Root operators

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

Last update: 28 February 2012

Root operators for the rank 1 case

Let Bk = {b1bk  |  biB}, where B= {+1,-1,0}. Define f˜: Bk Bk{0} and e˜: Bk Bk{0} as follows. Let b= b1bkBk. Ignoring 0s successively pair adjacent unpaired (+1,-1) pairs to obtain a sequence of unpaired -1s and +1s -1 -1 -1 -1 -1 -1 -1  +1 +1 +1 +1 (after pairing and ignoring 0s). Then f˜b = same as  b  except the leftmost unpaired  +1  is changed to  -1, e˜b = same as  b  except the rightmost unpaired  -1  is changed to  +1. (Ro 30) If there is no unpaired  +1  after pairing then   f˜b=0. If there is no unpaired  -1  after pairing then   e˜b=0.

These operators coincide with the operators used in the type A case by Lascoux and Schützenberger [LS] (see the nice exposition in [Ki]). The (+1,-1) 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 𝔥*=. By identifying +1, -1, 0 with the straight line paths p1 p-1 p0 p1: [0,1]𝔥* p-1: [0,1]𝔥* p0: [0,1]𝔥* tt, t-t, t0, respectively, the set Bk is viewed as a set of maps p:[0,k] 𝔥*. Let B0 ={φ} with f˜φ=0 and e˜φ=0. Then T(B) = k0 Bk (Ro 31)

Hαi,-1 Hαi t sip p h
Note that since pi(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 and p(t) = q(t). Thus siq=p and si (sip) =p. Since swt(sip) = ssiwt(p) = -swt(p) the terms swt(sip) and swt(p) cancel in the sum in (5.29). Thus char(B) = pB pC-ρ swt(p).

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 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+qC-ρ sμ+wt(q), and sλ = pB(λ) pCJ-ρJ swt(p)J.

  1. The path pλ+ is the unique highest weight path in B(λ). Thus, by Theorem 5.5, char(B(λ))=sλ.
  2. By the "Leibnitz formula" for the root operators in Theorem 2.1c the set B(μ)B(ν) = {pq  |  pB(μ),  qB(ν)} is a crystal. Since wt(pq) = wt(p)+wt(q), sμsν = char(B(μ)) char(B(ν)) = char((B(μ)B(ν))) = pqB(μ)B(ν) pqC-ρ s wt(p)+wt(q) = qB(ν) pμ+qC-ρ s μ+wt(q), is a set of paths closed under products, reverses and r-stretches (r0) and the action of e˜ and f˜. For pB let d+(p) = (number of unpaired +1s after pairing), d-(p) = (number of unpaired -1s after pairing). (Ro 32) See the picture in [Ro 33]. These are the nonnegative integers such that f˜d+(p)p0 and f˜d+(p)+1p=0, and e˜d-(p)p0 and e˜d-(p)+1p=0.

Use notations for T(B) as in [Ro 30]-[Ro 32].

  1. If pT(B) and f˜p0 then e˜f˜p =p. If pT(B) and e˜p0 then f˜e˜p =p.
  2. If pT(B) and r0 then f˜r(rp) = r(f˜p) and e˜r(rp) = r(e˜p).
  3. If p,qT(B) then f˜(pq) = { f˜pq, if   d+(p) >d-(q), pf˜q, if   d+(p) d-(q), } and e˜(pq) = { e˜pq, if   d+(p) d-(q), pe˜q, if   d+(p) <d-(q). }
  4. If pT(B) then f˜(p*) = (e˜p)* and e˜(p*) = (f˜p)*.

(a), (b) and (d) are direct consequences of the definition of the operators e˜ and f˜ and the definitions of r-stretching and reversing.

(c) View p and q as paths. After pairing, p and q have the form p= d+(p) d-(p) and q= d+(p) d-(p) (Ro 33) where the left traveling portion of the path corresponds to -1s and the right traveling portion of the path corresponds to +1s. Then f˜i (p1p2) = { f˜pq, if   pq = ,   i.e.   d+(p) > d-(q), pf˜q, if   pq = ,  i.e.   d+(p) d-(q), since in the first case, the leftmost unpaired +1 is from p and, in the second case, it is from q.

Use property (b) in Proposition 1.2 to extend the operators e˜ and f˜ to operators on T(B), the set of maps p:[0,l] generated by B under the operations of concatenation, reversing and r-stretching (r0). Then, by completion, the operators e˜ and f˜ extend to operators on T(B), the set of maps   p:[0,l]   generated by  B  by operations of concatenation, reversing and  r-stretching   (r0). (Ro 34) A rank 1 path is an element of T(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] 𝔥* in Buniv (see (5.23)).

Let p:[0,l] be a path and let αR+ be a positive root. The map pα:[0,l] given by pα(t) = p(t),α (Ro 35) is a rank 1 path (an element of T(B)). The rank 1 path pα is the projection of p onto the line perpendicular to the hyperplane Hα. Define operators e˜α: BunivBuniv{0} and f˜α: BunivBuniv{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   1in. (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˜.

Hαi p Hαi di+(p) di-(p) pαi Hαi f˜ip Hαi f˜pαi
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

  1. If   pBuniv   and   f˜ip0   then   e˜if˜ip=p. If   pBuniv   and   e˜ip0   then   f˜ie˜ip=p.
  2. If λP and λ,αi>0 then f˜iλ,αi pλ = psiλ.
  3. If p,qBuniv then f˜i(pq) = { f˜ipq, if   di+(p) > di-(q), pf˜iq, if   di+(p) di-(q), } and e˜i(pq) = { e˜ipq, if   di+(p) di-(q), pe˜iq, if   di+(p) < di-(q), } and where di±(p) = d±(pαi) with d± as in (Ro 33) and pαi as in (Ro 35).
  4. If pBuniv and r0 then f˜ir(rp) = r(f˜ip) and e˜ir(rp) = r(e˜ip).
  5. If pBuniv then f˜ip* = (e˜ip)* and e˜ip* = (f˜ip)*.
  6. If   pBuniv   and   f˜ip0   then   wt(f˜ip) = wt(p)-αi. If   pBuniv   and   e˜ip0   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.


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[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.

[Ha] T. Haines, Structure constants for Hecke and representation rings, Int. Math. Res. Not. 39 (2003), 2103-2119.

[IM] N. Iwahori and H. Hatsumoto, On some Bruhat decomposition and the structure of the Hecke rings of 𝔭-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5-48.

[Ki] K. Killpatrick, A Combinatorial Proof of a Recursion for the q-Kostka Polynomials, J. Comb. Th. Ser. A 92 (2000), 29-53.

[KM] M. Kapovich and J.J. Millson, A path model for geodesics in Euclidean buildings and its applications to representation theory, arXiv: math.RT/0411182.

[LS] A. Lascoux and M.P. Schützenberger, Le monoïde plaxique, Quad. Ricerce Sci. 109 (1981), 129-156.

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