Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu
Last updates: 28 January 2009
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≤i≤n,p is the head of the i-string Sip. Thus ptαi∨>-1 for all t and all 1≤i≤n.
So a path p is a highest weight path iff
p⊆C-ρ,whereC-ρ=μ-ρ|μ∈C.
Following the example at the end of Section 2, for the root system of the type C2 the picture is
the region C-ρ
If p is a highest weight path with wtp∈P then, necessarily, wtp∈P+.
The following theorem gives an expression for the character of a crystal in terms of the basis sλ|λ∈P+ of ℤPW.
Let B be a crystal. Let charB be as defined in 'paths and i-strings' and sλ as in 'Schur functions'. Then
charB=∑p∈B,p⊆C-ρswtp,
wherer the sum is over highest weight paths p∈B.
Proof.
Fix i,1≤i≤n .
If p∈B let sip be the element of the -string of p which satisfies
wtsip=siwtp.
Then sisip=p and
sicharB=∑p∈BXsiwtp=∑p∈BXwtsip=charB.
Hence charB∈ℤPW
Let
ε=∑w∈Wdetwwso thataμ=εXμ,forμ∈P
Since charB∈ℤPW,charBaρ=charBεXρ=εcharBXρ
and
charB=1aρcharBaρ=εcharBXρaρ
=∑p∈BεXwtp+ρaρ=∑p∈Bawtp+ρaρ=∑p∈Bswtp.
There is some cancellation which can occur in this sum. Assume p∈B such that p⊈C-ρ and let t be the first time that p leaves the cone C-ρ.
In other words, let t∈ℝ>0 be minimal such that there exists an i with
pt∈Hαi,-1 where Hαi,-1=λ∈𝔥ℝ*|λαi∨=-1.
Let i be the minimal index such that the point pt∈Hαi,-1 and define si∘p to be the element of the i-string if p such that
wtsi∘p=si∘p
Note that since pitαi∨=-1,p is not the head of its i-string and si∘p is well defined. If q=si∘p 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 pt=qt.
Thus
si∘q=p and si∘si∘p=p Since
swtsi∘p=ssi∘wtp=swtp,
the terms ssi∘wtp and swtp cancel in the sum in (3). Thus
charB=∑p∈B,p⊆C-ρswtp.□
Recall the notations for Weyl characters, tensor product multiplicities, restriction multiplicities and paths from ELSEWHERE. For each λ∈P+ fix a highest weight path pλ+ with endpoint λ and let
Bλ be the crystal generated bypλ+.
Let λ,μ,ν∈P+ and let J⊆12...n.
Then
sλ=∑p∈BλXwtp,sμsν=∑q∈Bν,pμ+⊗q⊆C-ρsμ+wtq,and sλ=∑p∈Bλ,p⊆Cj-ρJswtpJ.
Proof.
a) The path pλ+ is the unique ighest weight path in Bλ.
Thus, by the previous theorem, charBλ=sλ.
b) By the "Leibnitz formula" for the root operators in Theorem ELSEWHERE, the set
Bμ⊗Bν=p⊗q|p∈Bμ,q∈Bν is a crystal. Since wtp⊗q=wtp+wtq,sμsν=charBμcharBν=charBμ⊗Bν=∑p⊗q∈Bμ⊗Bν,p⊗q⊆C-ρswtp+wtq=∑q∈Bν,pμ+⊗q⊆C-ρsμ+wtq,
where the third equality is from the previous theorem and the last equality is because the path pμ+ has wtpμ+=μ is the only highest weight path in Bμ.
c) A J-crystal is a set of paths B which is closed under the operators e∼i, for j∈J.
Since sλ=charBλ the statement applies by applying the previous theorem to Bλ viewed as a J- crystal.□
