## Highest weight paths

Last update: 1 December 2012

## Highest weight paths

A highest weight path is a path $p$ such that

 $e∼ip =0, for all 1≤i≤n.$
A highest weight path is a path $p$ such that, for each $1\le i\le n$, $p$ is the head of the $i$-string ${S}_{i}\left(p\right)$. Thus $⟨p\left(t\right),{\alpha }_{i}^{\vee }⟩>-1$ for all $t$ and all $1\le i\le n$. So a path $p$ is a highest weight path if and only if
 $p⊆C-ρ, where C-ρ={μ-ρ | μ∈C}.$
Following the example at the end of Section 2, for the root system of the type ${C}_{2}$ the picture is

the region $C-\rho$

If $p$ is a highest weight path with $\mathrm{wt}\left(p\right)\in P$ then, necessarily, $\mathrm{wt}\left(p\right)\in {P}^{+}$. The following theorem gives an expression for the character of a crystal in terms of the basis $\left\{{s}_{\lambda }\phantom{\rule{.3em}{0ex}}|\phantom{\rule{.3em}{0ex}}\lambda \in {P}^{+}\right\}$ of ${ℤ\left[P\right]}^{W}$.

Let $B$ be a crystal. Let $\mathrm{char}\left(B\right)$ be as defined in (5.24) and ${s}_{\lambda }$ as in (5.25). Then

 $char(B) =∑ p∈B p⊆C-ρ swt(p),$
where the sum is over highest weight paths $p\in B$.

Proof.
Fix $i$, $1\le i\le n$. If $p\in B$ let ${s}_{i}p$ be the element of the $i$-string of $p$ which satisfies
 $wt(sip) =siwt(p).$

Then ${s}_{i}\left({s}_{i}p\right)=p$ and $sichar(B) =∑p∈B Xsi wt(p) =∑p∈B Xwt(sip) =char(B).$ Hence $\mathrm{char}\left(B\right)\in {\mathrm{ℤ\left[P\right]}}^{W}$.

Let $ε= ∑w∈W det(w)w so that aμ =ε(Xμ), for μ∈P.$

Since $\mathrm{char}\left(B\right)\in {ℤ\left[P\right]}^{W}$, $char(B)aρ =char(B) ε(Xρ) =ε(char(B) Xρ)$ and $char(B) =1aρ char(B)aρ =ε(char(B) Xρ) aρ$ $=∑p∈B ε( Xwt(p)+ρ ) aρ =∑p∈B awt(p) +ρ aρ =∑p∈B swt(p).$

There is some cancellation which can occur in this sum. Assume $p\in B$ such that $p⊈C-\rho$ and let $t$ be the first time that $p$ leaves the cone $C-\rho$. In other words, let $t\in {ℝ}_{>0}$ be minimal such that there exists an $i$ with $p(t)∈ H αi,-1 where Hαi, -1 ={λ∈ 𝔥ℝ* | ⟨λ, αi∨ ⟩=-1}.$

Let $i$ be the minimal index such that the point $p\left(t\right)\in {H}_{{\alpha }_{i},-1}$ and define ${s}_{i}\circ p$ to be the element of the $i$-string of $p$ such that $wt(si∘p) =si∘p$

Note that since $⟨p\left(t\right),{\alpha }_{i}^{\vee }⟩=-1,p$ is not the head of its $i$-string and ${s}_{i}\circ p$ is well defined. If $q={s}_{i}\circ p$ then the first time $t$ that $q$ leaves the cone $C-\rho$ is the same as the first time that $p$ leaves the cone $C-\rho$ and $p\left(t\right)=q\left(t\right)$. Thus ${s}_{i}\circ q=p$ and ${s}_{i}\circ \left({s}_{i}\circ p\right)=p$. Since $swt(si∘ p) =ssi∘ wt(p) =-swt(p) ,$ the terms ${s}_{\mathrm{wt}\left({s}_{i}\circ p\right)}$ and ${s}_{\mathrm{wt}\left(p\right)}$ cancel in the sum in (5.29). Thus $char(B) =∑ p∈B p⊆C-ρ swt(p) . □$

Recall the notations for Weyl characters, tensor product multiplicities, restriction multiplicities and paths from (5.5), (5.11), (5.17), (5.22). For each $\lambda \in {P}^{+}$ fix a highest weight path ${p}_{\lambda }^{+}$ with endpoint $\lambda$ and let $B(λ) be the crystal generated by pλ+.$ Let $\lambda ,\mu ,\nu \in {P}^{+}$ and let $J\subseteq \left\{1,2,\dots ,n\right\}.$ Then $sλ =∑ p∈B(λ) Xwt(p), sμsν =∑ q∈B(ν) pμ+⊗q ⊆C-ρ sμ+ wt(q), and sλ =∑ p∈B(λ) p⊆CJ -ρJ swt(p)J .$

 Proof. (a) The path ${p}_{\lambda }^{+}$ is the unique ighest weight path in $B\left(\lambda \right)$. Thus, by Theorem 5.5, $\mathrm{char}\left(B\left(\lambda \right)\right)={s}_{\lambda }.$ (b) By the "Leibnitz formula" for the root operators in Theorem 5.8c, the set $B(μ)⊗ B(ν) ={p⊗q | p∈B(μ), q∈B(μ)}$ is a crystal. Since $\mathrm{wt}\left(p\otimes q\right)=\mathrm{wt}\left(p\right)+\mathrm{wt}\left(q\right),$ $sμsν = char(B(μ)) char(B(ν)) =char(B(μ) ⊗B(ν)) =∑ p⊗q∈ B(μ)⊗ B(ν) p⊗q⊆ C-ρ swt(p) +wt(q) =∑ q∈B(ν) pμ+⊗q ⊆C-ρ sμ +wt(q),$ where the third equality is from Theorem 5.5 and the last equality is because the path ${p}_{\mu }^{+}$ has $\mathrm{wt}\left({p}_{\mu }^{+}\right)=\mu$ and is the only highest weight path in $B\left(\mu \right).$ (c) A $J$-crystal is a set of paths $B$ which is closed under the operators ${\stackrel{\sim }{e}}_{i},$, for $j\in J$. Since ${s}_{\lambda }=\mathrm{char}\left(B\left(\lambda \right)\right)$ the statement applies by applying Theorem 5.5 to $B\left(\lambda \right)$ viewed as a $J$-crystal.$\square$

## Notes and References

These notes are intended to supplement various lecture series given by Arun Ram. This is a typed version of section 5.4 of the paper [Ram2006].

## References

[Ram2006] Alcove walks, Hecke algebras, Spherical functions, crystals and column strict tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (Special Issue: In honor of Robert MacPherson, Part 2 of 3) (2006) 963-1013. MR2282411