LECTURE 13: YANG–BAXTER EQUATIONIGOR RUMANOVRecall from previous lecture the definition of the divided difference operator∂i=1xi− xi+1(1 − si).We showed:Proposition 0.1. Let w0be the longest element, then∂w0= a−1δXw∈Snε(w)w,where aδ=Q1≤i<j≤n(xi− xj) and δ = (n − 1, n − 2, . . . , 1, 0)One can defineaα=Xw∈Snε(w)w(xα),(α – any n-tuple of integers).The Schur functions (Schur polynomials):sα−δ=aαaδ,standard form:sλ=aλ+δaδ.Remark 0.2. ∂w0xα= sα−δ– the cause of using non-s tandard notation.Definition 0.3. Define isobaric divided difference operators πi:πif = ∂i(xif), f ∈ Z[x1, . . . , xn].This satisfies relations:πiπj= πjπi, if |i − j| > 1,πiπi+1πi= πi+1πiπi+1,π2i= πi.Exercise: Check that these relations are satisfied.Date: February 4, 2009.1Remark 0.4. πiare used to define Grothendieck polynomials similarly to ∂ibeingused to define Schubert polynomials.Asssociate to every permutation w ∈ Snan operator of degree 0:πw= πa1. . . πak, where a1. . . ak∈ R(w)Remark 0.5. This is independent of the r e duced word since the graph Γ(w) isconnected and πisatisfy the braid and commutation relations.Proposition 0.6.πw0f = a−1δXw∈Snε(w)w(xδf),in particular,πw0xα= sα.Proof. π1f = ∂1(x1f), π1π2f = ∂1(x1∂2(x2f)) = (can move ∂2past x1) = ∂1∂2(x1x2f),. . . . . . . . .π1. . . πrf = ∂1. . . ∂r(x1. . . xrf), also(1 . . . n − 1)(1 . . . n − 2) . . . (12)(1) ∈ R(w0)⇒ πw0(f) = ∂w0(xδf) Definition 0.7. x = (x1, . . . , xn), y = (y1, . . . , yn) – sets of indeterminates. Partialresultant∆(x, y) =Yi+j≤n(xi− yj).Definition 0.8. To each w ∈ Snassociate a double Schubert polynomial σw(x, y):σw(x, y) = ∂w−1w0∆(x, y),where the divided difference operators are taken w.r.t. x varia ble. The simpleSchubert polynomials are the specializationσw(x) = σw(x, 0) = ∂w−1w0xδ.Remark 0.9. Thus there exist recursive formulas for Schubert polynomials:∂uσw=σwu−1, if l(wu−1) = l(w) − l(u)0, else∂uσw= ∂u∂w−1w0∆ = ∂uw−1w0∆,if l(uw−1w0) = l(u) + l(w−1w0) (see previous lecture, = 0 otherwise)= ∂(wu−1)−1w0∆ = σwu−12⇐⇒l(u) = l(uw−1w0) − l(w−1w0) = l(w0) − l(uw−1) − l(w0) + l(w−1).1. Yang-Baxter Equations(see Fomin, Kirillov, “The Yang-Baxter equations, symmetric functions andSchubert polynomials”, Discrete Math. 153 (1996), 123).Goal – combinatorial formulas for Schubert polynomials.Definition 1.1. The Iwahori-Hecke algebra Hnabis generated by u1, . . . , un−1, sat-isfying the relations:uiuj= ujui, if |i − j| > 1,uiui+1ui= ui+1uiui+1,u2i= aui+ b.Example 1.2. : Hn0,1= C[Sn],Hn1,0= algebra of isobaric divided differences,H = Hn0,0– nil-Coxeter algebra. H has a basis indexed by permutations withmultiplication rule:u · w =uw, if l(uw) = l(u) + l(w)0, elseSet hi(x) = 1 + xui.Lemma 1.3.hi(x)hi(y) = h(x + y),hi(x)hj(y) = hj(y)hi(x), if |i − j| > 1,hi(x)hj(x + y)hi(y) = hj(y)hi(x + y)hj(x), |i − j| = 1(the Yang-Baxter Equation).Exercise: Check it.Definition 1.4. A configuratio n is a family C of n c ontinuous strands w hich cuteach vertical line at a unique p oint.3Example 1.5. :1234x1x2x3x4s3s1s2s1s3s2Each vertical line crosses every strand. Each pair o f strands crosses at most onceand at distinct x-coordinates.w = s3s1s2s1s3s2is reduced since strands do not cross twice.For w = s3s2s1s2s3s2, a1= 3, a2= 2, . . . , xi– weight:φ(C) = ha1(xk1− xl1)ha2(xk2− xl2) . . . ham(xkm− xlm)(subtracted argument is weight o f the strand with the steeper slope).Lemma 1.6. The weights of the st rands being fixed, the polynomial φ(C) dependsonly on the permutation w underlying C.Proof. G(w) - graph of reduced words is connected. Hence it suffices to show thatφ(C) remains unchanged under the commutation and braid relations.4Commutation relations:xyztyxtzssssij jihi(x − y)hj(z − t) = hj(z − t)hi(x − y), if |i − j| > 1Braid relation:xyzzyxsssssii+1i+1ii+1sihi(y − z)hi+1(x − z)hi(x − y) = hi+1(x − y)hi(x − z)hi+1(y − z),– the Yang-Baxter equation.