## Root operators

Last update: 28 February 2012

## Root operators for the rank 1 case

Let Define $f˜: B⊗k→ B⊗k∪{0} and e˜: B⊗k→ B⊗k∪{0}$ as follows. Let $b={b}_{1}\otimes \cdots \otimes {b}_{k}\in {B}^{\otimes k}.$ Ignoring 0s successively pair adjacent unpaired $\left(+1,-1\right)$ pairs to obtain a sequence of unpaired $-1s$ and $+1s$ (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 $\left(+1,-1\right)$ 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 ${𝔥}_{ℝ}^{*}=ℝ.$ By identifying with the straight line paths $p1 p-1 p0 p1: [0,1]→𝔥ℝ* p-1: [0,1]→𝔥ℝ* p0: [0,1]→𝔥ℝ* t↦t, t↦-t, t↦0,$ respectively, the set ${B}^{\otimes k}$ is viewed as a set of maps $p:\left[0,k\right]\to {𝔥}_{ℝ}^{*}.$ Let ${B}^{\otimes 0}=\left\{\phi \right\}$ with $\stackrel{˜}{f}\phi =0$ and $\stackrel{˜}{e}\phi =0.$ Then $Tℤ(B) = ⨆ k∈ℤ≥0 B⊗k (Ro 31)$

Note that since 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 $C-\rho$ 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 notation for Weyl characters, tensor products multiplicities, restriction multiplicities and paths from (5.5), (5.11), (5.17) and (5.22). For each $\lambda \in {P}^{+}$ fix a highest weight path ${p}_{\lambda }^{+}$ with endpoint $\lambda$ and let Let and let $J\subseteq \left\{1,2,...,n\right\}.$ 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. The path ${p}_{\lambda }^{+}$ is the unique highest weight path in $B\left(\lambda \right).$ Thus, by Theorem 5.5, $\mathrm{char}\left(B\left(\lambda \right)\right)={s}_{\lambda }.$ By the "Leibnitz formula" for the root operators in Theorem 2.1c the set is a crystal. Since $\mathrm{wt}\left(p\otimes q\right)=\mathrm{wt}\left(p\right)+\mathrm{wt}\left(q\right),$ $sμsν = char(B(μ)) char(B(ν)) = char((B(μ)⊗B(ν))) = ∑ p⊗q∈B(μ)⊗B(ν) p⊗q⊆C-ρ s wt(p)+wt(q) = ∑ q∈B(ν) pμ+⊗q⊆C-ρ s μ+wt(q),$ is a set of paths closed under products, reverses and $r-$stretches $\left(r\in {ℤ}_{\ge 0}\right)$ and the action of $\stackrel{˜}{e}$ and $\stackrel{˜}{f}.$ For $p\in B$ 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 $f˜d+(p)p≠0 and f˜d+(p)+1p=0, and e˜d-(p)p≠0 and e˜d-(p)+1p=0.$ $\square$

Use notations for ${T}_{ℤ}\left(B\right)$ as in [Ro 30]-[Ro 32].

1. $\begin{array}{l}\\ \\ \\ \\ \\ Ifp\in {T}_{ℤ}\left(B\right)and\stackrel{˜}{f}p\ne 0then\stackrel{˜}{e}\stackrel{˜}{f}p=p.\\ Ifp\in {T}_{ℤ}\left(B\right)and\stackrel{˜}{e}p\ne 0then\stackrel{˜}{f}\stackrel{˜}{e}p=p.\end{array}$
2. If $p\in {T}_{ℤ}\left(B\right)$ and $r\in {ℤ}_{\ge 0}$ then $f˜r(rp) = r(f˜p) and e˜r(rp) = r(e˜p).$
3. If $p,q\in {T}_{ℤ}\left(B\right)$ then and
4. If $p\in {T}_{ℤ}\left(B\right)$ then $f˜(p*) = (e˜p)* and e˜(p*) = (f˜p)*.$

 Proof. (a), (b) and (d) are direct consequences of the definition of the operators $\stackrel{˜}{e}$ and $\stackrel{˜}{f}$ 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 since in the first case, the leftmost unpaired $+1$ is from $p$ and, in the second case, it is from $q.$ $\square$

Use property (b) in Proposition 1.2 to extend the operators $\stackrel{˜}{e}$ and $\stackrel{˜}{f}$ to operators on ${T}_{ℚ}\left(B\right),$ the set of maps $p:\left[0,l\right]\to ℝ$ generated by $B$ under the operations of concatenation, reversing and $r-$stretching $\left(r\in {ℚ}_{\ge 0}\right).$ Then, by completion, the operators $\stackrel{˜}{e}$ and $\stackrel{˜}{f}$ extend to operators on A rank 1 path is an element of ${T}_{ℝ}\left(B\right).$

## 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 $p:\left[0,l\right]\to {𝔥}_{ℝ}^{*}$ in ${B}_{\mathrm{univ}}$ (see (5.23)).

Let $p:\left[0,l\right]\to ℝ$ be a path and let $\alpha \in {R}^{+}$ 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 ${T}_{ℝ}\left(B\right)$). The rank 1 path ${p}_{\alpha }$ is the projection of $p$ onto the line perpendicular to the hyperplane ${H}_{\alpha }.$ 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 The operators ${\stackrel{˜}{e}}_{i}$ and ${\stackrel{˜}{f}}_{i}$ are designed so that after projection onto the line perpendicular to ${H}_{{\alpha }_{i}}$ they are the operators $\stackrel{˜}{e}$ and $\stackrel{˜}{f}.$

The dark parts of that path $p$ are reflected (in a mirror parallel to ${H}_{{\alpha }_{i}}$) to form the path ${\stackrel{˜}{f}}_{i}p.$ The left dotted line is the affine hyperplane parallel to ${H}_{{\alpha }_{i}}$ which intersects the path $p$ at its leftmost (most negative) point (relative to ${H}_{{\alpha }_{i}}$) and the distance between the dotted lines is exactly the distance between lines of lattice points in $P$ parallel to ${H}_{{\alpha }_{i}}.$

The following theorem is a consequence of Proposition 1.2 and the definition in [Ro 34]. The uniqueness of the operators ${\stackrel{˜}{f}}_{i}$ and ${\stackrel{˜}{e}}_{i}$ is forced by the properties (b), (c) and (d) in Theorem 2.1.

The operators ${\stackrel{˜}{e}}_{i}$ and ${\stackrel{˜}{f}}_{i}$ defined in (Ro 38) are the unique operators such that

1. If $\lambda \in P$ and $⟨\lambda ,{\alpha }_{i}^{\vee }⟩\in {ℤ}_{>0}$ then $f˜i⟨λ,αi∨⟩ pλ = psiλ.$
2. If $p,q\in {B}_{\mathrm{univ}}$ then where ${d}_{i}^{±}\left(p\right)={d}^{±}\left({p}_{{\alpha }_{i}}\right)$ with ${d}^{±}$ as in (Ro 33) and ${p}_{{\alpha }_{i}}$ as in (Ro 35).
3. If $p\in {B}_{\mathrm{univ}}$ and $r\in {ℤ}_{\ge 0}$ then $f˜ir(rp) = r(f˜ip) and e˜ir(rp) = r(e˜ip).$
4. If $p\in {B}_{\mathrm{univ}}$ then $f˜ip* = (e˜ip)* and e˜ip* = (f˜ip)*.$

## 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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