The ring [P]W

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

Last update: 27 July 2012

The ring [P]W

[P]W is a polynomial ring, i.e. there are algebraically independent elements e1,...,en [P]W such that [P]W = [e1,...,en].

Let e1,...,en 𝕂[P]W be such that ei = xωi + ( lower terms in dominance order ). If λ = l1ω1 ++ lnωn P+ then e1l1 e2l2 enln = xλ + ( lower terms in dominance order ). Thus { e1l1 enln | l1,...,ln 0 } is a basis of 𝕂[P]W and the result follows.

The Shi arrangement 𝒜- is the arrangement of (affine) hyperplanes given by 𝒜- = { Hα, Hα-δ | αR+ } where Hα = { xn | x, α = 0 }, Hα-δ = { xn | x, α = -1 }. (ZPW 1) Consider the partition of 𝔥* determined by the Shi arrangement. Each chamber w-1C, wW, contains a unique region of 𝒜- which is a cone, and the vertex of this cone is the point λw = w-1 siw<w ωi . (ZPW 2) Hα1+α2 Hα1 Hα2 Hα1+2α2 Hα1+α2-δ Hα1-δ Hα1-δ Hα2-δ Hα1+2α2-δ C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C λ1 λs1 λs2s1 λs2 λs1s2 λs1s2s1 λs2s1s2 λs1s2s1s2 The arrangement 𝒜-

[P] is a free [P]W module of rank |W| with basis {xλw | wW}.

The proof is accomplished by establishing three facts:
  1. Let fy, yW, be a family of elements of [P]. Then det(zfy) is divisible by αR+ (1-x-α) |W|2 .
  2. det(zxλy)z,yW = α>0 xρ (1-x-α) |W| 2 .
  3. If f[P] then there is a unique solution to the equation wW awxλw=f, with aw[P]W.
  1. For each αR+ subtract row zfy from row sαzfy. Then this row is divisible by ( 1-x-α ). Since there are |W|2 pairs of rows ( zfy, sαzfy ) the whole determinant is divisible by ( 1-x-α ) |W|2 . For α,βR+ the factors (1-x-α) and (1-X-β) are coprime, and so det(zfy) is divisible by αR+ (1-x-α) |W|2.
  2. Since yλy is dominant, yλy zyλy. So all the entries in the yth column are (weakly) less than the entry on the diagonal. If yλy = zyλy then z is in the stabilizer of αiR(y) ωi. Thus l(y-1) = l(z-1) + l(zy-1). If l(y) l(z) then l(zy-1)=0 and so z=y. Thus, if the rows are ordered so that the yth row is above the zth row when l(y) l(z) then all terms above the diagonal are strictly less than the diagonal entry.

    Thus the top coefficient of det(zxλy) is equal to zW zxλz = zW i siz<z xωi = i=1n x |W|2 ωi = (xρ) |W|2 . Since si det(zxλy) = -det(zxλy) the lowest term of det(zxλy) is w0 (xρ) |W|2 = (xρ) - |W|2 . These are the same as the highest and lowest terms of xρ αR+ (1-x-α) and so (2) follows from (1).
  3. Assume that ay [P]W are solutions of the equation yW xλy ay = f. Act on this equation by the elements of W to obtain the system of |W| equations yW (zxλy)ay = zf, zW. By (1) the matrix (zxλy) z,yW is invertible and so this system has a unique solution with ay [P]W . Cramer's rule provides an expression for ay as a quotient of two determinants. By (1) and (2) the denominator divides the numerator to give an element of [P]. Since each determinant is an alternating function (an element of Fock space), the quotient is an element of [P]W.

In [Sb2] Steinberg proves this type of result in full generality without the assumptions that W acts irreducibly on 𝔥* and L=P. Note also that the proof given above is sketchy, particularly in the aspect that the top coefficient of the determinant is what we have claimed it is. See [Sb2] for a proper treatment of this point.

From Verma at Magdeburg meeting August 1998

R(T) = [P] (P the weight lattice) is R(G) = R(T)W = [P]W free with {eε(w) | wW} as a basis, where ε(w) = w-1 (ρw) where ρw = iD(w)wi, D(w) = {i | siw<w}. There is an explicit expansion formula eλ = wW χλ,w eε(w).

eλ+μ = wW χλ,w eμ+ε(w) for all μP. We shall use this for a set of |W| values μ below.

Recall that Δw0: R(T) R(T)W eλ ch(λ) the Reynolds/Demazure operator. Then Δw0 (eλ+μ) = wW χλ,w Δw0( eμ+ε(w) ). So let μ {-ε(yw0)-ρ | yW} and we get a |W|×|W| matrix ( ch( e ε(w) - ε(yw0) -ρ ) ) y,wW which Hulsurkar proved to be unipotent. The diagonal entries are sgn(w).

By definition ρw0w = ρ-ρw and so w-1 ρw0w = w-1ρ - w-1ρw w-1 w0-1 w0 ρw0w = -w-1 w0-1 ρw0w = -ε(w0w). So -ε(w0w) = w-1ρ - ε(w). So ε(w) - ε(w0w) = w-1ρ. But one can look at these elements as follows:

Remove the strips {xn | x,α<1} Hα1 Hα2 Hβ 0 The endpoints of the resulting cones in each chamber are the ε(w) (up to a possible multiplication by w or w-1).

This is not at all unrelated to

  1. Steinberg, "On a theorem of Pittie", Topology 14 (1975), 173-177.
There he defines ew = w-1λw = w-1 iD(w) ewi where D(w) = {i | siw<w}.

det(uev)u,vW = α>0 ( eα2 - e-α2 ) nα where nαi = | {vW | v-1αi<0} | and nwαi = nαi.

Let fv [X]. Then det(ufv) is divisible by α>0 ( eα2 - e-α2 ) nα .

If f[X] then there is a unique solution to awew = f with aw [X]W . Here X is the weight lattice.

Proof of Theorem 2.3.
Subtract row ufv from row sαufv. Then this row is divisible by 1-e-α. So det(ufv) is divisible by α ( 1-e-α ) nα .

Proof of Theorem 2.2.
The top coefficient of det(uev) is vW vev = vW λv = vW iD(v) ewi = i=1n ewini. The top coefficient of α (1-eα)nα is e α>0 nαα . Now si( α>0 nαα ) = α>0 nαα - 2nαi αi. So ( α>0 nαα , αi ) = 2nαi = 2ni. So α>0 nαα = i=1n 2niwi.

Proof of Theorem 2.4.
The system (uev)av = uf has a unique solution with av [X]W.
So evav = f has a unique solution.

Example for S2. W={1,s1} z1=x1 z1 = 1 e1 = 1 zs1 = z1 es1 = x2 D = det 1 x2 1 x1 = x1-x2 and 1 x2 1 x1 -1 = 1 x1-x2 x1 -x2 -1 1 .

Example for S3. W = { 1,s1, s2, s1s2, s2s1, s1s2s1 } z1 = x1 z2 = x1x2 z1 = 1 e1 = 1 zs1 = z1=x1 es1 = x2 zs2 = z2 = x1x2 es2 = x1x3 zs1s2 = z1=x1 es1s2 = x3 zs2s1 = z2 = x1x2 es2s1 = x2x3 zs1s2s1 = z1z2 = x12x2 es1s2s1 = x2x32 D = 1 x2 x1x3 x3 x2x3 x2x32 1 x1 x2x3 x3 x1x3 x1x32 1 x3 x1x2 x2 x2x3 x22x3 1 x3 x1x2 x1 x1x3 x12x3 1 x1 x1x3 x2 x1x2 x1x22 1 x2 x2x3 x1 x1x2 x12x2 1 s1 s2 s1s2 s2s1 s1s2s1 This is divisible by (x2-x1)3 (x3-x2)3 (x3-x1)3 and both things have top term x16x23.

In the proof of Theorem 2.4 we get av = det uew uf uf uf vth column det(uew) = α>0 ( eα2 - e-α2 ) -nα det uew uf uf uf from Cramer's Rule.

Notes and References

These notes are a retyping, into MathML, of the notes at

§2 is taken from Verma at the Magdeburg meeting in August 1998.


[GL] S. Gaussent, P. Littelmann, LS galleries, the path model, and MV cycles, Duke Math. J. 127 (2005), no. 1, 35-88.

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