Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 28 February 2012
Root operators for the rank 1 case
Let
Define
as follows. Let
Ignoring 0s successively pair adjacent unpaired
pairs to obtain a sequence of unpaired and
(after pairing and ignoring 0s). Then
These operators coincide with the operators used in the type A case by Lascoux and Schützenberger [LS] (see the nice exposition in [Ki]). The
pairing procedure is equivalent to the process of taking the "outer edge" of the path (Psf 19)-(Psf 20). In the context of Section 4 this is natural since only the outer edge of the path contributes nontrivially to the image of the path in the affine Hecke algebra.
Let
By identifying with the straight line paths
respectively, the set is viewed as a set of maps
Let
with
and
Then
Note that since
is not the head of its string and
is well defined. If
then the first time that leaves the cone and
Thus
and
Since
the terms
and
cancel in the sum in (5.29). Thus
Recall the notation for Weyl characters, tensor products multiplicities, restriction multiplicities and paths from (5.5), (5.11), (5.17) and (5.22). For each
fix a highest weight path with endpoint and let
Let
and let
Then
Proof.
The path is the unique highest weight path in Thus, by Theorem 5.5,
By the "Leibnitz formula" for the root operators in Theorem 2.1c the set
is a crystal. Since
is a set of paths closed under products, reverses and stretches
and the action of and For let
See the picture in [Ro 33]. These are the nonnegative integers such that
(a), (b) and (d) are direct consequences of the definition of the operators and and the definitions of stretching and reversing.
(c) View and as paths. After pairing, and have the form
where the left traveling portion of the path corresponds to and the right traveling portion of the path corresponds to Then
since in the first case, the leftmost unpaired is from and, in the second case, it is from
Use property (b) in Proposition 1.2 to extend the operators and to operators on
the set of maps
generated by under the operations of concatenation, reversing and stretching
Then, by completion, the operators and extend to operators on
A rank 1 path is an element of
Tℝ(B).
The root operators in the general case
Recall from (5.23) that
Buniv
is the set of maps generated by the straight line paths
by operations of concatenation, reversing and stretching,
and a path is an element
p:[0,l]→𝔥ℝ*
in Buniv (see (5.23)).
Let
p:[0,l]→ℝ
be a path and let α∈R+ be a positive root. The map
pα:[0,l]→ℝ given by pα(t)=⟨p(t),α∨⟩ (Ro 35)
is a rank 1 path (an element of Tℝ(B)). The rank 1 path pα is the projection of p onto the line perpendicular to the hyperplane Hα. Define operators
e˜α:Buniv→Buniv∪{0} and f˜α:Buniv→Buniv∪{0} (Ro 36)
by
e˜αp=p+12(e˜pα-pα)α and f˜αp=p-12(pα-f˜pα)α, (Ro 37)
and set
e˜i=e˜αi and f˜i=f˜αi, for
1≤i≤n. (Ro 38)
The operators e˜i and f˜i are designed so that after projection onto the line perpendicular to Hαi they are the operators e˜ and f˜.
The dark parts of that path p are reflected (in a mirror parallel to Hαi) to form the path f˜ip. The left dotted line is the affine hyperplane parallel to Hαi which intersects the path p at its leftmost (most negative) point (relative to Hαi) and the distance between the dotted lines is exactly the distance between lines of lattice points in P parallel to Hαi.
The following theorem is a consequence of Proposition 1.2 and the definition in [Ro 34]. The uniqueness of the operators f˜i and e˜i is forced by the properties (b), (c) and (d) in Theorem 2.1.
The operators e˜i and f˜i defined in (Ro 38) are the unique operators such that
If p∈Buniv and
f˜ip≠0 then
e˜if˜ip=p.
If p∈Buniv and
e˜ip≠0 then
f˜ie˜ip=p.
If λ∈P and
⟨λ,αi∨⟩∈ℤ>0
then
f˜i⟨λ,αi∨⟩pλ=psiλ.
If p,q∈Buniv then
f˜i(p⊗q)={f˜ip⊗q,
if
di+(p)>di-(q),p⊗f˜iq,
if
di+(p)≤di-(q),} and
e˜i(p⊗q)={e˜ip⊗q,
if
di+(p)≥di-(q),p⊗e˜iq,
if
di+(p)<di-(q),}and
where
di±(p)=d±(pαi)
with d± as in (Ro 33) and pαi as in (Ro 35).
If p∈Buniv and r∈ℤ≥0 then
f˜ir(rp)=r(f˜ip) and e˜ir(rp)=r(e˜ip).
If p∈Buniv then
f˜ip*=(e˜ip)* and e˜ip*=(f˜ip)*.
If p∈Buniv and
f˜ip≠0 then
wt(f˜ip)=wt(p)-αi.
If p∈Buniv and
e˜ip≠0 then
wt(e˜ip)=wt(p)+αi.
Notes and References
The above notes are taken from section 5 of the paper
[Ram]
A. Ram,
Alcove Walks, Hecke Algebras, Spherical Functions, Crystals and Column Strict Tableaux, Pure and Applied Mathematics Quarterly 2 no. 4 (2006), 963-1013.
specifically sections 5.5 and 5.6.
References
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Y. Billig and M. Dyer,
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M. Brion,
Positivity in the Grothendieck group of complex flag varieties, J. Algebra 258 no. 1 (2002), 137-159.
[GL]
S. Gaussent and P. Littelmann,
LS galleries, the path model, and MV cycles, Duke Math. J. 127 no. 1 (2005), 35-88.
[GR]
S. Griffeth and A. Ram,
Affine Hecke algebras and the Schubert calculus, European J. Combin. 25 no. 8 (2004), 1263-1283.
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Structure constants for Hecke and representation rings, Int. Math. Res. Not. 39 (2003), 2103-2119.
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A Combinatorial Proof of a Recursion for the q-Kostka Polynomials, J. Comb. Th. Ser. A 92 (2000), 29-53.
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M. Kapovich and J.J. Millson,
A path model for geodesics in Euclidean buildings and its applications to representation theory, arXiv: math.RT/0411182.
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Le monoïde plaxique, Quad. Ricerce Sci. 109 (1981), 129-156.