Flagged Grothendieck polynomials

0.21 MB·

by  Tomoo Matsumura

FLAGGED GROTHENDIECK POLYNOMIALS
TOMOO MATSUMURA
7
Abstract. We show that the flagged Grothendieck polynomials defined as generating functions
1
0 of flagged set-valued tableaux of Knutson–Miller–Yong [11] can be expressed by a Jacobi–Trudi
2
type determinant formula generalizing the work of Hudson–Matsumura [9]. We also introduce

n
theflagged skew Grothendieck polynomials inthesetwoexpressionsandshowthattheycoincide.
a
J

3
1
1. Introduction

]
O
Lascoux–Schu¨tzenberger ([12], [14]) introduced the Grothendieck polynomials to represent the
C
K-theory classes of the structure sheaves of Schubert varieties and Fomin–Kirillov ([5], [4]) gave
.
h
their combinatorial description in terms of pipe dreams or rc graphs. Knutson–Miller–Yong [11]
t
a
m expressed the Grothendick polynomial associated to a vexillary permutation as the generating
[ function of flagged set-valued tableaux, unifying the work of Wachs [16] on flagged tableaux, and

1 Buch [2] on set-valued tableaux (cf. [10]). On the other hand, in the joint work [9] with Hudson,
v
1 the author proved the Jacobi–Trudi type formula for the vexillary Grothendieck polynomials in
6
5 the context of degeneracy loci formula, generalizing his joint work [7] and [8] with Hudson, Ikeda,
3
and Naruse for the Grassmannian case (see also [1] and [15]).
0
1. Motivated by these results, we study the generating functions of flagged set-valued tableaux
0
in general beyond the ones given by vexillary permutations. For a given partition λ = (λ ≥
7 1
1 ··· ≥ λ > 0) of length r, a flagging f of λ is a nondecreasing sequence of natural numbers
r
:
v
(f ,...,f ). A flagged set-valued tableau of shape λ with a flagging f is nothing but a set-valued
i 1 r
X
tableaux of shape λ of Buch [2] satisfying extra conditions that the numbers used in the i-th row
r
a are at most f for all i. Let FSVT(λ,f) be the set of all flagged set-valued tableaux of shape λ
i
with the flagging f. Let x = (x ,x ,...) be a set of infinitely many indeterminants. Following
1 2
Knutson–Miller–Yong’s work, we define the flagged Grothendieck polynomials G (x) by
λ,f
G (x):= β|T|−|λ| x .
λ,f k
T∈FSVT(λ,f) k∈T
X Y
The main goal of this paper is to show that G (x) is given by the following Jacobi-Trudi type
λ,f
determinant formula (Theorem 2.8) :
∞
i−j
(1.1) G (x)= det βsG[fi] (x) .
λ,f s λi+j−i+s !
Xs=0(cid:18) (cid:19) 1≤i,j≤r
1
2 TOMOOMATSUMURA
[p]
Here G (x) is defined by the generating function
m
1 1+βx
G[p](x)um = i,
m 1+βu−1 1−x u
m∈Z 1≤i≤p i
X Y
which geometrically corresponds to the Segre classes of vector bundles (see [7]). Since not all
flagged Grothendieck polynomials are Grothendieck polynomials of Lascoux and Schu¨tzenberger,
our result generalizes the work of Knutson–Miller–Yong and Hudson–Matsumura mentioned
above.
The flagged Schur polynomials are the specialization of the flagged Grothendieck polynomials
G (x) at β = 0. These polynomials were introduced by Lascoux and Schu¨tzenberger in [13] to
λ,f
identify Schubertpolynomials associated to vexillary permutations. Theyare also generalizations
of Schur polynomials and satisfy the generalized Jacobi–Trudi formula, due to Gessel and Wachs.
In this paper, we closely follow Wachs’ inductive proof in [16], which makes the use of the
divided difference operators. In particular, our proof shows that the above determinant formula
[p]
in terms of the one row Grothendieck polynomials G (x) behaves nicely under the action of
m
those symmetrizing operators.
In Section 3, we give an inductive proof that the Grothendieck polynomial associated to a
vexillary permutation is a flagged Grothendieck polynomial. This gives a combinatorial proof
of Knutson–Miller–Yong’s result and of the determinant formula in [9]. We also show that any
flagged Grothendieck polynomial can be obtained from a monomial by applying the divided
difference operators.
In Section 4, we introduce the flagged skew Grothendieck polynomials. Their tableaux and
determinant expressions are natural extension of the (row) flagged skew Schur functions studied
by Wachs [16]. We show those two expressions coincide.
It is worth pointing out that the formulas of Knutson–Miller–Yong and Hudson–Matsumura
are also for double Grothendieck polynomials defined for two sets of variables. Thus it would
be natural to extend our result to its double version (see the work [3] of Chen–Li–Louck for the
flagged double Schur functions). This extension will be studied elsewhere.
2. Flagged Grothendieck polynomials
A partition is a weakly decreasing finite sequence λ = (λ ,...,λ ) of positive integers and r is
1 r
called its length. We often identify a partition λ of length r with its Young diagram {(i,j) | 1 ≤
i ≤ r,1 ≤ j ≤ λ } in English notation. A flagging f of a partition λ of length r is a weakly
i
increasing sequence f = (f ,...,f ) of positive integers. A flagged set-valued tableau T of shape
1 r
λ with a flagging f is a set-valued tableau of shapeλ such that each filling in thei-th row consists
of the numbers not greater than f . Namely, T is a filling of the boxes of λ such that the box
i
at (i,j) in λ is filled by a non-empty subset of {1,...,f } and the maximum number at (i,j) is
i
at most the minimum number at (i,j +1) for j +1 ≤ λ and less than the minimum number
i
FLAGGED GROTHENDIECK POLYNOMIALS 3
at (i+1,j) for j ≤ λ . In particular, if f = ··· = f , then it coincides with the definition of
i+1 1 r
set-valued tableaux defined by Buch in [2]. We denote by FSVT(λ,f) the set of all flagged set-
valued tableaux of shape λ with the flagging f. If f = f = 1 and r > 1, then FSVT(λ,f)= ∅.
1 2
Thus we assume that if f = 1 and r > 1, then f > 1.
1 2
Example 2.1. Let λ = (2,1) and f = (2,4). Then FSVT(λ,f) contains tableaux such as
1 1 12 2 1 12 2 2
23 34 23 4
.
If we chance f to f′ = (2,3), then FSVT(λ,f′) doesn’t contain the second and forth tableaux.
Let x = (x ,x ,...) be the set of infinitely many variables. Let Z[β] be the polynomial ring of
1 2
the variable β where we set degβ = −1. Let Z[β][x] and Z[β][[x]] be the ring of polynomials and
offormalpowerseries inx respectively. We definetheflagged Grothendieck polynomial associated
to a partition λ and a flagging f by
(2.1) G (x):= β|T|−|λ| x ,
λ,f k
T∈FSVT(λ,f) k∈T
X Y
where |T| is the total number of entries in T, |λ| is the number of boxes in λ, and k ∈ T denotes
an entry in T. We also define an element G (x) of Z[β][[x]] by
λ,f
∞ e
i−j
(2.2) G (x) := det βsG[fi] (x) ,
λ,f s λi+j−i+s !
Xs=0(cid:18) (cid:19) 1≤i,j≤r
e
where G[p](x) ∈ Z[β][[x]] is defined by the generating function
m
1 1+βx
(2.3) G[p](x;u) = G[p](x)um = i.
m 1+βu−1 1−x u
m∈Z 1≤i≤p i
X Y
For the empty partition, we set both of the functions G (x) and G (x) to be 1. We also
λ,f λ,f
remark that G[p] = (−β)m and G[1] = xm for all integer m ≥ 0, which follow from the direct
−m m 1
e
computation.
At β = 0, G (x) specializes to the flagged Schur polynomial of Wachs in [16], and G (x)
λ,f λ,f
[p]
is nothing but the corresponding Jacobi–Trudi formula since G (x) becomes the complete sym-
m
e
metric function of degree m.
The main goal of this section is to show that G (x) and G (x) coincide (Theorem 2.8). We
λ,f λ,f
closely follow Wachs’ proof of the analogous statement for the flagged Schur polynomials in [16].
e
The key for the proof is the action of the divided difference operators, which makes the induction
proof possible.
4 TOMOOMATSUMURA
2.1. Divided difference operators and basic formulas. We recall the definition of divided
difference operators and show a few formulas that will be used in the proof of the propositions to
follow.
Let S be the permutation group of the set {1,...,n}. We have the action of S on Z[β][[x]]
n n
permuting the variables. For example, let s denote the i-th transposition i.e. s (i) = i+1,s (i+
i i i
1) = i and s (j) = j for j 6= i,i + 1, then s (f(x)) is defined by exchanging x and x in
i i i i+1
f(x)∈ Z[β][[x]] and leave other variables unchanged.
Definition 2.2. For an element f(x) of Z[β][[x]], we define
(1+βx )f(x)−(1+βx )s (f(x))
i+1 i i
π (f(x)) := .
i
x −x
i i+1
The following Leibniz rule can be checked by a direct computation: for f(x),g(x) ∈ Z[β][[x]],
we have
(2.4) π (fg) = π (f)g+s (f)π (g)+βs (f)g.
i i i i i
It is also easy to check directly that, if f(x) is symmetric in x and x , then we have
i i+1
−βf(x) (k = 0)
(2.5) π (xkf(x)) =  k−1 k−1
i i
 xsixik+−11−s+β xsixik+−1s f(x) (k > 0).
!
s=0 s=1
X X

Lemma 2.3. For each m ∈ Zand p ∈ Z , we have
≥1
[p+1]
G (x) (i = p),
π (G[p](x)) = m−1
i m  [p]
−βG (x) (i 6= p).
 m
Proof. If i = p, it follows from the identity π G[p](x;u) = uG[p+1](x;u) which can be proved by
p
[p]
a direct computation. If i 6=p, then G (x) is symmetric in x and x , and hence (2.5) implies
m i i+1
the claim. (cid:3)
[p] [p+1]
Lemma 2.4. If f(x) is symmetric in x and x , then we have π (G f)= G f.
p p+1 p m m−1
Proof. It follows from the Leibniz rule (2.4) and Lemma 2.3. (cid:3)
Lemma 2.5. For each m ∈ Z and p ∈ Z , we have
≥1
x x
G[p]− 1 G[p] − 1 βG[p] = G[p] .
m 1+βx1 m−1 1+βx1 m m x1=0
(cid:12)
(cid:12)
Proof. It follows from the identity (cid:12)
x x
G[p]− 1 G[p] − 1 βG[p] um = G[p](x;u) ,
mX∈Z(cid:18) m 1+βx1 m−1 1+βx1 m(cid:19) (cid:12)x1=0
(cid:12)
which can be checked by a direct computation. (cid:12) (cid:3)
FLAGGED GROTHENDIECK POLYNOMIALS 5
2.2. The main theorem. We prove the main theorem (Theorem 2.8) below by induction based
on the following two propositions.
Proposition 2.6. Let λ be a partition of length r with a flagging f. If λ > λ and f < f , then
1 2 1 2
we have
(i) πf1(Gλ,f)= Gλ′,f′,
(ii) πf1(Gλ,f)= Gλ′,f′,
e e
where λ′ = (λ −1,λ ,...,λ ) and f′ = (f +1,f ,...,f ).
1 2 r 1 2 r
Proof. For (i), we recall from [7, §3.6] that we can write
(2.6) Gλ,f = asGλ[f11+]s1···Gλ[frr+]sr,
s∈Zr
X
e
where as ∈ Z[β] is the coefficient of ts in the Laurent series expansion
(1−t¯i/t¯j)= asts11···tsrr.
1≤Yi<j≤r s=(s1,X...,sr)∈Zr
Here we denoted t¯= −t = −t (−β)sts. Since f < f , one can apply Lemma 2.4 to the
1+βt s≥0 1 2
expression (2.6) and obtains (i). Indeed, we have
P
πf1(Gλ,f) = asπf1 Gλ[f11+]s1···Gλ[frr+]sr = asGλ[f11−+11+]s1Gλ[f22+]s2···Gλ[frr+]sr = Gλ′,f′.
sX∈Zr (cid:16) (cid:17) sX∈Zr
e e
Nextweprove(ii). Lett := f andt′ := f +1. Defineanequivalencerelation∼onFSVT(λ,f)
1 1
as follows: for T ,T ∈ FSVT(λ,f), let T ∼ T if the collection of boxes that contain either t or
1 2 1 2
t′ is the same for T and T . We have
1 2
G = M(T) ,
λ,f
A∈FSVT(λ,f)/∼ T∈A !
X X
where M(T):= β|T|−|λ| x for each T ∈ FSVT(λ,f). Let A be the equivalence class whose
k∈T k
tableaux have the configuration of t and t′ as shown in Figure 1.
Q
t...t tt...t
t...t ∗···∗ t′...t′ m1
t′...t′ m2 r1
r2
·
·
·
t...t ∗···∗
t′...t′ mk
r
k
Figure 1.
6 TOMOOMATSUMURA
Each rectangle with ∗ has m boxes and each box contains t or t′ so that the total number of
i
entries t and t′ in the rectangle is m or m +1 for i≥ 2. Note that m and r may be 0 and the
i i i i
rectangles in Figure 1 may not be connected. Then we see that
k mi mi
M(T) = xmt 1(xtxt′)r1+···+rk xvtxtm′i−v +β xvtxtm′i+1−v R(A),
T∈A i=2 v=0 v=1 !!
X Y X X
where R(A) is the polynomial contributed from the entries other than t and t′. Let
k mi mi
R′(A) := xvxmi−v +β xvxmi+1−v R(A).
t t′ t t′
!!
i=2 v=0 v=1
Y X X
Observe that the factor (xtxt′)r1+···+rkR′(A) is symmetric in xt and xt′. Thus, by Lemma 2.5, if
m = 0, then we have r = 0 and
1 1
(2.7) πt M(T) = −β(xtxt′)r2+···+rkR′(A),
T∈A !
X
if m = 1, we have
1
(2.8) πt M(T) = (xtxt′)r1+···+rkR′(A),
T∈A !
X
and if m ≥ 2, we have
1
m1−1 m1−1
(2.9) π M(T) = xsxm1−1−s+β xsxm1−s (x x )r1+···+rkR′(A).
t t t′ t t′ t t+1
T∈A ! s=0 s=1 !
X X X
We consider the decomposition
FSVT(λ,f)/∼ = F ⊔F ⊔F ⊔F
1 2 3 4
where F ,...,F are the sets of equivalence classes whose configurations of the boxes containing
1 4
t or t′ respectively satisfy
(1) m = 0 (so that r = 0),
1 1
(2) m = 1 and the box at (1,λ ) in λ contains more than one entry (so that r = 0),
1 1 1
(3) m = 1 and the box at (1,λ ) in λ contains contains only t,
1 1
(4) m ≥ 2.
1
By the expressions (2.7) and (2.8), we have
π M(T) = 0.
t
A∈XF1⊔F2 TX∈A !
Now we consider the equivalence class A′ in FSVT(λ′,f′)/∼ whose associated skew diagram of
boxes containing t or t′ is as shown in Figure 2 below.
FLAGGED GROTHENDIECK POLYNOMIALS 7
t...t t...t
t...t ∗···∗ t′...t′ m1−1
t′...t′ m2 r1
r2
·
·
·
t...t ∗···∗
t′...t′ mk
r
k
Figure 2.
One can see that, if m = 1, then M(T) is exactly the right hand side of (2.8) under the
1 T∈A′
condition (3), and if m ≥ 2, then M(T) is exactly the right hand side of (2.9). It is also
1 PT∈A′
clear that F and F are in bijection to the equivalence classes of FSVT(λ′,f′) such that m = 1
3 4 P 1
and m ≥ 2 respectively. Thus the desired identity holds. (cid:3)
1
Proposition 2.7. Let λ be a partition of length r with a flagging f. If f = 1, then we have
1
(i) Gλ,f = xλ11(Gλ′,f′|x1=0),
(ii) Gλ,f = xλ11(Gλ′,f′|x1=0),
e e
where λ′ = (λ ,...,λ ) and f′ = (f ,...,f ).
2 r 2 r
Proof. First we observe that (ii) holds clearly since f = 1. We prove (i). Since f = 1, we see
1 1
that the first row of the determinant for G is
λ,f
r−1
x x
xλ1,xλ1 1e ,...,xλ1 1 .
1 1 1+βx 1 1+βx
1 (cid:18) 1(cid:19) !
Indeed, since G[1] = xm for m ≥ 0, we have
m 1
∞ 1−j ∞ 1−j xλ1+j−1
βsG[f1] (x) = xλ1+j−1 βsxs = 1 .
s λ1+j−1+s 1 s 1 (1+βx )j−1
s=0(cid:18) (cid:19) s=0(cid:18) (cid:19) 1
X X
We do the column operation to G by subtracting x1 times the (j −1)-st column from the
λ,f 1+βx1
j-th column for each j = 2,...,r so that the first row becomes (xλ1,0,...,0). Then we observe
1
e
that the (i,j)-entry for i,j ≥ 2 equals to
∞ ∞
i−j x i−j +1
βsG[fi] − 1 βsG[fi]
s λi+j−i+s 1+βx s λi+j−1−i+s
s=0(cid:18) (cid:19) 1 s=0(cid:18) (cid:19)
X X
∞ ∞
i−j x i−j i−j
= βsG[fi] − 1 + βsG[fi]
s λi+j−i+s 1+βx s s−1 λi+j−1−i+s
s=0(cid:18) (cid:19) 1 s=0(cid:18)(cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
X X
∞
i−j x x
= βs G[fi] − 1 G[fi] − 1 βG[fi]
s λi+j−i+s 1+βx λi+j−1−i+s 1+βx λi+j−i+s
s=0(cid:18) (cid:19) (cid:18) 1 1 (cid:19)
X
∞
i−j
= βsG[fi] ,
s λi+j−i+s!(cid:12)
Xs=0(cid:18) (cid:19) (cid:12)x1=0
(cid:12)
(cid:12)
(cid:12)
8 TOMOOMATSUMURA
where the first equality is by an identity of the binomial coefficients and the last equality follows
from Lemma 2.5. Finally the cofactor expansion with respect to the first row gives the desired
identity for (i). (cid:3)
Theorem 2.8. For each partition λ and a flagging f, we have G (x) = G (x). In particular,
λ,f λ,f
G (x) is a polynomial in x.
λ,f
e
Peroof. By induction on (r,|f| := f +···+f ), Proposition 2.6 and 2.7 imply the claim: for the
1 r
base case (r,|f|) = (0,0), the claim holds trivially. If f =1, apply Proposition 2.7 and if f > 1,
1 1
apply Proposition 2.6. In both cases, the claim follows from the induction hypothesis. (cid:3)
3. Grothendieck polynomials
In this section, we show that the Grothendieck polynomials associated to a vexillary permu-
tation is a flagged Grothendieck polynomial, recovering the results of Knutson–Miller–Yong [11]
and Hudson–Matsumura [9]. We also show that any flagged Grothendieck polynomial can be
obtained from a monomial in the same way as any Grothendieck polynomial is defined.
The Grothendieck polynomial G = G (x ,...,x ) associated to a permutation w ∈ S is
w w 1 n n
defined as follows. We use the same convention as in [2]. For the longest element w in S , we
0 n
set
n−1
G = xn−1xn−2···x2 x = xn−i.
w0 1 2 n−2 n−1 i
i=1
Y
If w is not the longest, there is i such that ℓ(ws ) = ℓ(w)+1. We then define
i
G := π (G ).
w i wsi
This definition is independent of the choice of s because the operators π satisfy the Coxeter
i i
relations. By the same reason we can define π by π = π ···π where w = s ···s with
w w ik i1 i1 ik
ℓ(w) = k.
Nowwerecallhowtoobtainapartitionλ(w)andaflaggingf(w)foreachvexillarypermutation
w ∈ S . We follow [6] and [11] (cf. [9]). Let r be the rank function of w ∈ S defined by
n w n
r (p,q) := ♯{i ≤ p| w(i) ≤ q} and we define the diagram D(w) of w by
w
D(w) := {(p,q) ∈ {1,...,n}×{1,...,n} | π(p) > q, and π−1(q) > p}.
We call an element of the grid {1,...,n}×{1,...,n} a box. The essential set Ess(w) of w is the
subset of D(w) given by
Ess(w) := {(p,q) | (p+1,q),(p,q +1) 6∈ D(w)}.
A permutation w ∈ S is called vexillary if it avoids the pattern (2143), i.e. there is no a <
n
b < c < d such that w(b) < w(a) < w(d) < w(c). In [16], a vexillary permutation was called a
single-shape permutations. Fulton showed in [6] that w ∈ S is vexillary if and only if the boxes
n
in Ess(w) are placed along the direction going from northeast to southwest. We can assign a
FLAGGED GROTHENDIECK POLYNOMIALS 9
partition λ(w) to each vexillary permutation w as follows: let the number of boxes (i,i+k) in
the k-th diagonal of the Young diagram of λ(w) be equal to the number of boxes in the k-th
diagonal of D(w) for each k (see [11, 10]). This defines a bijection φ from D(w) to λ, namely
φ(p,q) = (p−r (p,q),q −r (p,q)) for each (p,q) ∈ D(w). In particular, φ restricted to Ess(w)
w w
is a bijection onto the set of the southeast corners of λ(w). Let r be the length of λ(w). The
flagging f(w)= (f(w) ,...,f(w) ) associated to w is defined as follows. We can choose a subset
1 r
{(p ,q ),i = 1,...,r} of {1,...,n}×{1,...,n} containing Ess(w) and satisfying
i i
(3.1) p ≤ p ≤ ··· ≤ p , q ≥ q ≥ ··· ≥ q ,
1 2 r 1 2 r
(3.2) p −r (p ,q ) = i, ∀i= 1,...,r.
i w i i
In [9], we called this subset {(p ,q )} a flagging set of w and used it to express the double
i i
Grothendieck polynomials as a determinant. We set f(w) by letting f(w) := p . We can always
i i
express λ(w) by λ = q −p +i for each i = 1,...,r. Remark that the set FSVT(λ(w),f(w))
i i i
doesn’t depend of the choice of flagging sets.
Example 3.1. Consider a vexillary permutation w = (w(1)···w(5)) = (23541) in S . We
5
represent the corresponding permutation matrix M by (M ) = δ . In the picture below,
w w ij w(i),j
we represent 1 in M by a dot and the boxes in D(w) by squares. We make hooks by drawing
w
lines from each dot going south and east, and then D(w) is the collection of boxes that are not on
the hooks. We see that λ(w) = (λ ,...,λ ) = (2,1,1,1). For a flagging set of w, we must have
1 4
(p ,q ) = (3,4) and (p ,q ) = (4,1) since they consist Ess(w). Then the conditions (3.1) and
1 1 4 4
(3.2) require that we must choose (p ,q ) from (3,2) and (4,3) and (p ,q ) from (3,1) and (4,2)
2 2 3 3
in such a way that p ≤ p and q ≥ q . Thus f(w) = (3,3,3,4), (3,3,4,4) or (3,4,4,4). By the
2 3 2 3
column strictness, each of the flaggings gives the same collection of flagging set-valued tableaux.
•
0 1 1 1 1
•
 0 1 2 2 2 
•
r = 0 1 2 2 3
w  
•  
 0 1 2 3 4 
 
•  
 1 2 3 4 5 
 
 
Theorem 3.2. If w is a vexillary permutation, then G = G .
w λ(w),f(w)
Proof. By Propositions 2.6 and 2.7, the proof is almost identical to the one given by Wachs for
the flagged Schur functions in [16]. We write the proof for completeness since we have slightly
different terminologies and notations.
10 TOMOOMATSUMURA
First we observe that for the longest element w ∈ S , we have λ(w ) = (n−1,n−2,...,1)
0 n 0
and f(w )= (1,2,3,...,n−1). Therefore by definition we have
0
n−1
G = xn−i = G .
λ(w0),f(w0) i w0
i=1
Y
We show the claim by induction on (n,ℓ(w ) −ℓ(w)) with the lexicographic order where ℓ(w)
0
denotes the length w ∈ S . We say that w has a descent at i if w > w . Let d be the leftmost
n i i+1
descent of w. We consider the following three cases.
Case 1. Assume d > 1. Let w′ = ws , then it is easy to see that w′ is vexillary. Since
d−1
ℓ(w′) > ℓ(w), we have Gw′ = Gλ(w′),f(w′) by the induction hypothesis. We can also observe that
λ(w′) = λ(w) +1 and λ(w′) = λ(w) for i ≥ 2. Furthermore f (w′) = d−1 = f (w)−1 and
1 1 i i 1 1
f(w′) = f(w) for i ≥2. Thus, by Proposition 2.6, we have
i i
Gw = πd−1Gw′ = πd−1Gλ(w′),f(w′) = Gλ(w),f(w).
This completes the case d> 1.
Case 2. Assume d = 1 and w(1) < n. Let w′ := s w so that w′(1) = w(1) + 1 and
w(1)
w′(l) = w(1) for l such that w(l) = w(1)+1. It is clear that w′ is vexillary. Since the leftmost
descent of w is 1, we have ℓ(w′) = ℓ(w)+1. Consider the vexillary permutation
u= (w(1)+1,w(1),w(1)+2,w(1)+3,...,n,w(1)−1,w(1)−2,...,1).
Since w is vexillary, all numbers greater than w(1) appear in ascending order in w, and therefore
we can find integers i ,...,i greater than 1, satisfying w′s s ···s = u and ℓ(w′)+k = ℓ(u).
1 k i1 i2 ik
For such a choice, we also have ws s ···s s = u. Thus by definition we have
i1 i2 ik 1
Gw = πi1···πikπ1Gu and Gw′ = πi1···πikGu.
By induction we have Gu = Gλ(u),f(u) and Gw′ =Gλ(w′),f(w′). We observe that
λ(u) = (w(1),w(1)−1,w(1)−1,...,w(1)−1,w(1)−2,w(1)−3,...,2,1)
and f(u) = (1,2,3,...,n), from which we find
G = xw(1)xw(1)−1···xw(1)−1 xw(1)−2 ···x2 x1 .
u 1 2 n−w(1)+1 n−w(1)+2 n−2 n−1
Lemma 2.5 implies that π G = (1/x )G . Since i ,...,i are greater than 1, again Lemma 2.5
1 u 1 u 1 k
implies that
Gw = πi1···πik((1/x1)Gu) =(1/x1)Gw′ = Gw = (1/x1)Gλ(w′),f(w′).
Furthermore, we observe that
λ(w′) = λ(w) +1, λ(w′) = λ(w) , ∀i= 2,...,r,
1 1 i i
f(w′) = f(w) = 1, f(w′) = f(w) , ∀i= 2,...,r.
1 1 i i
This implies that Gλ(w),f(w) = (1/x1)Gλ(w′),f(w). Therefore we conclude that Gw = Gλ(w),f(w).