## Polynomials and symmetric functions

Last update: 25 February 2012

## Satake isomorphism

Recall, from (3.6) of [Ram], that the finite Hecke algebra is the subalgebra of the affine Hecke algebra $\stackrel{˜}{H}$ given by Let ${\mathbf{1}}_{0}$ be the element of $H$ given by $102 =10 and Tw-1-1 10= q-l(w)10, (Psf 1)$ for $w\in W.$ Two explicit formulas for ${\mathbf{1}}_{0}$ are $10= 1 W0(q-2) ∑ w∈W q-l(w) Tw-1-1= 1 W0(q2) ∑ w∈W ql(w) Tw,$ where ${W}_{0}\left(t\right)=\sum _{w\in W}{t}^{l\left(w\right)}$ is the Poincaré polynomial of $W$.

As observed in (3.6), is a subalgebra of $\stackrel{˜}{H}.$ The vector space $𝕂\left[P\right]$ also sits inside $\stackrel{˜}{H}$ in a different way. Since is a basis of $\stackrel{˜}{H}{\mathbf{1}}_{0}$ there is $a vector space isomorphism 𝕂[P] → H˜10 f ↦ f10. (Psf 2)$

The ring of symmetric functions is By a theorem of Bernstein (see [NR, Theorem 1.4]) this subalgebra of $\stackrel{˜}{H}$ is the center of $\stackrel{˜}{H},$ $𝕂[P]W = Z(H˜). (Psf 4)$ The spherical Hecke algebra is the ring ${\mathbf{1}}_{0}{X}^{\lambda }{\mathbf{1}}_{0}$ and the restriction of the map (Psf 2) to $Z\left(\stackrel{˜}{H}\right)$ is the Satake isomorphism of the following theorem.

Let ${\mathbf{1}}_{0}$ and $𝕂{\left[P\right]}^{W}$ be as in (Psf 1) and (Psf 3), respectively. Then

 Proof. The map is a well defined homomorphism since, if $f,{f}_{1},{f}_{2} \in Z\left(\stackrel{˜}{H}\right)$ then $f{\mathbf{1}}_{0}=f{{\mathbf{1}}_{0}}^{2}={\mathbf{1}}_{0}f{\mathbf{1}}_{0}$ and ${f}_{1}{f}_{2}{\mathbf{1}}_{0}={f}_{1}{f}_{2}{{\mathbf{1}}_{0}}^{2}={f}_{1}{\mathbf{1}}_{0}{f}_{2}{\mathbf{1}}_{0}.$ Suppose that $⟨\lambda ,{\alpha }_{i}^{\vee }⟩>0$ so that $\lambda$ is on the positive side of ${H}_{{\alpha }_{i}}.$ Then, by Proposition 3.2e, $10 Xsiλ 10 = q-110 TsiXλ 10 = q-110Xλ Tsi10 - q-1 (q-q-1)10 (Zsiλ+αi +⋯+ Xλ-αi+Xλ) 10 = 10Xλ 10 - (1-q-2)10 (Xsiλ+αi +⋯+ Xλ-αi+Xλ) 10.$ so that $10 Xsiλ 10 = q-210Xλ10 - (1-q-2)10 (Xsiλ+αi +⋯+ Xλ-αi)10 (Psf 5)$ or, equivalently, $10 (Xsiλ +⋯+ Xλ-αi)10 = q-210 (Xsiλ+αi +⋯+ Xλ) 10. (Psf 6)$ For the relation in (Psf 5), $10Xsi10 - 10Xsiλ+αi10 = q-210Xλ10 - (1-q-2)10 (Xsiλ+αi +⋯+ Xλ-αi)10 -q-210Xλ-αi10 + (1-q-2)10 (Xsiλ+2αi +⋯+ Xλ-2αi) 10,$ so that $10Xsiλ10 = q-210Xλ10 + q-210Xsiλ+αi10 - 10Xλ-αi10. (Psf 7)$ It follows from these relations that any element of ${\mathbf{1}}_{0}\stackrel{˜}{H}{\mathbf{1}}_{0}$ can, inductively, be written as a linear combination of the elements Using Theorem 3.3 to expand ${\mathbf{1}}_{0}{X}^{\lambda }$ in terms of the basis produces $10Xλ = Xw0λTw0-1 + ∑ μ>w0λ dμ,vXμTv-1-1,$ and, since these leading terms are all different (as $\lambda$ runs over ${P}^{+}$), it follows that The orbit sums $mλ = ∑ γ∈Wλ Xγ, λ∈P+, (Psf 9)$ for a basis of $𝕂{\left[P\right]}^{W}.$ The relation in Proposition 3.2e implies that, if $f\in 𝕂{\left[P\right]}^{W}$ then ${T}_{w}f=f{T}_{w}$ for all $w\in W,$ and so the ${m}_{\lambda }{\mathbf{1}}_{0}={\mathbf{1}}_{0}{m}_{\lambda }{\mathbf{1}}_{0}$ are in ${\mathbf{1}}_{0}\stackrel{˜}{H}{\mathbf{1}}_{0}.$ Viewing these in terms of the basis of $\stackrel{˜}{H}$ one sees that the are linearly independent and so The point is that the transition matrix between the basis in (Psf 8) and the basis in (Psf 10) is triangular. $\square$

## Hall-Littlewood polynomials

For $\mu \in P$ let ${W}_{\mu }=\mathrm{Stab}\left(\mu \right)$ be the stabilizer of $\mu .$ The Poincaré polynomial of ${W}_{\mu }$ is $Wμ(t) = ∑ w∈Wμ tl(w). (Psf 11)$ For $\mu \in P,$ the Hall-Littlewood polynomial or Macdonald spherical function ${P}_{\mu }\left(X;t\right)$ is the element of $𝕂{\left[P\right]}^{W}$ defined by $Pμ (X;q-2)10 = ( ∑ w∈Wμ q-l(w) Tw-1-1) Xμ10, (Psf 12)$ where ${W}^{\mu }$ is a set of minimal length coset representatives for the cosets in $W/{W}_{\mu }.$ Since every element $w\in W$ has a unique expression $w=uv$ with $u\in {W}^{\mu }$ and $v\in {W}_{\mu },$ $∑ w∈Wμ q-l(w) Tw-1-1 Xμ10 = 1 Wμ(q-2) ( ∑ u∈Wμ q-l(u) Tu-1-1) Xμ ( ∑ v∈Wμ q-l(v) Tv-1-1) 10 = 1 Wμ(q-2) ( ∑ u∈Wμ q-l(u) Tu-1-1) ( ∑ v∈Wμ q-l(v) Tv-1-1) Xμ10 = W0(q-2) Wμ(q-2) 10Xμ10,$ and hence $Pμ (X,q-2)10 is exactly 10Xμ10 except normalised$ so that the coefficient of ${X}^{\mu }{\mathbf{1}}_{0}$ is 1.

Macdonald's formula for the spherical functions on a $p-$adic group [Mac1, Theorem 4.1.2] is $Pμ (X;q-2) = 1 Wμ(q-2) ∑ w∈W w(Xμ ∏ α∈R+ 1-q-2X-α 1-X-α ). (Psf 13)$ See [NR, Theorem 2.9a] for a proof in this context.

The following theorem gives additional formulas for ${P}_{\lambda }\left(X;{q}^{-2}\right).$ $A positively folded alcove walk is an alcove walk with no negative folds. (Psf 14)$ In the following theorem we shall consider alcove walks which do not necessarily begin at $A.$ This is the natural way to account for the sum over ${W}^{\lambda }$ which appears in the definition of ${P}_{\lambda }$ is (Psf 12). The type of $p$ is the sequence of labels of the folds and the wall crossings of $p.$

For $\lambda \in {P}^{+}$ let ${t}_{\lambda }\in \stackrel{˜}{W}$ be the translation in $\lambda$ and let ${n}_{\lambda }$ be the maximal length element in the double coset $W{t}_{\lambda }W.$

1. [Sc, Theorem 1.1] Let $\lambda \in {P}^{+}$ and fix a minimal length walk ${p}_{\lambda }={c}_{{i}_{1}}^{+}\cdots {c}_{{i}_{l}}^{+}$ from $A$ to $\lambda +A.$ Let Then $Pλ(X;q-2) = ∑ p∈Bq(pλ) q -( l(ι(p))+l(φ(p))-f(p) ) (1-q-2)f(p) Xwt(p),$ where $\iota \left(p\right)$ is the alcove where $p$ begins, $\mathrm{wt}\left(p\right)+\phi \left(p\right)A$ is the alcove where $p$ ends, and $f\left(p\right)$ is the number of folds in $p.$
2. Let $\lambda \in {P}^{+}.$ Then $ql(w0) W0(q-2) Pλ10 = ∑ x∈WtλW ql(x)-l(nλ) Tx.$

 Proof. The proof is accomplished by using Theorem 3.3 to expand the sum in (Psf 12). Since all crossings in the walk ${p}_{\lambda }$ are positive crossings Theorem 3.3 gives $∑ w∈Wλ q-l(w) Tw-1-1 Xλ = ∑ w∈Wλ q-l(w) ∑ p∈Bq(pλ) ι(p)=w (q-q-1)f(p) Xwt(p) Tφ(p)-1.$ Hence $Pλ10 = ∑ p∈Bq(λ) q-l(ι(p)) (q-q-1)f(p) Xwt(p) q-l(φ(p)) 10.$ Let $\lambda \in {P}^{+}.$ Let ${W}_{\lambda }=\mathrm{Stab}\left(\lambda \right)$ and let ${w}_{0}$ and ${w}_{\lambda }$ be the maximal length elements in $W$ and ${W}_{\lambda },$ respectively. Let ${m}_{\lambda }$ and ${n}_{\lambda }$ be the minimal and maximal length elements respectively in the double coset $W{t}_{\lambda }W.$ If $\lambda =2{\omega }_{2}$ in type ${C}_{2},$ then and $l\left({n}_{\lambda }\right)=10.$ Labeling the alcove $wA$ by the element $w,$ the 32 alcoves $wA$ with $w\in W{t}_{\lambda }W$ make up the four shaded diamonds. $Hα1+α2$ $Hα1$ $Hα2$ $Hα1+2α2$ $tλ$ $mλ$ $wλ$ $w0$ $nλ$ Then $ql(w0) W0(q-2) Pλ10 = ql(w0) W0(q-2) q-2l(w0)+2l(wλ) ∑ u∈Wλ ql(u) Tu Xλ10 = ql(w0) q-2l(w0) W0(q2) q-2l(w0)+2l(wλ) ∑ u∈Wλ ql(u) Tu Tmλ Tw0wλ10 = q-3l(w0)+2l(wλ) ∑ u∈Wλ ql(u) Tu Tmλ ql(w0)-l(wλ) ∑ w∈W ql(w) Tw = q-2l(w0)+l(wλ) ∑ u∈Wλ w∈W ql(u)+l(w) Tumλw = q-2l(w0)+l(wλ) ∑ x∈WtλW ql(x)-l(mλ) Tx ,$ and the result follows from the identity $l\left({n}_{\lambda }\right)=l\left({w}_{0}\right)+l\left({t}_{\lambda }\right)=l\left({w}_{0}\right)+l\left({w}_{0}\right)-l\left({w}_{\lambda }\right)+l\left({m}_{\lambda }\right).$ $\square$

The set ${B}_{q}\left({p}_{\lambda }\right)$ appearing in Theorem 2.1a is heavily dependent on the choice of ${p}_{\lambda }.$ One way to make this seem less dependent on this choice is as follows. For convenience assume that $\lambda$ is regular (not on any wall). Similar definitions can be made in the general case. Consider the region This region is a union of alcoves and any minimal length walk ${p}_{\lambda }$ from $A$ to $\lambda +{w}_{0}A$ lies in $\left[\lambda \right].$ Foldings of the walk ${p}_{\lambda }$ are then produced by folding the region $\left[\lambda \right]$ along the "creases" formed by the hyperplanes. This process produces a bijection between the paths in ${B}_{q}\left({p}_{\lambda }\right)$ and the set ${B}_{q}\left(\left[\lambda \right]\right)$ of "positively folded foldings" of the region $\left[\lambda \right]$, and the set ${B}_{q}\left(\left[\lambda \right]\right)does not depend on the choice of the initial path{p}_{\lambda }is to remember all the possible initial paths all at once. This translation of foldings was explained to me by J. Ramagge in Fall 2000.$

Let ${B}_{q}\left({p}_{\lambda }\right)$ be as in Theorem 2.1a and let $p\in {B}_{q}\left({p}_{\lambda }\right).$ Suppose that $p$ has $f$ folds. For $0\le i\le f,$ let ${p}^{\left(i\right)}$ be the positively folded walk in ${B}_{q}\left({p}_{\lambda }\right)$ which coincides with $p$ up to the $i$th fold and is nonfolded thereafter. Then ${p}^{\left(0\right)},...,{p}^{\left(f\right)}$ is a sequence of positively folded walks such that $p(f)=p, ι(p(i))=ι(p), φ(p(0))=ι(p), and φ(p(i))=sαφ(p(i-1))$ if the $i$th fold is on the hyperplane ${H}_{\alpha ,j}.$ Since $\phi \left({p}^{\left(i-1\right)}\right)>\phi \left({p}^{\left(i\right)}\right)$ and $(-1)l(φ(p(i))) = (-1)l(sα) (-1)φ(p(i-1)) = (-1) (-1)φ(p(i-1)),$ $l\left(\phi \left({p}^{\left(i-1\right)}\right)\right)-l\left(\phi \left({p}^{\left(i\right)}\right)\right)-1$ is an even integer $\ge 0.$ It follows that $l(ι(p))+ l(φ(p))- f(p) = l(ι(p))- lφp(1)))-1 +lφ(p(1)))- l(φ(p(2)))-1 + l(φ(p(2)))- l(φ(p(3)))-1 +⋯+ l(φ(p(f-1)))- l(φ(p(f)))-1+ 2l(φ(p))$ is an even integer $\ge 0.$ This proves that $f\left(p\right)\le l\left(\iota \left(p\right)\right)-l\left(\phi \left(p\right)\right)$ and that ${P}_{\lambda }\left(X;{q}^{-2}\right)$ really is a polynomial in the variable ${q}^{-2}.$

## Demazure operators

The group $W$ acts on by For each $1\le i\le n,$ define Demazure operators $Δi: 𝕂[P]→𝕂[P] and Δ˜i: 𝕂[P]→𝕂[P]$ by $Δif = 1 1-X-αi (1-si)f and Δ˜if = 1 Xαi-1 (Xαi-si)f, (Psf 16)$ respectively.

Via the isomorphism in (Psf 2), the vector space $𝕂\left[P\right]$ is an $\stackrel{˜}{H}-$module. Let $Ci = q-2+q-1Tsi = 1+q-1Tsi-1 = (1+q-2)1i, (Psf 17)$ where ${\mathbf{1}}_{i}$ is the element of $H$ such that ${{\mathbf{1}}_{i}}^{2}={\mathbf{1}}_{i}$ and ${T}_{{s}_{i}}^{-1}{\mathbf{1}}_{i}={q}^{-1}{\mathbf{1}}_{i}.$ The element ${\mathbf{1}}_{i}$ is the rank 1 version of the element ${\mathbf{1}}_{0}$ in (Psf 1).

The following proposition shows that, at ${q}^{-2}=0,$ the action of ${C}_{i}$ on $𝕂\left[P\right]$ is the Demazure operator ${\stackrel{˜}{\Delta }}_{i}.$ In geometry, the Demazure operators arise naturally as push-pull operators on the K-theory of the flag variety (see [PR, Proposition]).

Let $\rho ={\omega }_{1}+\cdots +{\omega }_{n}$ as in (2.16). As operators on $𝕂\left[P\right],$

1. ${\stackrel{˜}{\Delta }}_{i}={X}^{-\rho }{\Delta }_{i}{X}^{\rho }={\Delta }_{i}+{s}_{i},$
2. ${C}_{i}=\left(1-{q}^{-2}\right){\Delta }_{i}+\left({s}_{i}+{q}^{-2}\right)=\left(1+{s}_{i}\right)\left(\frac{1-{q}^{-2}{X}^{-{\alpha }_{i}}}{1-{X}^{-{\alpha }_{i}}}\right).$

 Proof. Let $\lambda \in P.$ Since ${s}_{i}\rho =\rho -⟨\rho ,{\alpha }_{i}^{\vee }⟩{\alpha }_{i}=\rho -{\alpha }_{i},$ $(X-ρΔiXρ) (Xλ) = X-ρ Xλ+ρ-Xsiλ+ρ-αi 1-X-αi = Xλ-Xsiλ-αi 1-X-αi = Δ˜i (Xλ),$ and, as operators, $Δi+si = 1 1-X-αi (1-si)+si = 1 1-X-αi (1-si+si-X-αisi) = Δ˜i.$ Using Proposition 3.2e, $q-1TsiXλ10 = ( q-1 Xsiλ Tsi+ (1-q-2) Xλ-Xsiλ 1-X-αi )10 = ( Xsiλ+ (1-q-2) Xλ-Xsiλ 1-X-αi )10 = ( Xsiλ- Xsiλ-αi+ Xλ-Xsiλ-q-2 (Xλ-Xsiλ) 1-X-αi )10 = ( Δ˜i- q-2Δi ) (Xλ)10,$ and the first formula in (b) now follows from the second formula in (a). Then $Ci = (q-2+Δ˜i-q-2Δi) = 1 1-X-αi ( q-2- q-2X-αi+1- X-αisi- q-2+ q-2si ) = 1-q-2X-αi 1-X-αi +si q-2-Xαi 1-Xαi = (1+si) ( 1-q-2X-αi 1-X-αi ).$ $\square$

Slightly renormalizing the generators of the affine Hecke algebra by setting ${\stackrel{˜}{T}}_{i}={q}^{-1}{T}_{{s}_{i}}$ allows one to let ${q}^{-1}=0$ so that ${\stackrel{˜}{T}}_{i}$ acts on $𝕂\left[P\right]$ by ${\stackrel{˜}{\Delta }}_{i}.$ This is the action of the nil affine Hecke algebra on $𝕂\left[P\right].$ Since the ${\stackrel{˜}{T}}_{i}$ satisfy the braid relations so do the ${\stackrel{˜}{\Delta }}_{i}.$ The first formula in Proposition 1.1 shows that ${\Delta }_{i}$ is a conjugate of ${\stackrel{˜}{\Delta }}_{i}$ and so the ${\Delta }_{i}$ also satisfy the braid relations. Although ${C}_{i}$ equals ${\stackrel{˜}{\Delta }}_{i}$ at ${q}^{-2}=0,$ the operators ${C}_{i}$ do not satisfy the braid relations. Furthermore, if ${w}_{0}={s}_{{i}_{1}}\cdots {s}_{{i}_{l}}$ is a reduced word for the longest element then $Ci1⋯Cil = W0(q-2)10+ q-2(extra terms).$ In contrast to the case for Weyl characters (when ${q}^{-2}=0\right),$ because of the ${q}^{-2}$(extra terms) the Hall-Littlewood polynomial cannot be generated by applying the product ${C}_{{i}_{1}}\cdots {C}_{{i}_{l}}$ unless one somehow knows how to throw away the extra terms.

As operators on $𝕂\left[P\right],$ $q-Tsi = ( q-1-qX-αi 1-X-αi )(1-si) and$ $10 = 1 W0(q-2) ∑ w∈W w ∏ α∈R+ 1-q-2X-α 1-X-α .$ The second formula is equivalent to Macdonald's spherical function formula (Psf 13).

## Root operators

The idea of root operators is to give an alcove walk interpretation of the action of the operator ${C}_{i}$ on $𝕂\left[P\right]$ by considering the projections of the alcove walks onto the line perpendicular to ${H}_{{\alpha }_{i}}.$ The main point is the identity (Psf 22) which gives a combinitorial description of the action of ${C}_{i}$ on $𝕂\left[P\right].$ The appropriate combinatorics is more or less forced by the Leibnitz rule or tensor product rule for the operator ${C}_{i}$ given in (Psf 28).

Let $p$ be a positively folded walk. $INSERT DIAGRAM (Psf 18)$ Let $1\le i\le n.$ The projection of $p$ onto a line perpendicular to ${H}_{{\alpha }_{i}}$ is positively folded alcove walk $\stackrel{_}{p}$ "with respect to ${\alpha }_{i}$" (the only important information in the projection is the relative position of the walk to each of the hyperplanes parallel to ${H}_{{\alpha }_{i}}$). $di+(p) = d+(p_) di-(p) = d-(p_) (Psf 19)$ Because $\stackrel{_}{p}$ is positively folded it is a concatenation of negative-positive sections of the form ${c}^{-}{c}^{-}\cdots {c}^{-}f{c}^{+}{c}^{+}\cdots {c}^{+},$ where ${c}^{+}$ denotes a positive crossing, ${c}^{-}$ a negative crossing, and $f$ a (positive) fold. The outer edge (bottom most negative traveling portion and topmost traveling portion) of the walk is a single negative-positive walk If $\stackrel{_}{p}$ is the walk in (Psf 18), the outer edge is the darkened portion of the path, ${d}^{+}\left(\stackrel{_}{p}\right)=7$ and ${d}^{-}\left(\stackrel{_}{p}\right)=3.$

The root operators $\stackrel{˜}{e}$ and $\stackrel{˜}{f}$ change the outer edge of the path $\stackrel{_}{p}$ and leave all other parts of the walk unchanged. Define $\stackrel{˜}{e}\stackrel{_}{p}$ and $\stackrel{˜}{f}\stackrel{_}{p}$ to be the positively folded alcove walks which are the same as $\stackrel{_}{p}$ except that If $\stackrel{_}{p}$ is the walk in (Psf 18) then $f˜p_ =$ and $e˜p_ =$ The precise rules for the limiting cases, when ${d}^{+}\left(\stackrel{_}{p}\right)$ or ${d}^{-}\left(\stackrel{_}{p}\right)=0,$ are illustrated by the following example, where the dashed arrows indicate the action of $\stackrel{˜}{e}$ and $\stackrel{˜}{f}.$

with $\stackrel{˜}{f}\left( \right)=0$ and $\stackrel{˜}{e}\left( \right)=0.$

With notations for ${d}^{+}\left(\stackrel{_}{p}\right)$ and ${d}^{-}\left(\stackrel{_}{p}\right)$ as in (Psf 19) and (Psf 20), define $di+(p) = d+(p_) and di-(p) = d-(p_), (Psf 21)$ where $\stackrel{_}{p}$ is the projection of $p$ onto the line perpendiculat to ${\alpha }_{i}.$ The walks ${\stackrel{˜}{e}}_{i}p$ and ${\stackrel{˜}{f}}_{i}p$ are the walks obtained from $p$ by changing the corresponding edges $p$ (so that the projections of ${\stackrel{˜}{e}}_{i}p$ and ${\stackrel{˜}{f}}_{i}p$ are $\stackrel{˜}{e}\stackrel{_}{p}$ and $\stackrel{˜}{f}\stackrel{_}{p},$ respectively). $INCLUDE 2 PICTURES HERE$

The i-string of $p$ ${S}_{i}\left(p\right)$ is the set of paths generated from $p$ by applying the root operators ${\stackrel{˜}{e}}_{i}$ and ${\stackrel{˜}{f}}_{i}.$ The head of the $i-$string is the path $h$ in ${S}_{i}\left(p\right)$ which has ${d}_{i}^{-}\left(h\right)=0.$ If $\mathrm{wt}\left(h\right)=\lambda$ and $\lambda$ is on the positive side of ${H}_{{\alpha }_{i}}$ then $CiXλ = q-1XsiλTsi-1 + (1-q-2) ( Xsiλ + Xsiλ+αi +⋯+ Xλ-αi ) +Xλ, (Psf 22)$ and the terms in this sum correspond to the paths in the $i-$string ${S}_{i}\left(h\right).$

## $q-$Crystals

The $q-$crystals provide a combinatorial model for the spherical Hecke algebra ${\mathbf{1}}_{0}\stackrel{˜}{H}{\mathbf{1}}_{0}$ in the basis of Hall-Littlewood polynomials. Three structural properties motivate the definition of $q-$crystals:

1. The Hall-Littlewood polynomials are normalised versions of the basis ${\mathbf{1}}_{0}{X}^{\lambda }{\mathbf{1}}_{0}.$
2. The element ${\mathbf{1}}_{0}$ is characterised by the property that ${C}_{i}{\mathbf{1}}_{0}=\left(1+{q}^{-2}\right){\mathbf{1}}_{0}$ for all $1\le i\le n.$
3. The action of $Ci$ on $\stackrel{˜}{H}{\mathbf{1}}_{0}\cong 𝕂\left[P\right]$ is captured in the combinatorics of $i-$strings.
These properties indicate that the combinatorics of Hall-Littlewood polynomials can be captured with the root operators.

Let If $B$ is a finite subset of ${B}_{\mathrm{univ}}$ the character of $B$ is $char(B) = ∑ p∈B q -( ι(p) + φ(p) - f(p) ) (1-q-2) f(p)-c(p) Xwt(p), (Psf 24)$ where $p$ has $f\left(p\right)$ folds, $\iota \left(p\right)A$ is the alcove where $p$ begins, $\mathrm{wt}\left(p\right)+\phi \left(p\right)A$ is the alcove where $p$ ends and A q-crystal is a finite subset $B$ of ${B}_{\mathrm{univ}}$ which is closed under the action of the root operators.

A positively folded alcove walk is i-dominant if it never touches the hyperplane ${H}_{{\alpha }_{i},-1}.$ The head $h$ of an $i-$string ${S}_{i}\left(p\right)$ is $i-$dominant and ${S}_{i}\left(h\right)={S}_{i}\left(p\right).$ A positively folded alcove walk where In other words, a positively folded alcove walk $p$ is dominant if it is $i-$dominant for all $i,$ $1\le i\le n.$

Let $B$ be a $q-$crystal. Then, with notations as in (Psf 24)-(Psf 26), $char(B) = ∑ p∈B p⊆C-ρ q-( ι(p) + φ(p) - f(p) ) (1-q-2) f(p)-c(p) Pwt(p).$

 Proof. If $p$ is a dominant walk let under the action of the root operators ${\stackrel{˜}{e}}_{i}$ and ${\stackrel{˜}{f}}_{i}.$ The point is that the set of all positively folded alcove walks is partitioned into "equivalence classes" given by the sets ${B}_{q}\left(p\right)$ such that $p\in {B}_{\mathrm{univ}}$ is dominant and $p$ is the unique dominant walk in ${B}_{q}\left(p\right).$ Because ${\mathbf{1}}_{0}$ is characterized by the property that ${C}_{i}{\mathbf{1}}_{0}=\left(1+{q}^{-2}\right){\mathbf{1}}_{0}$ and the action of ${C}_{i}$ is modeled by the combinatorics of $i-$strings (Psf 25), this equivalence relation is generated by the relations $p\sim {\stackrel{˜}{f}}_{i}p$ and $p\sim {\stackrel{˜}{e}}_{i}p.$ $\square$

## Products and restrications

The results in this section are generalizations of the Littlewood-Richardson rules. These are obtained as corollaries of Theorem 5.1.

The combinatorial definition of the root operators given above is essentially a consequence of the Leibniz rule for the Demazure operator, The corresponding rule for the operators ${C}_{i}$ is This identity is implicit in the additivity in $\lambda$ of the relation in Proposition 3.2e (the product ${T}_{{s}_{i}}{X}^{\lambda +\mu }=\left({T}_{{s}_{i}}{X}^{\lambda }\right){X}^{\mu }$ can be expanded in two different ways using Proposition 3.2e).

In order to define the product of ${p}_{1}\otimes {p}_{2}$ of walks ${p}_{1},{p}_{2}\in {B}_{\mathrm{univ}}$ the final direction $\phi \left({p}_{1}\right)$ of ${p}_{1}$ and the initial direction $\iota \left({p}_{2}\right)$ of ${p}_{2}$ need to be taken into account. (To properly model the multiplication of Hall-Littlewood polynomials we must account for the effect of ${\mathbf{1}}_{0}$ in the product ${P}_{\mu }{\mathbf{1}}_{0}{P}_{\nu }{\mathbf{1}}_{0}$ and we cannot just concatenate walks as in the alcove walk algebra). Define ${p}_{1}\otimes {p}_{2},$ recursively, by where ${p}_{1}^{\text{'}}$ is the alcove walk constructed by the following procedure. Let ${c}_{{i}_{1}}^{{\epsilon }_{1}}\cdots {c}_{{i}_{r}}^{{\epsilon }_{r}}$ be a minimal length walk from $\phi \left({p}_{1}\right)$ to $\iota \left({p}_{2}\right).$ If ${\epsilon }_{1}=-$ let ${p}_{1}^{\text{'}}={p}_{1}{c}_{{i}_{1}}^{-}.$ If ${\epsilon }_{1}=+$ let ${H}_{\alpha ,j}$ be the hyperplane crossed by the last step of ${p}_{1}{c}_{{i}_{1}}^{-}$ and change the last negative crossing of ${H}_{\alpha ,j}$ in ${p}_{1}$ to a fold to obtain a new path ${p}_{1}^{\text{'}}$ with $\phi \left({p}_{1}^{\text{'}}\right)=\phi \left({p}_{1}{c}_{{i}_{1}}^{+}\right).$

In terms of root operators, the Leibniz rule (Psf 28) translates to the property (Psf 29) for ${p}_{1},{p}_{2}\in {B}_{\mathrm{univ}}.$ It follows from this version of the Leibnitz rule that if ${B}_{1}$ and ${B}_{2}$ are $q-$crystals then the product and $char(B1⊗B2) = char(B1) char(B2). (Psf 31)$ This last property is not completely trivial. The general case follows from the rank one case (projecting onto the line perpendicular to ${H}_{\alpha }$). More importantly, the definition of the product $\otimes$ is forced by (Psf 29)-(Psf 31).

The following theorem is a version of [Sc, Theorem 1.2] and results in [KM] and [Ha].

Recall the notations from Theorem 2.1. If $\lambda \in {P}^{+}$ let ${B}_{q}\left({p}_{\lambda }\right)$ be the $q-$crystal generated by ${p}_{\lambda },$ where ${p}_{\lambda }$ is a fixed minimal length alcove walk from $A$ to $\lambda +A.$ For $\mu ,\nu \in {P}^{+},$ $PμPν = ∑ p∈Bq(pν) pμ⊗p⊆C-ρ q -( l(ι(p))+l(φ(p))-f(p) ) (1-q-2) f(p)-c(p) Pμ+wt(p).$

 Proof. Using (Psf 30) and (Psf 31) and applying Theorem 5.1 to the $q-$crystal ${B}_{q}\left({p}_{\mu }\right)\otimes {B}_{q}\left({p}_{\nu }\right)$ gives $PμPν10 = char(Bq(pμ)) char(Bqpν)) 10 = char((Bq(pμ)⊗Bq(pν)) 10 = ∑ p=p1⊗p2∈Bq(pμ)⊗Bq(pν) p1⊗p2⊆C-ρ q -( l(ι(p)) + l(φ(p)) - f(p) ) (1-q-2) f(p)-c(p) Pμ+wt(p2) 10$ since every path in ${p}_{1}\in {B}_{q}\left({p}_{\mu }\right)$ which is constrained in $C-\rho$ has weight $\mu$ (so that $\mathrm{wt}\left({p}_{1}\otimes {p}_{2}\right)=\mu +\mathrm{wt}\left({p}_{2}\right).\right)$ $\square$

Fix $J\subseteq \left\{1,2,...,n\right\}.$ The subgroup of $W$ generated by the reflections in the hyperplanes as a fundamental chamber. Let and let ${\mathbf{1}}_{J}\in {H}_{J}$ be given by For $\mu \in P$ let ${W}_{\mu }^{J}$ be the stabilizer of $\mu$ under the ${W}_{J}$ action on $P$ and where ${W}_{J}^{\mu }$ is a set of minimal length coset representatives for the cosets in ${W}_{J}/{W}_{\mu }^{J}.$ Then and with

Recall the notations from Theorem 2.1. If $\mu \in {P}^{+}$ let ${B}_{q}\left({p}_{\mu }\right)$ be the $q-$crystal generated by ${p}_{\mu },$ where ${p}_{\mu }$ is a fixed minimal length alcove walk from $A$ to $\mu +A.$ Let $\lambda \in {P}^{+}$ and let $J\subseteq \left\{1,2,...,n\right\}.$ Then $Pλ = ∑ p∈B(λ) p⊆CJ-ρJ q -( l(ι(p)) + l(φ(p)) - f(p) ) (1-q-2) f(p)-cJ(p) Pwt(p)J,$ where and ${c}_{J}\left(p\right)$ is the number of folds of $p$ which touch a hyperplane ${H}_{{\alpha }_{j}}$ with $j\in J.$

 Proof. A J-crystal is a set of positively folded alcove walks $B$ which is closed under the operators for $j\in J.$ Since ${P}_{\lambda }{\mathbf{1}}_{0}=\mathrm{char}\left({B}_{q}\left({p}_{\lambda }\right)\right){\mathbf{1}}_{0},$ the statement follows by applying Theorem 5.1 to ${B}_{q}\left({p}_{\lambda }\right)$ viewed as a $J-$crystal. $\square$

## Notes and References

The above notes are taken from section 4 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.

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