## Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras

Last update: 11 September 2013

## Classical symmetric functions

This section gives a brief summary of the classical symmetric function theory. See [Mac] Chapter 1 for a complete treatment.

Fix a positive integer $n\text{.}$ A partition $\lambda =\left({\lambda }_{1},{\lambda }_{2},\dots ,{\lambda }_{n}\right)$ is a sequence ${\lambda }_{1}\ge {\lambda }_{2}\ge \cdots \ge {\lambda }_{n}\ge 0$ of nonnegative integers. Let $𝒫$ denote the set of partitions. We have the following sequence of inclusions $𝒫⊂ℤn⊃ℂn. (1.1)$ There is a partial ordering, the dominance ordering, on elements of ${ℤ}^{n}$ given by $γ≥κif γ1+γ2+⋯ +γi≥κ1+ κ2+⋯κi, for all i. (1.2)$

Let ${S}_{n}$ denote the symmetric group. The sign $\epsilon \left(w\right)$ of a permutation $w\in {S}_{n}$ is the determinant of the corresponding permutation matrix. ${S}_{n}$ acts on elements of ${ℤ}^{n}$ by permuting the positions.

For each $1\le i the raising operator ${R}_{ij}$ is the operator which acts on elements of ${ℤ}^{n}$ by $Rij (γ1,γ2,…,γn) =(γ1,γ2,…,γi+1,…,γj-1,…,γn) . (1.4)$

Let ${x}_{1},{x}_{2},\dots ,{x}_{n}$ be commuting variables and for each $\gamma =\left({\gamma }_{1},{\gamma }_{2},\dots ,{\gamma }_{n}\right)\in {ℤ}^{n}$ define $xγ= x1γ1 x2γ2 ⋯ xnγn. (1.5)$ Define an action of ${S}_{n}$ on monomials by $wxγ= xwγ. (1.6)$ The ring $Λn=ℚ [x1,x2,…,xn]Sn$ is the ring of symmetric functions.

Bases of symmetric functions

For each partition $\lambda$ define the monomial symmetric function by $mλ= ∑γ∈Snλ xγ, (1.7)$ where the sum runs over all $\gamma \in {ℤ}^{n}$ in the ${S}_{n}$ orbit of $\lambda ,$ i.e., over all distinct permutations of $\lambda \text{.}$

The Schur functions are given by $sλ= ∑w∈Sn ε(w) xw(λ+ρ) ∑w∈Sn ε(w) xwρ , (1.8)$ where $\rho =\left(n-1,n-2,\dots ,1,0\right)\text{.}$

The elementary symmetric functions are given by defining $e0 = 1, er = ∑1≤i1<⋯ for each positive integer $r,$ and $eλ=eλ1 eλ2⋯eλn , (1.9)$ for all partitions $\lambda \text{.}$

The homogeneous symmetric functions are given by defining $h0 = 1, hr = ∑1≤i1≤⋯≤ir≤n xi1xi2⋯xir,$ for each positive integer $r,$ and $hγ=hγ1 hγ2⋯hγn,$ for all sequences $\gamma \in {ℤ}^{n}\text{.}$

Define integers ${K}_{\lambda \mu }$ by $sλ=∑μ∈𝒫 Kλμmμ. (1.10)$ One has the following (nontrivial) facts:

 (a) The ${K}_{\lambda \mu }$ are nonnegative integers. (b) ${K}_{\lambda \lambda }=1$ for all $\lambda \in 𝒫\text{.}$ (c) ${K}_{\lambda \mu }=0$ if $\mu \nleqq \lambda \text{.}$

Each of the sets ${\left\{{m}_{\lambda }\right\}}_{\lambda \in 𝒫},$ ${\left\{{s}_{\lambda }\right\}}_{\lambda \in 𝒫},$ ${\left\{{e}_{\lambda }\right\}}_{\lambda \in 𝒫},$ ${\left\{{h}_{\lambda }\right\}}_{\lambda \in 𝒫},$ forms a $ℤ\text{-basis}$ of ${\Lambda }_{n}\text{.}$

Inner product

There is an inner product on the ring of symmetric functions given by making the Schur functions orthonormal, $⟨sλ,sμ⟩ =δλμ. (1.11)$

Further facts

The homogeneous symmetric functions are the dual basis to the basis of monomial symmetric functions, $⟨hλ,mμ⟩ =δλμ. (1.12)$ A consequence of this is that $hμ=∑λ sλKλμ. (1.13)$

One has the following "Jacobi-Trudi" formula for the Schur functions in terms of the homogeneous symmetric functions, $sλ=∑w∈Sn ε(w) hλ+ρ-wρ. (1.14)$ There is also a formula for the Schur function in terms of raising operators and the homogeneous symmetric function. $sλ=∏i

## Notes and references

This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.