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LECTURE 12: DIVIDED DIFFERENCE OPERATORS
WANG, QIANG
Definition 0.1. Let X = {x 1 , · · · , xn}, then si ∈ Sn acts on f ∈ Z[X] by switching the xi and xi+1. That is,
si. f (x 1 , · · · , xi, xi+1, · · · , xn) = f (x 1 , · · · , xi+1, xi, · · · , xn)
For si ∈ Sn, define ∂i : Z[X] → Z[X] by
(∂if )(x 1 , · · · , xn) =
f (x 1 , · · · , xn) − si. f (x 1 , · · · , xn) xi − xi+
In another word, ∂i = (xi − xi+1)−^1 (1 − si).
- Graph of reduced words Given w ∈ (W, S), we can define a colored graph Γ(w), called the graph of reduced words of w, as follow. The nodes in this graph are the set of all reduced words of w. Let u, v be two nodes of this graph, (i.e, two reduced work of w) then u, v are connected by an edge colored (labeled) by a defining relation of (W, S) if and only if u can be transformed to v (or vise versa) by one application of the given defining relation. It is clear that the defining relations s^2 i = e are never used in Γ(w), for all nodes in Γ(w) are reduced words.
Example 1.1. Let the Coxeter system be (S 5 ), {s 1 , s 2 , s 3 , s 4 }), and let w = [31542] in one-line notation, then one reduced word for w could be s 2 s 3 s 4 s 3 s 1 , we will just code this word by its indices 23431. Let us show the connected component of Γ(w) that contains this word. (As we will see later this is the whole graph)
23431
ttt
tt ttt
t JJJ
JJJ
JJJ
JJ
JJJ
JJJ
J
J $ $
JJJ
JJJ
JJ
JJ J
JJJ
JJJ
JJ
JJJ
JJJ
J 24134
ttt
ttt
tt t 42134
Proposition 1.2. Given w ∈ (W, S), Γ(w) is connected.
Date: January 23, 2009.
Proof. We prove the statement for the case of (Sn, {s 1 , · · · , sn− 1 }), the spirits are the same for all other types of Coxeter group. Induct on (w). For the case(w) ≤ 2 the statement clearly holds. Now assume `(w) ≥ 3, let a = a 1 · · · al and b = b 1 · · · bl be two reduced words of w. By E.P. b 1 w has a reduced expression of the form a 1 · · · aˆk · · · al, for some 1 ≤ k ≤ l. Thus b 1 a 1 · · · aˆk · · · al is another reduced expression of w. First note that by induction, b 2 · · · bl and a 1 · · · aˆk · · · al are connected by a se- quence of edges in Γ(b 1 w), thus b 1 b 2 · · · bl and b 1 · · · aˆk · · · al are connected by a sequence of edges in Γ(w). If k < l, then in Γ(wal), a 1 · · · al− 1 and b 1 · · · aˆk · · · al− 1 are connected by a sequence of edges, thus in Γ(w), a 1 · · · al and b 1 · · · aˆk · · · al are connected by a sequence of edges. Therefore a and b are connected in Γ(w). If k = l, then either a 1 and b 1 are consecutive or not. If they are not, then b 1 a 1 · · · al− 1 and a 1 b 1 a 2 · · · al− 1 are connected by a single edge labeled by a 1 b 1 = b 1 a 1. Now in Γ(w), a and a 1 b 1 a 2 · · · al− 1 are connected by a sequence of edges ”lifted” from Γ(a 1 w), thus a and b are connected in Γ(w). Finally, if k = l but a 1 and b 1 are consecutive, then by E.P. a 1 b 1 a 1 · · · aˆj · · · al− 1 for some 1 ≤ j ≤ l − 1 is another reduced word of w. If j = 1, · · · , l − 2 then we just repeat about argument for the case k < l with b′^ = a and a′^ = b 1 a 1 · · · al− 1. Otherwise if j = l − 1, in particular j > 1, then a 1 b 1 a 1 · · · aˆj · · · al− 1 is connected to b 1 a 1 b 1 · · · aˆj · · · al− 1 by an edge labeled by a 1 b 1 a 1 = b 1 a 1 b 1 in Γ(w). Now we note that b and b 1 a 1 b 1 · · · aˆj · · · al− 1 are connected in Γ(w) by lifting a path from Γ(b 1 w), and a and a 1 b 1 a 1 · · · aˆj · · · al− 1 are connected in Γ(w) by lifting a path from Γ(a 1 w), we are done.
- Properties of divided difference operators
Lemma 2.1. If f, g ∈ Z[X] then
∂i(f ∗ g) = (∂if ) ∗ g + (si. f ) ∗ (∂ig)
Proof.
∂i(f ∗ g) =
f ∗ g − si. (f ∗ g) xi − xi+
= f ∗ g − (si. f ) ∗ g + (si. f ) ∗ g − si. (f ∗ g) xi − xi+
=
f ∗ g − (si. f ) ∗ g xi − xi+
(si. f ) ∗ g − (si. f ) ∗ (si. g) xi − xi+ = (∂if ) ∗ g + (si. f ) ∗ (∂ig)
Theorem 2.2 (Nil-Coxeter relations).
∂i∂j = ∂j ∂i for |i − j| > 1 ∂i∂i+1∂i = ∂i+1∂i∂i+1 for i = 1, · · · , n − 1 ∂ i^2 = 0
Proof. In lecture note 1, Steven and Alex gave a detailed proof.
Definition 2.3. If a 1 · · · al is a reduced word of w ∈ Sn, then define ∂w = ∂a 1 · · · ∂al.
We are interested in the coefficient of w 0 after ”multiply out” the rhs. For n = 3, the least non-trivial case we see
∂w 0 = ∂ 2 ∂ 1 ∂ 2 =
x 2 − x 3
(1 − s 2 )
x 1 − x 2
(1 − s 1 )
x 2 − x 3
(1 − s 2 )
clearly cw 0 = (^) x 2 −^1 x 3 x 1 −^1 x 2 x 2 −^1 x 3 (−1)^3 , claim shown. The general case can be checked explicitly in a similar fashion.
