Highest weight paths

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 1 December 2012

Highest weight paths

A highest weight path is a path p such that

eip =0, for all 1in.
A highest weight path is a path p such that, for each 1in, p is the head of the i-string Si(p). Thus p(t), αi >-1 for all t and all 1in . So a path p is a highest weight path if and only if
pC-ρ, where C-ρ={μ-ρ | μC}.
Following the example at the end of Section 2, for the root system of the type C2 the picture is

Hα1 +α2 Hα1 Hα2 Hα1 +2α2 -ρ 0 C-ρ
the region C-ρ

If p is a highest weight path with wt(p)P then, necessarily, wt(p) P+. The following theorem gives an expression for the character of a crystal in terms of the basis {sλ | λP+} of [P]W.

Let B be a crystal. Let char(B) be as defined in (5.24) and sλ as in (5.25). Then

char(B) = pB pC-ρ swt(p),
where the sum is over highest weight paths pB.

Fix i, 1in. If pB let sip be the element of the i-string of p which satisfies
wt(sip) =siwt(p).

t h sip p Hαi

Then si (sip) =p and sichar(B) =pB Xsi wt(p) =pB Xwt(sip) =char(B). Hence char(B) [P]W .

Let ε= wW det(w)w so that aμ =ε(Xμ), for μP.

Since char(B) [P]W , char(B)aρ =char(B) ε(Xρ) =ε(char(B) Xρ) and char(B) =1aρ char(B)aρ =ε(char(B) Xρ) aρ =pB ε( Xwt(p)+ρ ) aρ =pB awt(p) +ρ aρ =pB swt(p).

There is some cancellation which can occur in this sum. Assume pB such that pC-ρ and let t be the first time that p leaves the cone C-ρ. In other words, let t >0 be minimal such that there exists an i with p(t) H αi,-1 where Hαi, -1 ={λ 𝔥* | λ, αi =-1}.

Let i be the minimal index such that the point p(t) Hαi, -1 and define sip to be the element of the i-string of p such that wt(sip) =sip

t h sip p Hαi, -1 Hαi

Note that since p(t) ,αi =-1,p is not the head of its i-string and sip is well defined. If q=sip then the first time t that q leaves the cone C-ρ is the same as the first time that p leaves the cone C-ρ and p(t) =q(t) . Thus siq =p and si( sip) =p. Since swt(si p) =ssi wt(p) =-swt(p) , the terms swt(si p) and swt(p) cancel in the sum in (5.29). Thus char(B) = pB pC-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 λP+ fix a highest weight path pλ+ with endpoint λ and let B(λ) be the crystal generated by pλ+. Let λ,μ,νP+ and let J{1,2, ,n}. Then sλ = pB(λ) Xwt(p), sμsν = qB(ν) pμ+q C-ρ sμ+ wt(q), and sλ = pB(λ) pCJ -ρJ swt(p)J .


(a) The path pλ+ is the unique ighest weight path in B(λ). Thus, by Theorem 5.5, char(B(λ)) =sλ.

(b) By the "Leibnitz formula" for the root operators in Theorem 5.8c, the set B(μ) B(ν) ={pq | pB(μ), qB(μ)} is a crystal. Since wt(pq) =wt(p) +wt(q), sμsν = char(B(μ)) char(B(ν)) =char(B(μ) B(ν)) = pq B(μ) B(ν) pq C-ρ swt(p) +wt(q) = qB(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 pμ+ has wt(pμ+) =μ and is the only highest weight path in B(μ).

(c) A J-crystal is a set of paths B which is closed under the operators ei, , for jJ. Since sλ =char(B(λ)) the statement applies by applying Theorem 5.5 to B(λ) viewed as a J-crystal.

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].


[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

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