## Rank two and a positivity conjecture

Last update: 25 February 2013

## Rank two and a positivity conjecture

In this section we will give explicit formulas for the rank two root systems. The data supports the following positivity conjecture which generalizes the theorems of Brion [Bri2002, formula before Theorem 1] and Graham [Gra9908172, Corollary 4.1].

Conjecture 5.9. For $\beta \in {R}^{+}$ let ${y}_{\beta }={e}^{-\beta }$ and ${\alpha }_{\beta }={e}^{-\beta }-1$ and let $d\left(w\right)=\ell \left({w}_{0}\right)-\ell \left(w\right)$ for $w\in W\text{.}$ Let ${c}_{wv}^{z}$ be the structure constants of ${K}_{T}\left(G/B\right)$ with respect to the basis $\left\{\left[{𝒪}_{{X}_{w}}\right] \mid w\in W\right\}$ as defined in (0.1). Then

$cwvz= (-1)d(w)+d(v)-d(z) f(α,y),where f(α,y)∈ ℤ≥0 [ αβ,yβ ∣ β∈R+ ] ,$

that is, $f\left(\alpha ,y\right)$ is a polynomial in the variables ${\alpha }_{\beta }$ and ${y}_{\beta },$ $\beta \in {R}^{+},$ which has non-negative integral coefficients.

In the following, for brevity, use the following notations:

$in KT(G/B), [w]= [𝒪Xw], αrs= e-(rα1+sα2)-1 ,andyrs= e-(rα1+sα2), in K(G/B), [w]= [𝒪Xw], αrs=0 ,andyrs=1, in HT*(G/B), [w]= [Xw], αrs=rα1+sα2, ,andyrs=1, in H*(G/B), [w]= [Xw], αrs=0, ,andyrs=1,$

and in ${H}_{T}^{*}\left(G/B\right)$ and in ${H}^{*}\left(G/B\right)$ the terms in $\left\{\right\}$ brackets do not appear.

Type ${A}_{2}\text{.}$ For the root system $R$ of type ${A}_{2}$

$α1=-ω1+2ω2, λ1=ρ, λs1=ω2=13α1+23α2, λs2s1=s2ω2=13α1-13α2, α2=2ω1-ω2, λw0=0, λs2=ω1=23α1+13α2, λs1s2=s1ω1=-13α1+13α2.$

Formulas for the Schubert classes in terms of homogeneous line bundles can be given by

$[s1s2s1]=1, [1]= ( 1-es1ω1 X-ω1 ) [s1]= ( 1-es2ω2 X-ω2 ) [s2], [s2s1]= 1-e-ω1 X-ω2, [s1s2]=1- e-ω2 X-ω1 [s1]= ( 1-es2ω2 X-ω2 ) [s2s1], [s2]= ( 1-es1ω1 X-ω1 ) [s1s2],$

and

$[s1s2s1]=1, [s1s2]=1- e-ω2X-ω1, [s2s1]=1- e-ω1X-ω2, [s1]=1-e-ω2 X-s1ω1- e-ω2X-ω1+ e-2ω2X-ω2, [s2]=1-e-ω1 X-s2ω2- e-ω1X-ω2 +e-2ω1X-ω1, [1]=1-e-ω2 X-s1ω1- e-ω1 X-s2ω2+ e-2ω1X-ω1 +e-2ω2 X-ω2- e-ρX-ρ.$

The multiplication of the Schubert classes is given by

$[1]2=- α10α01α11[1] ,[s1]2= α01α11[s1], [s2]2=α10 α11[s2], [1][s1]= α01α11[1], [s1][s2]= -α11[1], [s2][s1s2] =-α11[s2], [1][s2]= α10α11[1], [s1][s1s2] =y01[1]-α01[s1] ,[s2][s2s1] =y10[1]-α10[s2], [1][s1s2]= -α11[1], [s1][s2s1]= -α11[s1], [1][s2s1]= -α11[1], [s1s2]2= y01[s2]-α01 [s1s2], [s1s2] [s2s1]= {-[1]}+ [s1]+[s2], [s2s1]2= y10[s1]-α10 [s2s1].$

Type ${B}_{2}\text{.}$ For the root system $R$ of type ${B}_{2}$

$α1=2ω1-ω2, λ1=ρ=2α1+32α2, λs1=ω2=α1+α2, α2=-2ω1+2ω2, λw0=0, λs2=ω1=α1+12α2, λs2s1=s2ω2=α1, λs1s2s1=s1s2ω2=-α1, λs1s2=s1ω1=12α2, λs2s1s2=s2s1ω1=-12α2.$

Formulas for the Schubert classes in terms of homogeneous line bundles can be given by

$[s1s2s1s2]=1, [1]=(1-es1ω1X-ω1) [s1]=(1-es2ω2X-ω2) [s2], [s1s2s1]=1- e-ω2X-ω2, [s2s1s2]=1- e-ω1X-ω1, [s2s1]= (1-e-ω1X-s1ω1) [s2s1s2], [s2s1]= (1-es2s1ω1X-ω1) [s2s1s2], [s1]= (1-es2ω2X-ω2) [s2s1],[s2] =(1-es1ω1X-ω1) [s1s2],$

and

$[s1s2s1s2]=1, [s1s2s1]=1- e-ω2X-ω2, [s2s1s2]=1- e-ω1X-ω1, [s1s2]= (1-e-ω2)- e-ω2X-ω2- e-ω2X-s2ω2+ (e-ρ+e-s1ρ) X-ω1, [s2s1]=1- e-ω1X-ω1- e-ω1X-s1ω1+ e-2ω1X-ω2, [s1]=(1-e-ω2)+ (e-ρ+e-s1ρ) X-s1ω1+ (e-ρ+e-s1ρ) X-ω1 -e-ω2 X-s1s2ω2- e-ω2 X-s2ω2- ( e-2ω2+ e-ω2 ) X-ω2, [s2]= (1+e-2ω1)+ e-2ω1 X-s2ω2+ e-2ω1 X-ω2 - e-ω1 X-s2s1ω1 - e-ω1 X-s1ω1 - ( e-3ω1 + e-ω1 ) X-ω1 , [1] = (1+e-2ω1) - e-ω1 X-s2s1ω1 + (e-ρ+e-s1ρ) X-s1ω1 - (e-3ω1+e-ω1) X-ω1 - e-ω2 X-s1s2ω2 + e-2ω1 X-s2ω2 - (e-2ω2+e-ω2) X-ω2 + e-ρX-ρ .$

The multiplication of the Schubert classes is given by

$[1]2=α10 α01α11α21 [1], [1][s1]=- α01α11α21 [1], [1][s2]- α10α11α21[1], [1][s1s2]= α11α21[1], [1][s2s1]= α11α21[1], [1][s1s2s1]= -α11(1+y11)[1], [1][s2s1s2] =-α21[1], [s1]2=- α01α11α21 [s1], [s1][s2]= α11α21[1], [s1][s1s2] =-α11 (y01+y11)[1]+ α01α11[s1], [s1][s2s1] =α11α21[s1], [s1][s1s2s1] =-α11(1+y11) [s1], [s1][s2s1s2] =y11[1]-α11[s1], [s2]2=-α10 α11α21[s2], [s2][s1s2]= α11α21[s2], [s2][s2s1]= -α21y10[1]+ α10α21[s2], [s2][s1s2s1] =y21[1]-α21[s2], [s2][s2s1s2] =-α21[s2], [s1s2]2=-α11 (y01+y11)[s2] +α01α11[s1s2], [s1s2] [s2s1]= ({α11}+y21) [1]-α11[s1] -α21[s2], [s1s2] [s1s2s1]= {-(y01+y11)[1]} +y01[s1]+ (y11+y12)[s2] -α01[s1s2], [s1s2] [s2s1s2]=y11 [s2]-α11 [s1s2], [s2s1]2=- α21y10[s1]+ α10α21[s2s1], [s2s1] [s1s2s1]=y21 [s1]-α21 [s2s1], [s2s1] [s2s1s2]= {-y10[1]}+ y10[s1]+y10 [s2]-α10 [s2s1], [s1s2s1]2= {-y11[s1]}+ (y01+y11) [s2s1]-α01 [s1s2s1], [s1s2s1] [s2s1s2]= {[1]-[s1]-[s2]} +[s1s2]+ [s2s1], [s2s1s2]2= y10[s1s2]-α10 [s2s1s2].$

Type ${G}_{2}\text{.}$ For the root system $R$ of type ${G}_{2}$

$λ1=ρ=5α+3α2, λs1s2s1=s1 s2ω2=α2, λs1=ω2=3α1 +2α2, λs2s1s2s1 =s2s1s2ω2=-α2, λs2=ω1=2α1 +α2, λs1s2s1s2= s1s2s1ω1=-α1, λs2s1=s2ω2 =3α1+α2, λs1s2s1s2s1 =s1s2s1s2ω2 =-3α1-α2, λs1s2=s1ω1 =α1+α2, λs2s1s2s1s2= s2s1s2s1ω1 =-α1-α2, λs2s1s2= s2s1ω1=α1, λw0=0.$

Formulas for the Schubert classes in terms of homogeneous line bundles can be given by

$[s1s2s1s2s1s2]=1, [1]=(1-es1ω1X-ω1) [s1]=(1-es2ω2X-ω2) [s2], [s1s2s1s2s1]= 1-e-ω2X-ω2, [s2s1s2s1s2]=1 -e-ω1X-ω1, [s2s1s2s1]= (1-e-ω1X-s1ω1) [s2s1s2s1s2], [s1s2s1s2]= (1-e-s1ω1X-ω1) [s2s1s2s1s2], [s1s2s1]=see below, [s2s1s2]= 1-es2s2ω1X-ω1 1+X-ω1 [s1s2s1s2], [s2s1]= ( 1-e-ω1 X-s1s2s1ω1 ) [s2s1s2], [s1s2]= ( 1-es2s1ω1 X-ω1 ) [s1s2], [s1]= ( 1-es2ω2 X-ω2 ) [s2s1], [s2]= ( 1-es1ω1 X-ω1 ) [s1s2], [s1s2s1]= (1-e-α2X-ω2) [s2s1s2s1]+ e-α2 (1+eω1X-ω2) [s2s1] 1+e-α2 ,$

and

$[w0] = 1 , [s2s1s2s1s2] = 1 - y21X-ω1 , [s1s2s1s2s1] = 1 - y32X-ω2 , [s2s1s2s1] = 1 - y21X-ω1 - y21X-s1ω1 + y42X-ω2 , [s1s2s1s2] = (1-y32) + (y22+y42+y43+y53) X-ω1 - y32X-s1ω1 - y32 X-s2s1ω1 - y32X-ω2 - y32X-s2ω2 , [s2s1s2] = (1-y21+y42) + (y42-y21-y52-y53-y63) X-ω1 + (y42-y21) X-s1ω1 + (y42-y21) X-s2s1ω1 + y42X-ω2 + y42X-s2ω2 , [s1s2s1] = (1-2y32) + (y22+y42+y43+y53) X-ω1 + (y22+y42+y43+y53) X-s1ω1 - y32X-s2s1ω1 - y32X-s1s2s1ω1 - (y32+y43+y53) X-ω2 - y32X-s2ω2 - y32X-s1s2ω2 , [s2s1] = (1-y21+2y42) + (y42-y21-y52-y53-y63) X-ω1 + (y42-y21-y32-y53-y63) X-s1ω1 + (y42-y21) X-s2s1ω1 + (y42-y21) X-s1s2s1ω1 + (y42+y63) X-ω2 + y42X-s2ω2 + y42X-s1s2ω2 , [s1s2] = 1 - y11 - y21 - y32 - y43 - y53 + (y22+y32) (1+y10+y20) X-ω1 + (y22+y32+y42) X-s1ω1 + (y22+y32+y42) X-s2s1ω1 - (y32+y43+y53) X-ω2 - (y32+y43+y53) X-s2ω2 - y32 X-s1s2ω2 - y32 X-s2s1s2ω2 , [s2] = (1+y31+y32+2y42+y63) - (y21+y52+y53+y84) X-ω1 - (y21+y52+y53) X-s1ω1 - (y21+y52+y53) X-s2s1ω1 - y21X-s1s2s1ω1 - y21 X-s2s1s2s1ω1 + (y42+y63)X-ω2 + (y42+y63) X-s2ω2 + y42 X-s1s2ω2 + y42 X-s2s1s2ω2 , [s1] = 1- (y11+y21+y32+2y43+2y53) + (y22+y54) (1+y10+y20) X-ω1 + (y22+y54) (1+y10+y20) X-s1ω1 + (y22+y32+y42) X-s2s1ω1 + (y22+y32+y42) X-s1s2s1ω1 - (y32+y43+y53+y64) X-ω2 - (y32+y43+y53) X-s2ω2 - (y32+y43+y53) X-s1s2ω2 - y32X-s2s1s2ω2 - y32 X-s1s2s1s2ω2 , [1] = ( 1 + y31 + y42 + y63 - y53 - y43 ) - y21(1+y32)2 X-ω1 + y22(1+y10+y20) (1+y21+y31) X-s1ω1 - (y21+y52+y53) X-s2s1ω1 + y22X-s1s2s1ω1 - y21X-s2s1s2s1ω1 - y32(1+y11) (1+y21)X-ω2 + (y42+y63) X-s2ω2 - (y32+y43+y53) X-s1s2ω2 + y42X-s2s1s2ω2 - y32 X-s1s2s1ω2ω2 + y53X-ρ .$

The multiplication of the Schubert classes is given by

$[1]2 = α10α01α11 α21α31α32 [1], [1][s1] = - α01α11α21 α31α32[1], [1][s2] = - α10α11α21 α31α32[1], [1][s1s2] = α11α21 α31α32[1], [1][s2s1] = α11α21 α31α32[1], [1] [s1s2s1] = -α11α21α32 (1+y11+y21) [1], [1] [s2s1s2] = -α21 α31α32[1], [1] [s1s2s1s2] = α31α32 (1+y21)[1], [1] [s2s1s2s1] = α31α32 (1+y21)[1], [1] [s1s2s1s2s1] = -α32 (1+y32)[1], [1] [s2s1s2s1s2] = -α21 (1+y21)[1], [s1]2 = -α01α11α21 α31α32[s1] [s1] [s2] = α11α21 α31α32[1] [s1][s1s2] = -α11α21 α31α32 (y01+y11+y21) [1]+α01α11 α21α32[s1] [s1][s2s1] = α11α21 α31α32[s1] [s1][s1s2s1] = -α11α21α32 (1+y11+y21) [s1] [s1] [s2s1s2] = α21α32 (y11+y21)[1] -α11α21α32[s1] [s1] [s1s2s1s2] = -α32 (y22+y32)[1] +α11α32 (1+y11)[s1] [s1] [s2s1s2s1] = α21α32 (1+y21)[s1] [s1] [s1s2s1s2s1] = -α32(1+y32) [s1] [s1] [s2s1s2s1s2] = y32[1]- α32[s1] [s2]2 = -α10α11 α21α31 α32[s2] [s2] [s1s2] = α11 α21α31 α32[s2] [s2] [s2s1] = -α21α31 α32y10[1] +α10α21α31 α32[s2] [s2] [s1s2s1] = α21α32 (y21+y31)[1] -α21α31α32 [s2] [s2] [s2s1s2] = -α21α31α32[s2] [s2] [s1s2s1s2] = α21α32 (1+y21)[s2] [s2] [s2s1s2s1] = -α21 (y31+y52)[1] +α21α31 (1+y21)[s2] [s2] [s1s2s1s2s1] = y63[1]-α21 (1+y21+y42) [s2] [s2] [s2s1s2s1s2] = -α21(1+y21) [s2] [s1s2]2 = - α11α21α32 (y01+y11+y21) [s2] + α01α11α21α32 [s1s2] [s1s2] [s2s1] = α21α32 (y11+y21+α31) [1] - α11α21α32 [s1] - α21α31α32 [s2] [s1s2] [s1s2s1] = - α32 ( y32+y42 {+α11(y01+2y11+y21)} ) [1] + α11α32 (y01+y11) [s1] + ( α31α32y11+ α11α32 (y01+y11+y21) ) [s2] - α01α11α32 [s1s2] [s1s2] [s2s1s2] = α21α32 (y11+y21) [s2]-α11 α21α32 [s1s2] [s1s2] [s1s2s1s2] = -α32 (y22+y32) [s2]+α11 α32(1+y11) [s1s2] [s1s2] [s2s1s2s1] = (y63{+α32(y11+y21)}) [1]-α32y11[s1] - ( α32(y11+y21) +α31y32 ) [s2]+α11α32 [s1s2] [s1s2] [s1s2s1s2s1] = {-(y33+y43+y53)[1]} +y33[s1]+ (y33+y43+y53) [s2] - α11(1+y11+y22) [s1s2] [s1s2] [s2s1s2s1s2] = y32[s2]- α32[s1s2] [s2s1]2 = -α21α31 α32y10[s1] α10α21α31 α32[s2s1] [s2s1] [s1s2s1] = α21α31 (y21+y31)[s1] -α21α31α32 [s2s1] [s2s1] [s2s1s2] = -α21 (y51+y52{+α31y10}) [1]+α21 (α10y31+α32y10) [s1] + α21α31(y10+y21) [s2]-α10α21 α31[s2s1] [s2s1] [s1s2s1s2] = (y62{+α31(y21+y31)}) [1]- (α31y21+α10(y31+y41)) [s1] - (α31y21+α32y31) [s2]+α21α31 [s2s1] [s2s1] [s2s1s2s1] = -α21 (y31+y52)[s1] +α21α31 (1+y21)[s2s1] [s2s1] [s1s2s1s2s1] = y63[s1]-α21 (1+y21+y42) [s2s1] [s2s1] [s2s1s2s1s2] = {-y31[1]}+ y31[s1]+ y31[s2]-α31 [s2s1] [s1s2s1]2 = -α32 ( y32+y42 {+α11(y11+y21)} ) [s1] + ( α11α32 (y01+y11+y21) +α31α32y11 ) [s2s1]-α01 α11α32 [s1s2s1] [s1s2s1] [s2s1s2] = ( 1 { +α11 (y11+y22+y33+y31+y42) +α31(y21+y32)+ α32y21 } ) [1] - ( α11(y21+α32) +α10(y31+y41+y32+y42) ) [s1] - ( α31(y21+y32) +α11(y21+y32+y31+α42) ) [s2] + α11+α32[s1s2] +α21α31 [s2s1] [s1s2s1] [s1s2s1s2] = { - ( y33+2y43+y53+ α11(y01+y11) +α21(y11+y21) ) [1] } + ( y33+y43 { +α11(y01+y11) +α21(y11+y21) } ) [s1] ( (y33+y43+y53) { +α11(y01+y11) +α21(y11+y21) } ) [s2] - α11 (y01+y11+y22) [s1s2] - ( α11(y01+y11)+ α21(y11+y21) ) [s2s1] + α01α11[s1s2s1] [s1s2s1] [s2s1s2s1] = (y62{+α32y21}) [s1]- (α31y32+α32(y11+y21)) [s2s1] + α11α32[s1s2s1] [s1s2s1] [s1s2s1s2s1] = {-(y43+y53)[s1]} +(y33+y43+y53) [s2s1] - α11(1+y11+y22) [s1s2s1] [s1s2s1] [s2s1s2s1s2] = { (y11+y21)[1]- (y11+y21)[s1]- (y11+y21)[s2] } + y11[s1s2] +(y11+y21) [s2s1]- α11[s1s2s1] [s2s1s2]2 = -α21 (y21+y42)[s2] + ( α11α21y31+ α21α31y10 ) [s1s2]- α10α21α31 [s2s1s2] [s2s1s2] [s1s2s1s2] = y53[s2]- ( α21y31+α11 α21α32y21 ) [s1s2]+ α21α31 [s2s1s2] [s2s1s2] [s2s1s2s1] = {-(y51+y52+α31y10)[1]} +(y41{+α31y10}) [s1] + ( y42+y52 {+α31y10} ) [s2]- (α11y31+α31y10) [s1s2] - α31y10 [s2s1]+ α10α31 [s2s1s2] [s2s1s2] [s1s2s1s2s1] = { (y31+y32+y42) [1]- (y31+y32)[s1] -(y31+y32+y42) [s2] } + (y31+y32)[s1s2] +y31[s2s1]- α31[s2s1s2] [s2s1s2] [s2s1s2s1s2] = y31[s1s2]- α31[s2s1s2] [s1s2s1s2]2 = {-y42[s2]}+ ( y31+y42 {+α01y21+α31y11} ) [s1s2] - ( α01(y11+y21) +α31(y01+y11) ) [s2s1s2]+ α01α11 [s1s2s1s2] [s1s2s1s2] [s2s1s2s1] = { (y21+y31+y32+y42+α11) [1]- (y21+y31+y32+α11) [s1]- (y21+y31+y32+y42+α11) [s2] } + (y31+y42{+α11}) [s1s2]+ (y21+y31{+α11}) [s2s1]-α11 [s1s2s1]- α31[s2s1s2] [s1s2s1s2] [s1s2s1s2s1] = { -(y01+y11+y21+y22+y32) [1]+ (y01+y11+y21+y22) [s1] + (y01+y11+y21+y22+y32) [s2]- (y01+y11+y21+y22) [s1s2] - (y01+y11+y21) [s2s1] } +y01[s1s2s1] +(y01+y11+y21) [s2s1s2]- α01[s1s2s1s2] [s1s2s1s2] [s2s1s2s1s2] = {-y21[s1s2]} +(y11+y21) [s2s1s2]- α11[s1s2s1s2] [s2s1s2s1]2 = { -y52[s1]+ (y42+y52) [s2s1] } - (α11y31+α31y10) [s1s2s1] + α10α31 [s2s1s2s1] [s2s1s2s1] [s1s2s1s2s1] = { y42[s1]- (y31+y41) [s2s1] } +(y31+y32) [s1s2s1] - α31[s2s1s2s1] [s2s1s2s1] [s2s1s2s1s2] = { -y10[1]+ y10[s1]+ y10[s2]- y10[s1s2] -y10[s2s1] } +y10[s1s2s1] +y10[s2s1s2] -α10[s2s1s2s1] [s1s2s1s2s1]2 = { -y32[s1]+ (y22+y32) [s2s1]- (y11+y21+y22) [s1s2s1] } + (y01+y11+y21) [s2s1s2s1]- α01[s1s2s1s2s1] [s1s2s1s2s1] [s2s1s2s1s2] = { [1]-[s1]- [s2]+ [s1s2]+ [s2s1]- [s1s2s1]- [s2s1s2] } + [s1s2s1s2]+ [s2s1s2s1] [s2s1s2s1s2]2 = y10[s1s2s1s2] -α10[s2s1s2s1s2]$

## Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and the Schubert calculus authored by Stephen Griffeth and Arun Ram. It was dedicated to Alain Lascoux.