Polynomials and symmetric functions

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

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 H˜ given by H=span {Tw-1-1  |  wW}. Let 10 be the element of H given by 102 =10 and Tw-1-1 10= q-l(w)10, (Psf 1) for wW. Two explicit formulas for 10 are 10= 1 W0(q-2) wW q-l(w) Tw-1-1= 1 W0(q2) wW ql(w) Tw, where W0(t)= wW tl(w) is the Poincaré polynomial of W.

As observed in (3.6), 𝕂[P]= span{Xλ  |  λP} is a subalgebra of H˜. The vector space 𝕂[P] also sits inside H˜ in a different way. Since {Xλ10  |  λP} is a basis of H˜10 there is a vector space isomorphism 𝕂[P] H˜10 f f10. (Psf 2)

The ring of symmetric functions is 𝕂[P]W = {f𝕂[P]  |  wf=f   for all   wW}. (Psf 3) By a theorem of Bernstein (see [NR, Theorem 1.4]) this subalgebra of H˜ is the center of H˜, 𝕂[P]W = Z(H˜). (Psf 4) The spherical Hecke algebra is the ring 10Xλ10 and the restriction of the map (Psf 2) to Z(H˜) is the Satake isomorphism of the following theorem.

Let 10 and 𝕂[P]W be as in (Psf 1) and (Psf 3), respectively. Then 𝕂[P]W =Z(H˜) 10 H˜ 10 f f10 is a   𝕂-algebra isomorphism.

The map is a well defined homomorphism since, if ff1f2 Z(H˜) then f10= f102= 10f10 and f1f210= f1f2102= f110 f210.

Suppose that λ,αi>0 so that λ is on the positive side of Hαi. 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 10H˜10 can, inductively, be written as a linear combination of the elements 10Xλ10,   λP+. Using Theorem 3.3 to expand 10Xλ in terms of the basis {XμTv-1-1  |  μP, vW} produces 10Xλ = Xw0λTw0-1 + μ>w0λ dμ,vXμTv-1-1, and, since these leading terms are all different (as λ runs over P+), it follows that 10H˜10 has basis {10Xλ10  |  λP+}. (Psf 8) The orbit sums mλ = γWλ Xγ, λP+, (Psf 9) for a basis of 𝕂[P]W. The relation in Proposition 3.2e implies that, if f𝕂[P]W then Twf = fTw for all wW, and so the mλ10 = 10mλ10 are in 10H˜10. Viewing these in terms of the basis {XμTv-1-1  |  μP, vW} of H˜ one sees that the mλ10,  λP+, are linearly independent and so 10H˜10 has basis {mλ10  |  λP+}. (Psf 10) The point is that the transition matrix between the basis in (Psf 8) and the basis in (Psf 10) is triangular.

Hall-Littlewood polynomials

For μP let Wμ= Stab(μ) be the stabilizer of μ. The Poincaré polynomial of Wμ is Wμ(t) = wWμ tl(w). (Psf 11) For μP, the Hall-Littlewood polynomial or Macdonald spherical function Pμ(X;t) is the element of 𝕂[P]W defined by Pμ (X;q-2)10 = ( wWμ q-l(w) Tw-1-1) Xμ10, (Psf 12) where Wμ is a set of minimal length coset representatives for the cosets in W/Wμ. Since every element wW has a unique expression w=uv with uWμ and vWμ, wWμ q-l(w) Tw-1-1 Xμ10 = 1 Wμ(q-2) ( uWμ q-l(u) Tu-1-1) Xμ ( vWμ q-l(v) Tv-1-1) 10 = 1 Wμ(q-2) ( uWμ q-l(u) Tu-1-1) ( vWμ 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 Xμ10 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) wW w(Xμ αR+ 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 Pλ (X;q-2). 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 Wλ which appears in the definition of Pλ is (Psf 12). The type of p is the sequence of labels of the folds and the wall crossings of p.

For λP+ let tλW˜ be the translation in λ and let nλ be the maximal length element in the double coset WtλW.

  1. [Sc, Theorem 1.1] Let λP+ and fix a minimal length walk pλ= ci1+cil+ from A to λ+A. Let Bq(pλ) = { | positively folded alcove walks of type   (i1,...,il)   which begin at   wA |  wWλ}. Then Pλ(X;q-2) = pBq(pλ) q -( l(ι(p))+l(φ(p))-f(p) ) (1-q-2)f(p) Xwt(p), where ι(p) is the alcove where p begins, wt(p)+ φ(p)A is the alcove where p ends, and f(p) is the number of folds in p.
  2. Let λP+. Then ql(w0) W0(q-2) Pλ10 = xWtλW ql(x)-l(nλ) Tx.

  1. The proof is accomplished by using Theorem 3.3 to expand the sum in (Psf 12). Since all crossings in the walk pλ are positive crossings Theorem 3.3 gives wWλ q-l(w) Tw-1-1 Xλ = wWλ q-l(w) pBq(pλ) ι(p)=w (q-q-1)f(p) Xwt(p) Tφ(p)-1. Hence Pλ10 = pBq(λ) q-l(ι(p)) (q-q-1)f(p) Xwt(p) q-l(φ(p)) 10.
  2. Let λP+. Let Wλ=Stab(λ) and let w0 and wλ be the maximal length elements in W and Wλ, respectively. Let mλ and nλ be the minimal and maximal length elements respectively in the double coset WtλW. If λ=2ω2 in type C2, then Wλ={1,s1},  wλ=s1,  w0=s1s2s1s2,  l(tλ)=6,  l(mλ)=3, and l(nλ)=10. Labeling the alcove wA by the element w, the 32 alcoves wA with wWtλW 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) uWλ ql(u) Tu Xλ10 = ql(w0) q-2l(w0) W0(q2) q-2l(w0)+2l(wλ) uWλ ql(u) Tu Tmλ Tw0wλ10 = q-3l(w0)+2l(wλ) uWλ ql(u) Tu Tmλ ql(w0)-l(wλ) wW ql(w) Tw = q-2l(w0)+l(wλ) uWλ wW ql(u)+l(w) Tumλw = q-2l(w0)+l(wλ) xWtλW ql(x)-l(mλ) Tx , and the result follows from the identity l(nλ) = l(w0)+l(tλ) = l(w0)+l(w0)-l(wλ)+l(mλ).

The set Bq(pλ) appearing in Theorem 2.1a is heavily dependent on the choice of pλ. One way to make this seem less dependent on this choice is as follows. For convenience assume that λ is regular (not on any wall). Similar definitions can be made in the general case. Consider the region [λ]= {x= i xiωi  |  0xiλi}, where λ= i λiωi. This region is a union of alcoves and any minimal length walk pλ from A to λ+w0A lies in [λ]. Foldings of the walk pλ are then produced by folding the region [λ] along the "creases" formed by the hyperplanes. This process produces a bijection between the paths in Bq(pλ) and the set Bq([λ]) of "positively folded foldings" of the region [λ], and the set Bq([λ]) does not depend on the choice of the initial path pλ 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 Bq(pλ) be as in Theorem 2.1a and let pBq(pλ). Suppose that p has f folds. For 0if, let p(i) be the positively folded walk in Bq(pλ) which coincides with p up to the ith fold and is nonfolded thereafter. Then p(0)... p(f) 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 ith fold is on the hyperplane Hα,j. Since φ(p(i-1))>φ(p(i)) and (-1)l(φ(p(i))) = (-1)l(sα) (-1)φ(p(i-1)) = (-1) (-1)φ(p(i-1)), l(φ(p(i-1))) - l(φ(p(i))) -1 is an even integer 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 0. This proves that f(p) l(ι(p))- l(φ(p)) and that Pλ(X;q-2) really is a polynomial in the variable q-2.

Demazure operators

The group W acts on 𝕂[P]= span{Xλ  |  λP} by wXλ = Xwλ, for   wW,  λP. (Psf 15) For each 1in, 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[P] is an H˜-module. Let Ci = q-2+q-1Tsi = 1+q-1Tsi-1 = (1+q-2)1i, (Psf 17) where 1i is the element of H such that 1i2=1i and Tsi-11i = q-11i. The element 1i is the rank 1 version of the element 10 in (Psf 1).

The following proposition shows that, at q-2=0, the action of Ci on 𝕂[P] is the Demazure operator Δ˜i. In geometry, the Demazure operators arise naturally as push-pull operators on the K-theory of the flag variety (see [PR, Proposition]).

Let ρ= ω1++ωn as in (2.16). As operators on 𝕂[P],

  1. Δ˜i = X-ρΔiXρ = Δi+si,
  2. Ci = (1-q-2)Δi+(si+q-2) = (1+si) ( 1-q-2X-αi 1-X-αi ).

  1. Let λP. Since siρ = ρ-ρ,αiαi = ρ-αi, (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.
  2. 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 ).

Slightly renormalizing the generators of the affine Hecke algebra by setting T˜i = q-1Tsi allows one to let q-1=0 so that T˜i acts on 𝕂[P] by Δ˜i. This is the action of the nil affine Hecke algebra on 𝕂[P]. Since the T˜i satisfy the braid relations so do the Δ˜i. The first formula in Proposition 1.1 shows that Δi is a conjugate of Δ˜i and so the Δi also satisfy the braid relations. Although Ci equals Δ˜i at q-2=0, the operators Ci do not satisfy the braid relations. Furthermore, if w0= si1sil is a reduced word for the longest element then Ci1Cil = W0(q-2)10+ q-2(extra terms). In contrast to the case for Weyl characters (when q-2=0), because of the q-2(extra terms) the Hall-Littlewood polynomial cannot be generated by applying the product Ci1Cil unless one somehow knows how to throw away the extra terms.

As operators on 𝕂[P], q-Tsi = ( q-1-qX-αi 1-X-αi )(1-si) and 10 = 1 W0(q-2) wW w αR+ 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 Ci on 𝕂[P] by considering the projections of the alcove walks onto the line perpendicular to Hαi. The main point is the identity (Psf 22) which gives a combinitorial description of the action of Ci on 𝕂[P]. The appropriate combinatorics is more or less forced by the Leibnitz rule or tensor product rule for the operator Ci given in (Psf 28).

Let p be a positively folded walk. INSERT DIAGRAM (Psf 18) Let 1in. The projection of p onto a line perpendicular to Hαi is positively folded alcove walk p_ "with respect to αi" (the only important information in the projection is the relative position of the walk to each of the hyperplanes parallel to Hαi). di+(p) = d+(p_) di-(p) = d-(p_) (Psf 19) Because p_ is positively folded it is a concatenation of negative-positive sections of the form c-c-c-fc+c+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 c-c-c-c- d-(p_)   factors f c+c+c+c+ d+(p_)   factors . (Psf 20) If p_ is the walk in (Psf 18), the outer edge is the darkened portion of the path, d+(p_)=7 and d-(p_)=3.

The root operators e˜ and f˜ change the outer edge of the path p_ and leave all other parts of the walk unchanged. Define e˜p_ and f˜p_ to be the positively folded alcove walks which are the same as p_ except that f˜p_   has outer edge c-c-c-c-c- d-(p_)+1   factors f c+c+c+c+ d+(p_)-1   factors ,   and e˜p_   has outer edge c-c-c- d-(p_)-1   factors f c+c+c+c+c+ d+(p_)+1   factors . If p_ is the walk in (Psf 18) then f˜p_ = and e˜p_ = The precise rules for the limiting cases, when d+(p_) or d-(p_)=0, are illustrated by the following example, where the dashed arrows indicate the action of e˜ and f˜.

e˜ f˜ e˜ f˜ e˜ f˜
with f˜( )=0 and e˜( )=0.

With notations for d+(p_) and d-(p_) as in (Psf 19) and (Psf 20), define di+(p) = d+(p_) and di-(p) = d-(p_), (Psf 21) where p_ is the projection of p onto the line perpendiculat to αi. The walks e˜ip and f˜ip are the walks obtained from p by changing the corresponding edges p (so that the projections of e˜ip and f˜ip are e˜p_ and f˜p_, respectively). INCLUDE 2 PICTURES HERE

The i-string of p Si(p) is the set of paths generated from p by applying the root operators e˜i and f˜i. The head of the i-string is the path h in Si(p) which has di-(h) =0. If wt(h)=λ and λ is on the positive side of Hαi 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 Si(h).


The q-crystals provide a combinatorial model for the spherical Hecke algebra 10H˜10 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 10Xλ10.
  2. The element 10 is characterised by the property that Ci10 = (1+q-2)10 for all 1in.
  3. The action of Ci on H˜10 𝕂[P] 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 Buniv   be the set of positively folded alcove walks which begin in the  0-polygon   WA. (Psf 23) If B is a finite subset of Buniv the character of B is char(B) = pB q -( ι(p) + φ(p) - f(p) ) (1-q-2) f(p)-c(p) Xwt(p), (Psf 24) where p has f(p) folds, ι(p)A is the alcove where p begins, wt(p) + φ(p)A is the alcove where p ends and c(p)   is the number of folds of   p touching one of the hyperplanes   Hα1...Hαn . (Psf 25) A q-crystal is a finite subset B of Buniv which is closed under the action of the root operators.

A positively folded alcove walk is i-dominant if it never touches the hyperplane Hαi,-1. The head h of an i-string Si(p) is i-dominant and Si(h) = Si(p). A positively folded alcove walk p   is dominant if   pC-ρ, (Psf 26) where C-ρ = {μ𝔥*  |  μ,αi>-1   for   1in}. In other words, a positively folded alcove walk p is dominant if it is i-dominant for all i, 1in.

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

If p is a dominant walk let Bq(p)   be the  q-crystal generated by   p under the action of the root operators e˜i and f˜i. The point is that the set of all positively folded alcove walks is partitioned into "equivalence classes" given by the sets Bq(p) such that pBuniv is dominant and p is the unique dominant walk in Bq(p). Because 10 is characterized by the property that Ci10 = (1+q-2)10 and the action of Ci is modeled by the combinatorics of i-strings (Psf 25), this equivalence relation is generated by the relations pf˜ip and pe˜ip.

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, Δi(fg) = Δi(f)g + (sif)(Δig), for   f,g 𝕂[P]. (Psf 27) The corresponding rule for the operators Ci is Ci(fg) = (Cif)g + (sif) ( (Ci-(1+q-2))g ), for   f,g 𝕂[P]. (Psf 28) This identity is implicit in the additivity in λ of the relation in Proposition 3.2e (the product TsiXλ+μ = (TsiXλ)Xμ can be expanded in two different ways using Proposition 3.2e).

In order to define the product of p1p2 of walks p1,p2Buniv the final direction φ(p1) of p1 and the initial direction ι(p2) of p2 need to be taken into account. (To properly model the multiplication of Hall-Littlewood polynomials we must account for the effect of 10 in the product Pμ10 Pν10 and we cannot just concatenate walks as in the alcove walk algebra). Define p1p2, recursively, by If   φ(p1) = ι(p2) then p1p2 = p1p2,   the concatenation of   p1   and   p2,  and if   φ(p1) ι(p2) then p1p2 = p1'p2, where p1' is the alcove walk constructed by the following procedure. Let ci1ε1 cirεr be a minimal length walk from φ(p1) to ι(p2). If ε1=- let p1' = p1ci1-. If ε1=+ let Hα,j be the hyperplane crossed by the last step of p1ci1- and change the last negative crossing of Hα,j in p1 to a fold to obtain a new path p1' with φ(p1') = φ(p1ci1+).

In terms of root operators, the Leibniz rule (Psf 28) translates to the property (Psf 29) e˜i (p1p2) = { (e˜ip1) p2, if   di+(p1) di-(p2), p1 (e˜ip2), if   di+(p1) < di-(p2), } and f˜i (p1p2) = { (f˜ip1) p2, if   di+(p2) > di-(p2) p1p2_ = , p (f˜iq), if   di+(p) di-(q) pq_ = , } for p1,p2 Buniv. It follows from this version of the Leibnitz rule that if B1 and B2 are q-crystals then the product B1B2 = {p1p2  |  p1B1,  p2B2} is also a q-crystal, (Psf 30) and char(B1B2) = 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 Hα). More importantly, the definition of the product 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 λP+ let Bq(pλ) be the q-crystal generated by pλ, where pλ is a fixed minimal length alcove walk from A to λ+A. For μ,νP+, PμPν = pBq(pν) pμpC-ρ q -( l(ι(p))+l(φ(p))-f(p) ) (1-q-2) f(p)-c(p) Pμ+wt(p).

Using (Psf 30) and (Psf 31) and applying Theorem 5.1 to the q-crystal Bq(pμ) Bq(pν) gives PμPν10 = char(Bq(pμ)) char(Bqpν)) 10 = char((Bq(pμ)Bq(pν)) 10 = p=p1p2Bq(pμ)Bq(pν) p1p2C-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 p1Bq(pμ) which is constrained in C-ρ has weight μ (so that wt(p1p2) = μ+wt(p2).)

Fix J {1,2,...,n}. The subgroup of W generated by the reflections in the hyperplanes Hαj,   jJ, WJ = sj  |  jJ, acts on   𝔥*, with   CJ = {μ𝔥*  |  μ,αj>0   for   jJ} as a fundamental chamber. Let HJ = span{Tw-1-1  |  wWJ} and let 1JHJ be given by 1J2 = J and Tw-1-11J = q-l(w)1J, for   wWJ. For μP let WμJ be the stabilizer of μ under the WJ action on P and define   PμJ (X;q-2) 𝕂 [P]WJ   by PμJ (X;q-2)10 = wWJμ q -l(w) Tw-1-1 Xμ10, (Psf 32) where WJμ is a set of minimal length coset representatives for the cosets in WJ/WμJ . Then up to normalization   PλJ (X;q-2)10   equals   1JXλ10, (Psf 33) and 1JH˜10   has basis   {PλJ10  |  λPJ+}, where   PJ+ = PCJ_, (Psf 34) with CJ_ = {μ𝔥*  |  μ,αj>0   for   jJ}.

Recall the notations from Theorem 2.1. If μP+ let Bq(pμ) be the q-crystal generated by pμ, where pμ is a fixed minimal length alcove walk from A to μ+A. Let λP+ and let J{1,2,...,n}. Then Pλ = pB(λ) pCJ-ρJ q -( l(ι(p)) + l(φ(p)) - f(p) ) (1-q-2) f(p)-cJ(p) Pwt(p)J, where CJ-ρJ = {μ𝔥*  |  μ,αj>-1   for   jJ} and cJ(p) is the number of folds of p which touch a hyperplane Hαj with jJ.

A J-crystal is a set of positively folded alcove walks B which is closed under the operators e˜j, f˜j, for jJ. Since Pλ10 = char(Bq(pλ))10, the statement follows by applying Theorem 5.1 to Bq(pλ) viewed as a J-crystal.

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.


[BD] Y. Billig and M. Dyer, Decompositions of Bruhat type for the Kac-Moody groups, Nova J. Algebra Geom. 3 no. 1 (1994), 11-31.

[Br] 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.

[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.

page history