.10)\end{array}$$ Since $$\begin{array}{rl}\sum _{1\le i<j\le r}{\epsilon }_i-{\epsilon }_j& +\sum _{1\le i<j\le r}{\epsilon }_i+{\epsilon }_j+\sum _{i=1}^r{\epsilon }_i+\sum _{i=1}^r{\epsilon }_i\\ & =\sum _{i=1}^r(r-2i+1){\epsilon }_i+\sum _{i=1}^r(r-1){\epsilon }_i+\sum _{i=1}^r{\epsilon }_i+\sum _{i=1}^r{\epsilon }_i,\end{array}$$ it follows that $$\begin{array}{lll}2\rho =\sum _{i=1}^r(y-2i+1){\epsilon }_i,& where\; y=⟨{\epsilon }_1,{\epsilon }_1+2\rho ⟩=\{\begin{array}{ll}r,& if\; 𝔤={\mathrm{𝔰𝔩}}_r,\\ 2r,& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1},\\ 2r+1,& if\; 𝔤={\mathrm{𝔰𝔭}}_{2r},\\ 2r-1,& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r},\end{array}\phantom{\}}& (Pctc\; 1.11)\end{array}$$ is the value by which the Casimir $\kappa$ acts on $L\left({\epsilon }_1\right).$ Letting $q=e^{\frac{h}{2}}$, the quantum dimension of $V$ is $$\begin{array}{lll}{\dim }_q\left(V\right)={\mathrm{tr}}_V\left(e^{h\rho }\right)=\epsilon +\left[y\right],& where\; \left[y\right]=\frac{q^y-q^{-y}}{q-q^{-1}},& (Pctc\; 1.12)\end{array}$$ since, with respect to a weight basis of $V,$ the eigenvalues of the diagonal matrix $e^{h\rho }$ are $e^{\frac{1}{2}h(y-2i+1)}=q^{(y-2i+1)}.$

A standard notation is to view a weight $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_r{\epsilon }_r$ as a configuration of boxes with ${\lambda }_i$ boxes in row $i.$ If $b$ is a box in position $(i,j)$ if $\lambda$ the content of $b$ is $$\begin{array}{lll}c\left(b\right)=j-i= \; the\; diagonal\; number\; of\; b,& so\; that\begin{array}{c}\\ 0 1 2 -1 0 1 -2 \\ \end{array}& (Pctc\; 1.13)\end{array}$$ are the contents of the boxes of $\lambda =3{\epsilon }_1+3{\epsilon }_2+{\epsilon }_3.$ If $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n,$ then $$⟨\lambda ,\lambda +2\rho ⟩-⟨\lambda -{\epsilon }_i,\lambda -{\epsilon }_i+2\rho ⟩=2{\lambda }_i+2{\rho }_i-1=y+2{\lambda }_i-2i=y+2c(\lambda /{\lambda }^{-}),$$ where $\lambda /{\lambda }^{-}$ is the box at the end of row $i$ in $\lambda .$ By induction $$\begin{array}{ll}⟨\lambda ,\lambda +2\rho ⟩=y\left|\lambda \right|+2\sum _{b\in \lambda }c\left(b\right),& (Pctc\; 1.14)\end{array}$$ for $\lambda ={\lambda }_1{\epsilon }_1+\cdots +{\lambda }_r{\epsilon }_r$ with ${\lambda }_i\in \mathbb{Z}.$

Let $L\left(\lambda \right)$ be the irreducible highest height $𝔤-$module with highest weight $\lambda ,$ and let $V=L\left({\epsilon }_1\right).$ Then for $𝔤={\mathrm{𝔰𝔬}}_{2r+1},{\mathrm{𝔰𝔭}}_{2r}$ or ${\mathrm{𝔰𝔬}}_{2r},$ $$\begin{array}{ll}V\cong V^{*}andV\otimes V\cong L\left(0\right)\oplus L\left(2{\epsilon }_1\right)\oplus L({\epsilon }_1+{\epsilon }_2).& (Pctc\; 1.15)\end{array}$$ For each component in the decomposition of $V\otimes V$ the values by which $\gamma =\sum _{b\in B}b\otimes b^{*}$ acts as in (1.17) this reference again are

The second symmetric and exterior powers of $V$ are $$\begin{array}{ll}{S}^2\left(V\right)=\{\begin{array}{ll}L\left(2{\epsilon }_1\right)\oplus L\left(0\right),& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1} \; or\; {\mathrm{𝔰𝔬}}_{2r},\\ L\left(2{\epsilon }_1\right),& if\; 𝔤={\mathrm{𝔰𝔭}}_{2r},\end{array}\phantom{\}}& (Pctc\; 1.17)\end{array}$$ and $$\begin{array}{ll}{\Lambda }^2\left(V\right)=\{\begin{array}{ll}L({\epsilon }_1+{\epsilon }_2),& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1} \; or\; {\mathrm{𝔰𝔬}}_{2r},\\ L({\epsilon }_1+{\epsilon }_2)\oplus L\left(0\right),& if\; 𝔤={\mathrm{𝔰𝔭}}_{2r}.\end{array}\phantom{\}}& (Pctc\; 1.18)\end{array}$$ For all dominant integral weights $\lambda$ $$\begin{array}{ll}L\left(\lambda \right)\otimes V=\{\begin{array}{ll}L\left(\lambda \right)\oplus \left(\underset{{\lambda }^{\pm }}{⨁}L\left({\lambda }^{\pm }\right)\right),& if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1} \; and\; {\lambda }_r>0,\\ \underset{{\lambda }^{\pm }}{⨁}L\left({\lambda }^{\pm }\right),& \begin{array}{c}if\; 𝔤={\mathrm{𝔰𝔭}}_{2r}, 𝔤={\mathrm{𝔰𝔬}}_{2r}, \; or\\ if\; 𝔤={\mathrm{𝔰𝔬}}_{2r+1} \; and\; {\lambda }_r=0.\end{array}\end{array}\phantom{\}}& (Pctc\; 1.19)\end{array}$$

Notes and References

Where are these from?

References

References?

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