Preliminaries on classical type combinatorics

Last update: 23 February 2012

Preliminaries on classical type combinatorics

The Lie algebras $𝔤={\mathrm{𝔤𝔩}}_{r}$ and ${\mathrm{𝔰𝔩}}_{r}$ are given by with bracket $\left[x,y\right]=xy-yx.$ Then 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 and the dual basis $\left\{{\epsilon }_{1},...,{\epsilon }_{r}\right\}$ of ${𝔥}_{\mathrm{𝔤𝔩}}^{*}$ is specified by $εi: 𝔥𝔤𝔩→ℂ given by εi(Ejj) =δij.$ The form $⟨,⟩: 𝔤×𝔤→ℂ given by ⟨x,y⟩= trV(xy) (Pctc 1.1)$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to $𝔥$ is a nondegenerate form $⟨,⟩:𝔥×𝔥\to ℂ$ on $𝔥.$ 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 the orthogonal subspace to $ℂ\left({E}_{11}+\cdots +{E}_{rr}\right).$ The dominant integral weights for ${\mathrm{𝔤𝔩}}_{r},$ 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 $L(λ_)= Res𝔰𝔩r𝔤𝔩r (L(λ)), where 𝔥𝔤𝔩* → 𝔥𝔰𝔩*=(ε1+⋯+εr)⊥ λ ↦ λ_$ is the orthogonal projection.

The matrix units form a basis of ${\mathrm{𝔤𝔩}}_{r}$ for which the dual basis with respect to the form in (Pctc 1.1) is so that $γ𝔤𝔩= ∑ 1≤i,j≤r Eij⊗Eji= ∑ 1≤i,j≤r i≠j Eij⊗Eji+ ∑ i=1 Eii⊗Eii, and (Pctc 1.2) γ𝔰𝔩= γ𝔤𝔩- E+⊗E+, where E+=E11+⋯+Err. (Pctc 1.3)$ If the Casimir for ${\mathrm{𝔤𝔩}}_{r},$ then the Casimir for ${\mathrm{𝔰𝔩}}_{r}$ where $|\lambda |={\lambda }_{1}+\cdots +{\lambda }_{r}.$

The Lie algebras and ${\mathrm{𝔰𝔬}}_{2r}$ are given by where $\left(,\right):V×V\to ℂ$ is a nondegenerate bilinear form such that

Choose so that the matrix of the bilinear form $\left(,\right):V×V\to ℂ$ is where $N=\mathrm{dim}\left(V\right).$ Then, as in Molev [Mo, (7.9)] and [Bou, Ch. 8 §13 2.I, 3.I, 4.I], where ${E}_{ij}$ is the matrix with 1 in the $\left(i,j\right)-$entry and 0 elsewhere, and

A Cartan subalgebra of $𝔤$ is The dual basis $\left\{{\epsilon }_{1},...,{\epsilon }_{r}\right\}$ of ${𝔥}^{*}$ is specified by $εi:𝔥→ℂ given by εi(Fjj)=δij.$ The form $⟨,⟩: 𝔤×𝔤→ℂ given by ⟨x,y⟩ =12trV(xy) (Pctc 1.8)$ is a nondegenerate ad-invariant symmetric bilinear form on $𝔤$ such that the restriction to $𝔥$ is a nondegenerate form $⟨,⟩:𝔥×𝔥\to ℂ$ 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 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 $Dij* =Fji if i≠-j, and Fi,-i* =12 F-i,i.$ Since the sets form alternate bases, and ${F}_{i,-i}=0$ when $𝔤={\mathrm{𝔰𝔬}}_{2r+1}$ or $𝔤={\mathrm{𝔰𝔬}}_{2r},$ $2γ= ∑ i,j∈V^ Fij⊗ Fji*+ ∑ i∈V^ Fi,-i⊗ Fi,-i*= ∑ i,j∈V^ Fij⊗ Fji. (Pctc 1.9)$

To compute the value in (1.17) Where does this reference? choose positive roots Since $∑ 1≤i it follows that 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 since, with respect to a weight basis of $V,$ the eigenvalues of the diagonal matrix ${e}^{h\rho }$ are ${e}^{\frac{1}{2}h\left(y-2i+1\right)}={q}^{\left(y-2i+1\right)}.$

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 $\left(i,j\right)$ if $\lambda$ the content of $b$ is 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 $⟨λ,λ+2ρ⟩- ⟨λ-εi,λ-εi+2ρ⟩ = 2λi+2ρi-1 = y+2λi-2i = y+2c(λ/λ-),$ where $\lambda /{\lambda }^{-}$ is the box at the end of row $i$ in $\lambda .$ By induction $⟨λ,λ+2ρ⟩ = y|λ|+2 ∑ b∈λ c(b), (Pctc 1.14)$ for $\lambda ={\lambda }_{1}{\epsilon }_{1}+\cdots +{\lambda }_{r}{\epsilon }_{r}$ with ${\lambda }_{i}\in ℤ.$

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 ${\mathrm{𝔰𝔬}}_{2r},$ $V≅V* and V⊗V≅L(0)⊕L(2ε1)⊕L(ε1+ε2). (Pctc 1.15)$ 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

$⟨0,0+2ρ⟩- ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩ = 0-y-y = -2y, ⟨2ε1,2ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩ = 4+2(y-1)-y-y=2, (Pctc 1.16) ⟨ε1+ε2,ε1+ε2+2ρ⟩- ⟨ε1,ε1+2ρ⟩- ⟨ε1,ε1+2ρ⟩ = 2+(y-1)+(y-3)-y-y=-2.$
The second symmetric and exterior powers of $V$ are and For all dominant integral weights $\lambda$

Notes and References

Where are these from?

References?