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LECTURE 1: COXETER GROUPS AND SCHUBERT CALCULUS
STEVEN PON, ALEXANDER WAAGEN
Class website: http://www.math.ucdavis.edu/ anne/WQ2009/280.html
Recommended references: Combinatorics of Coxeter Groups, by Bjorner and Brenti; Reáection 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.
- About Schubert Polynomials Schubert polynomials were Örst 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 the cohomology ring H(G=B). They are also related to áag varieties and Grassman- nians, etc.
- The Symmetric Group The symmetric group Sn is of primary importance in the study of Coxeter groups and Schubert polynomials. We deÖne Sn as follows:
DeÖnition 2.1. Let Sn be the group generated by si, for 1 i < n, with relations:
s^2 i = 1 for all 1 i < n; sisj = sj si if ji jj 2 ; and sisi+1si = si+1sisi+1 for all 1 i < n.
Alternatively, we can think of Sn as permuting the numbers f 1 ; 2 ; : : : ; ng. We can represent a permutation using 1-line notation, say! = [! 1 ;! 2 ; : : : ; !n] where !i = !(i). For example, the permutation of f 1 ; 2 ; 3 g that switches 1 and 2 and leaves 3 Öxed is! = [2; 1 ; 3]. We can then view the elements si as transpositions that either switch the numbers in positions i and i + 1, or switch the locations of i and i + 1, depending on whether si acts 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 necessarily unique. We let R(!) be the set of all reduced expressions of !. If w is a reduced expression of !, we let (w) = number of transpositions in w. By the following lemma,(!) = `(w) is well deÖned.
Lemma 2.2. Given w; v 2 R(!), (w) =(v).
Date : January 5, 2009.
One last thing we must note about the symmetric group is the existence of a unique longest element. In 1 -line notation, this element is [n; n 1 ; : : : ; 1], and it
has length (n 2 1) n. We denote this element by! 0.
- Divided Difference Operators
DeÖnition 3.1. Let K[X] := Z[x 1 ; x 2 ; : : : ; xn] be the polynomial ring over the integers in n variables.
If! 2 Sn, then Sn acts on K[X] by! (xi) = x!(i) for i = 1; 2 ; :::; n.
DeÖnition 3.2. We deÖne the divided di§erence operator, @i : K[X]! K[X], by
@if (x 1 ; : : : ; xn) =
f (x 1 ; : : : ; xn) sif (x 1 ; : : : ; xn) xi xi+
for 1 i < n.
Given this deÖnition, one can check the following relations: (1) @^2 i = 0 (2) @i@j = @j @i for ji jj 2 (3) @i@i+1@i = @i+1@i@i+ These three relations are checked explicitly below: (1) Let f 2 K[X]. Then
@ i^2 (f ) = @i(
f (x 1 ; : : : ; xn) sif (x 1 ; : : : ; xn) xi xi+
f (x 1 ;:::;xn) sif (x 1 ;:::;xn) xi xi+1 ^ si(^
f (x 1 ;:::;xn) sif (x 1 ;:::;xn) xi xi+1 ) xi xi+
=
(xi xi+1)^2
(f (x 1 ; : : : ; xn) sif (x 1 ; : : : ; xn) f (x 1 ; : : : ; xn) + sif (x 1 ; : : : ; xn))
= 0
(2) Let f 2 K[X]. Then
@i@j f =
f (x 1 ;:::;xn) sj f (x 1 ;:::;xn) xj xj+1 ^ si(^
f (x 1 ;:::;xn) sj f (x 1 ;:::;xn) xj xj+1 ) xi xi+
=
(xi xi+1)(xj xj+1)
[f (x 1 ; : : : ; xn) sj (f (x 1 ; : : : ; xn)) si(f (x 1 ; : : : ; xn)) + sisj (f (x 1 ; : : : ; xn))]
= @j @if
(3) Similar to above ñsimply expand using the deÖnition, and apply the relation
sisi+1si = si+1sisi+1.
Given the above three relations for divided di§erence operators, we can deÖne the divided di§erence operator corresponding to a general element of the symmetric group:
Figure 1. An algorithm to Önd 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 = x^31 + x^21 x 2 + x^21 x 3 + x^21 x 4.
The set of reduced words C (!) can be found by checking for descents in !. If there is a descent at! (i), multiply by si, and form a tree as in Ögure 1 to Önd the set of all inverses of reduced words of !, from which it is trivial to Önd the set of reduced words of !.
Note: For those interested in experimentation, SAGE (sagemath.org) can be very helpful. New functionality is being added daily, and itís free and open-source.
- Coxeter Groups
DeÖnition 5.1. Let S be a set. A matrix m : S S! f 1 ; 2 ; : : : ; 1g is called a Coxeter matrix if:
m(s; s^0 ) = m(s^0 ; s) for all s; s^0 2 S m(s; s^0 ) = 1 () s = s^0
DeÖnition 5.2. A Coxeter graph is a graph with vertex set S and an undirected edge fs; s^0 g if m(s; s^0 ) 3. Additionally, we label the edge fs; s^0 g by m(s; s^0 ) if m(s; s^0 ) 4.
Example 5.3. The following is a Coxeter matrix:
0 B B @
C
C
A
The Coxeter graph corresponding to this matrix is given in Ögure 2.
Figure 2. A Coxeter graph.
S. Billey, W. Jockusch, R. Stanley, Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), no. 4, 345ñ374. S. Fomin, R. Stanley, Schubert polynomials and the nil-Coxeter algebra, Adv. Math. 103 (1994), no. 2, 196ñ207.
