LECTURE 1: COXETER GROUPS AND SCHUBERT CALCULUSSTEVEN PON, ALEXANDER WA AGENClass website: http://www.math.ucdavis.edu/ anne/WQ2009/280.htmlRecommended references: Combinatorics of Coxeter Groups, by Bjorner and Brenti; Re‡ection Groups and Coxeter Groups, by Humphreys; Young Tableaux by Fulton; Symmetric Functions, Schubert Polynomials and Degeneracy Loci by Manivel; Notes on Schubert Polynomials by Macdonald.1. About Schubert PolynomialsSchubert polynomials were …rst introduced in 1982 by Lascoux and Schutzen-berger. They are of great interest in mathematics, as they relate to combinatorics,representation theory and geometry. For example, they form a natural basis of thecohomology ring H (G=B). They are also related to ‡ag varieties and Grassman-nians, etc.2. The Symmetric GroupThe symmetric group Snis of primary importance in the study of Coxeter groupsand Schubert polynomials. We de…ne Snas follows:De…nition 2.1. Let Snbe the group generated by si, for 1  i < n, with relations: s2i= 1 for all 1  i < n; sisj= sjsiif ji  jj  2; and sisi+1si= si+1sisi+1for all 1  i < n.Alternatively, we can think of Snas permuting the numbers f1; 2; : : : ; ng. Wecan represent a permutation using 1-line notation, say ! = [!1; !2; : : : ; !n] where!i= !(i). For example, the permutation of f1; 2; 3g that switches 1 and 2 andleaves 3 …xed is ! = [2; 1; 3]. We can then view the elements sias transpositionsthat either switch the numbers in positions i and i + 1, or switch the locations of iand i + 1, depending on whether siacts on the right or the left.Given an element ! of Sn, we can express ! as a minimal product of transposi-tions si. We call such an expression a reduced expression, which is not necessarilyunique. We let R(!) be the set of all reduced expressions of !. If w is a reducedexpression of !, we let `(w) = number of transpositions in w. By the followinglemma, `(!) = `(w) is well de…ned.Lemma 2.2. Given w; v 2 R(!), `(w) = `(v ).Date : January 5, 2009.1One last thing we must note about the symmetric group is the existence of aunique longest element. In 1-line n otation, this element is [n; n  1; : : : ; 1], and ithas length(n1)n2. We denote this element by !0.3. Divided Difference OperatorsDe…nition 3.1. Let K[X] := Z[x1; x2; : : : ; xn] be the polynomial ring over theintegers in n variables.If ! 2 Sn, then Snacts on K[X] by ! (xi) = x!( i)for i = 1; 2; :::; n.De…nition 3.2. We de…ne the divided di¤erence operator, @i: K[X] ! K[X], by@if(x1; : : : ; xn) =f(x1; : : : ; xn)  sif(x1; : : : ; xn)xi xi+1for 1  i < n.Given this de…nition, one can check the following relations:(1) @2i= 0(2) @i@j= @j@ifor ji  jj  2(3) @i@i+1@i= @i+1@i@i+1These three relations are checked explicitly below:(1) Let f 2 K[X]. Then@2i(f) = @i(f(x1; : : : ; xn)  sif(x1; : : : ; xn)xi xi+1)=f(x1;:::;xn)sif(x1;:::;xn)xixi+1 si(f(x1;:::;xn)sif(x1;:::;xn)xixi+1)xi xi+1=1(xi xi+1)2(f(x1; : : : ; xn)  sif(x1; : : : ; xn)  f(x1; : : : ; xn) + sif(x1; : : : ; xn))= 0(2) Let f 2 K[X]. Then@i@jf =f(x1;:::;xn)sjf(x1;:::;xn)xjxj+1 si(f(x1;:::;xn)sjf(x1;:::;xn)xjxj+1)xi xi+1=1(xi xi+1)(xj xj+1)[f(x1; : : : ; xn)  sj(f(x1; : : : ; xn))  si(f(x1; : : : ; xn)) + sisj(f(x1; : : : ; xn))]= @j@if(3) Similar to above –simply expand using the de…nition, and apply the relationsisi+1si= si+1sisi+1.Given the above three relations for divided di¤erence operators, we can de…nethe divided di¤erence operator corresponding to a general element of the symmetricgroup:2De…nition 3.3. Given ! 2 Snand w = si1si2   sik2 R(!), we let@!= @i1@i2   @ikBy the above relations, @!is well-de…ned and does not depend on the choice ofreduced word. The algebra generated by @ifor 1  i < n is known as the nil-Heckealgebra. Note that if we were to try to use a non-reduced word in the de…nition of@!, we would get 0 because @2i= 0.We can then de…ne Schubert polynomials:De…nition 3.4. For every ! 2 Sn, we de…ne the Schub e rt polynomial !by:!= @!1!0(xn11xn22   x1n1x0n)where !0is the unique longest element of Sn.This is a straightforward de…nition; however, it is not ideal from a combinatorialstandpoint since it involves applying a large number of divided di¤erence operators.Billey, Jockusch and Stanley derived a more combinatorial formula (based on workby Fomin and Stanley) for Schubert polynomials that is presented below.4. Combinatorial Definition of Schubert PolynomialsIn the following, we identify a reduced word with the indices of that reducedword. For example, if w = w1w2w3w4= s3s1s2s1is a reduced expression for anelement ! 2 Sn, we identify w with the word 3121, so statements such as 1  w1make sense.De…nition 4.1. Let a = a1   ap2 R(!). We say that a p-tuple  = (1; : : : ; p)of positive integers is a  compatible if: 0  1 2     p; j ajfor all 1  j  p; and j< j+1if aj< aj+1.Let C(a) denote the set of a-compatible sequences.Theorem 4.2 (Fomin, Stanley 1991; Billey, Jockusch, Stanley 1993).w=Xa2R(w)X2C(a)x1   xpProof of this theorem is withheld until later in the class.Example: Let ! = [3; 1; 2; 5; 4]. Then we have R(!) = f214; 241; 421g. Notethat we are writing reduced words as acting from left to right. We have to list alla-compatible sequences for each reduced word. w = 214: 0 < 2 1 so 2= 1. Since iare weakly increasing, 1= 1 aswell. Then 3can be 2,3, or 4 since we ne ed 2< 3 a3. w = 241: There are no a-compatible sequences because 3must be 1, butwe have an ascent a1< a2, so we must h ave 0 < 1< 2 3= 1.3Figure 1. An algorithm to …nd the set of reduced words of [3,1,2,5,4]. w = 421. 0 < 3 1 so 3= 1. This forces 1= 2= 1.Therefore, w= x31+ x21x2+ x21x3+ x21x4.The set of reduced words C (!) can be found by checking for descents in !. Ifthere is a descent at ! (i), multiply by si, and form a tree as in …gure 1 to …nd theset of all inverses of reduced words of !, from which it is trivial to …nd the set ofreduced words of !.Note: For those interested in experimentation, SAGE (sagemath.org) can b every helpful. New functionality is being added daily, and it’s f ree and open-source.5. Coxeter GroupsDe…nition 5.1. Let S be a set. A matrix m : S  S ! f1; 2; : : : ; 1g is