A short proof on the transition matrix from the Specht basis to the Kazhdan-Lusztig basis

Abstract

We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan-Lusztig basis, using Kazhdan-Lusztig theory for parabolic Hecke algebra.

Full text

ROCKY MOUNTAIN
JOURNAL OF MATHEMATICS
Volume 51 (2021), No. 5, 1671–1680
DOI: 10.1216/rmj.2021.51.1671 c
Rocky Mountain Mathematics Consortium
A SHORT PROOF ON THE TRANSITION MATRIX
FROM THE SPECHT BASIS TO THE KAZHDAN–LUSZTIG BASIS
MEE SE ONG IM
We provide a short proof on the change-of-basis coefficients from the Specht basis to the Kazhdan–
Lusztig basis, using Kazhdan–Lusztig theory for the parabolic Hecke algebra.
1. Introduction
It is well-known that the irreducible representations for the symmetric group
6d
are the Specht modules
parametrized by the set of standard Young tableaux consisting of
d
boxes. The Specht module has a
purely combinatorial basis (called the Specht basis) described in terms of certain alternating sums of
so-called tabloids.
For the Young tableaux consisting of two rows of equal size, Russell and Tymoczko in [8] compare the
Specht basis with another combinatorial basis (called the web basis) which arises from Temperley–Lieb
algebra and knot theory. In their context, the web basis is a reincarnation of the Kazhdan–Lusztig basis,
but this is not true in general. Their main result is a combinatorial model that gives a different proof
to a special case of a classic theorem by Naruse [7, Theorem 4.1]that the change-of-basis matrix is
unitriangular, altogether with some vanishing conditions. The unitriangularity result is also found in [4],
but without vanishing conditions.
In this paper, we give a short proof of [7, Theorem 4.1]using Kazhdan–Lusztig theory for the parabolic
Hecke algebra. We also notice that the argument applies to all classical types, with a nonstandard notion
of tableaux. For example, in type B we use certain centro-symmetric tableaux which are not the bi-
tableaux that parametrize the irreducibles. That is to say, the analog of the Specht modules here are not
the irreducibles. However, we hope that they correspond to the top nonvanishing cohomology of the
Springer fiber corresponding to the parabolic subgroup. Furthermore, we postpone the details for type D
to future work. In this manuscript, we introduce a Specht basis for this module using certain alternating
sums of centro-symmetric tabloids. We then prove a unitriangular theorem regarding the change-of-basis
matrix between the Specht basis and the Kazhdan–Lusztig basis (see Theorem 3.1).
I would like to thank Chun-Ju Lai, Arik Wilbert, and Jieru Zhu for insightful discussions. I would also like to thank the referee
for constructive feedback on this manuscript. I thank the general manager Jim Hutchinson’s front desk staff at Homewood
Suites by Hilton in New Windsor, NY for their unwavering support to the author and access to quiet work areas during the
covid-19 lockdown period in the United States. I was partially supported by the Summer Collaborators Program at the School
of Mathematics in the Institute for Advanced Study in Princeton, NJ.
2010 AMS Mathematics subject classification: primary 05E10; secondary 20C08, 20C30.
Keywords and phrases: Kazhdan–Lusztig basis, Specht module, parabolic Hecke algebra.
Received by the editors on December 9, 2019, and in revised form on February 9, 2021.
1671

1672 MEE SEONG IM
Figure 1. Dynkin diagrams of type Adand Bd.
1 2
...
d0 1
...
d−1
===
Type Ad(d≥1)Type Bd(d≥2)
2. Combinatorics
2.1. Weyl groups.
Throughout this manuscript, let
8
be the type A or B. Let
W8
d
be the Weyl group
of type 8dassociated with the Dynkin diagrams in Figure 1.
We define
(2-1) [d] := [1,d] ∩ Z,[±d] := [−d,d] ∩ Z.
Let
(2-2) IA
d:= [d]and IB
d:= [0,d−1] ∩ Z.
We denote by
S8
d= {s8
i:i∈I8
d}
, the corresponding generators of
W8
d
as Coxeter groups. Define
`8:W8
d→Z≥0
to be the length function, where
`8(u)
of an element
u
in the Weyl group
W8
d
is the
minimal number lsuch that ucan be written as a product of lgenerators.
For any set
I
, let
Perm(I)
be the group of permutations on
I
, and let
(i,j)∈Perm(I)
be the trans-
position for
i,j∈I
. It is a standard fact that
W8
d
can be identified as a group of certain permutations
(see [2]). Here we treat them as subgroups of Perm([±d])as follows:
(2-3) WB
d:= {g∈Perm([±d]):g(−i)= −g(i)for all i},
WA
d−1:= {g∈WB
d:neg(g)=0},
where the function neg counts the total number of negative entries of {1,...,d}, i.e.,
(2-4) neg :WB
d→Z≥0,g7→ #{k∈ [d] : g(k) < 0}.
The generators can be identified with the following permutations in [±d]:
(2-5) si=sA
i=sB
i=(i,i+1)(−i,−i−1)for 1 ≤i≤d−1,
s0=sB
0=(−1,1).
It is sometimes convenient to use the one-line notation
(2-6) w≡ |w(1), w(2), . . . , w(d)|for w∈W8
d.
We remark that the element
w
is uniquely determined by these
d
values due to the centro-symmetry
condition g(−i)= −g(i)for all i.
A partial Bruhat order
≤
on
W8
d
is defined as follows: for
u, w ∈W8
d
,
u≥w
if and only if
u=sra1· · · srab
, a reduced expression, for some 1
≤a1<· · · <ab≤m
, where
w=sr1· · · srm
is a

THE TRANSITION MATRIX FROM THE SPECHT BASIS TO THE KAZHDAN–LUSZTIG BASIS 1673
reduced decomposition for
w
. That is,
u≥w
if any reduced expression for
w
contains a subexpression
which is a reduced expressions for u.
Example 2.1. Let u=(1,2)and w=(1,2)(3,4). Then it is clear that u≥w.
Example 2.2. Let u=s3s1and w=s1s3s2s1. Then u≥w.
2.2. Parabolic Hecke algebras.
Let
H8=H(W8
d)
be the Hecke algebra of
W8
d
over
C[q,q−1]
, where
8=
A or B. As a free module,
H8
has a basis
{H8
w:w∈W8
d}
. The multiplication in
H8
is determined
by
(2-7) H8
wH8
x=H8
wxif `8(wx)=`8(w) +`8(x),
(H8
s)2=(q−1−q)H8
s+H8
eif s∈S8
d,
where
e∈W8
d
is the identity, and hence
H8
e
is the identity element in
H8
. By a slight abuse of notation,
we use the same symbol
H8
i
(1
≤i≤d−
1) to denote the element
H8
s8
i
∈H8
. We write
H8
0:= H8
s8
0
when 8=B. It is a standard fact that H8is generated as a C[q,q−1]-algebra by {H8
i:i∈I8
d}.
For each subset
J⊂I8
d
, denote the corresponding parabolic subgroup and parabolic Hecke algebra
of W8
dby
(2-8) W8
J= hs8
j:j∈Jiand H8
J:= H(W8
J)= hH8
j:j∈Ji,
respectively. We also denote the set of shortest right coset representatives for W8
J\W8
dby
(2-9) D8
J= {w∈W8
d:`8(wg)=`8(w) +`8(g)for all g∈W8
J}.
If x≤y∈W8
J, then we give it a total order x≤Dyon D8
J.
Denote the induced trivial module corresponding to J⊂I8
dby
(2-10) M8
J=C[q,q−1] ⊗H8
J
H8,
where C[q,q−1]is regarded as a right H8-module by setting
(2-11) f·H8
i=q−1ffor f∈C[q,q−1].
The module M8
Jadmits a standard basis
(2-12) {Mw=1⊗H8
w:w∈D8
J}.
We note that
M8
J
can also be identified as the right
H8
-module
yλH8
under the assignment
Mw7→ yJH8
w
,
where
yJ=Pw∈W8
Jq−`(w) H8
w
, on which
H8
acts by a right multiplication. The action of
H8
on
M8
J
below follows from (2-7) and a straight-forward calculation:
Mw·H8
i=


Mwsiif wsi∈D8
J, `8(wsi)>`8(w),
Mwsi+(q−1−q)Mwif wsi∈D8
J, `8(wsi)<`8(w),
q−1Mwif wsi6∈ D8
J.
Following [9, Theorem 3.1],M8
Jadmits a Kazhdan–Lusztig basis {Mw:w∈D8
J}such that
(2-13) Mw=X
x∈D8
J
m8
x,w(q−1)Mx,

1674 MEE SEONG IM
where m8
x,w ∈Z[q−1]. We also define polynomials p8
w,x(q−1)∈Z[q−1]such that
(2-14) Mw=X
x∈D8
J
p8
w,x(q−1)Mx.
Below we recall some properties of the (inverse) parabolic Kazhdan–Lusztig polynomials.
Lemma 2.3. We have the following:
(a) If x 6≤ wwith respect to the Bruhat order on W 8
d,then m8
x,w =0=p8
x,w.
(b) m8
x,x=1=p8
x,x.
Proof. It follows from [9, Theorem 3.1]that the matrix
(m8
x,w(q−1))x,w
is upper unitriangular, so its
inverse matrix (p8
x,w(q−1))x,w is also upper unitriangular. 
2.3. Young tableaux.
Now we want to associate to each parabolic subgroup
W8
J⊆W8
d
a “Young
tableau” beyond type A. The idea here is to use the corresponding composition (cf. [1;3]), where in type
A, a composition of dis an integer vector of positive integers summing to d. To be precise, we set
(2-15) 3A(d):= G
n≥0
3A(n,d)and 3B(d):= G
n≥0
3B(n,d),
where the legit n-part (signed) compositions are defined as
3A(n,d)=nλ=(λi)i∈[n]∈Zn
>0:
n
P
i=1
λi=do,(2-16)
3B(n,d)=


(λi)i∈[±r]∈Zn
>0:λ0∈1+2Z,Pr
i=−rλi=2d+1,
λ−i=λi∀iif n=2r+1,
{λ∈3B(2r+1,d):λ0=1}if n=2r.
(2-17)
In other words, the corresponding parabolic subgroups W8
Jare generated by
(2-18) SA
d− {sλ1,sλ1+λ2,...,sλ1+···+λn−1}for λ∈3A(n,d),
and generated by
(2-19) SB
d−{sλ1,sλ1+λ2,...,sλ1+···+λr−1}if n=2r,
sλ\
0
,sλ\
0+λ1,...,sλ\
0+λ1+···+λr−1if n=2r+1,for λ∈3B(n,d)
(letting
λ\
0:= bλ0/2c
, where
bxc
is the greatest integer less than or equal to
x
). We also write
W8
λ
and
H8
λ
to denote the corresponding parabolic subgroups and the parabolic Hecke algebras, respectively. Note
that rfor 8=B corresponds to nfor 8=A.
For type A, any composition λ=(λ1, . . . , λn)∈3A(n,d)defines a Young subgroup
WA
λ'6λ1×6λ2×···×6λn.
That is, any two compositions giving the same Young subgroup (and hence parabolic Hecke algebra)
correspond to the same partition. We denote the set of type A partitions by
5A(d)=Fn≥15A(n,d),
where
(2-20) 5A(n,d)= {λ∈3A(n,d):λ1≥λ2≥ · · · ≥ λn}.

THE TRANSITION MATRIX FROM THE SPECHT BASIS TO THE KAZHDAN–LUSZTIG BASIS 1675
However, for type B, a composition λ=(λ0, . . . , λr)∈3B(n,d)defines a Young subgroup
WB
λ'WB
λ\
0
×6λ1×6λ2×···×6λr.
In particular,
WB
λ
is always a product of symmetric groups when
n
is even. Hence, compositions
λ, µ
describe the same Young subgroup (and hence parabolic Hecke algebra) if
λ0=µ0
and also
(λ1, . . . , λr)
corresponds to the same partition as
(µ1, . . . , µr)
. We denote the set of type B “partitions” by
5B(d):=
Fn≥15B(n,d), where
(2-21) 5B(n,d)= {λ∈3B(n,d):λ1≥λ2≥ · · · ≥ λr=λd(n−1)/2e}
and dxeis the least integer greater than or equal to x.
Example 2.4.
Let
d=
3. There are 4 and 8 subsets of
IB
d
for
n
even and odd, respectively, and hence
we have
r n λ∈3B(3)WB
JJDB
J
1 2 (3,1,3)WB
3{0,1,2} {e}
2 4 (2,1,1,1,2)hsB
0,s2i ' 62×62{0,2} hs1i
2 4 (1,2,1,2,1)hsB
0,s1i ' WB
2{0,1} hs2i
3 6 (1,1,1,1,1,1,1)hsB
0i ' 62{0} hs1,s2i
0 1 (7)WB
3{0,1,2} {e}
1 3 (3,1,3)hs1,s2i ' 63{1,2} hsB
0i
1 3 (2,3,2)hsB
0,s2i ' 62×62{0,2} hs1i
1 3 (1,5,1)hsB
0,s1i ' WB
2{0,1} hs2i
2 5 (2,1,1,1,2)hs2i ' 62{2} hsB
0,s1i
2 5 (1,2,1,2,1)hs1i ' 62{1} hsB
0,s2i
2 5 (1,1,3,1,1)hsB
0i ' 62{0} hs1,s2i
3 7 (1,1,1,1,1,1,1){e}∅WB
3
Let
λ∈38(n,d)
. For now we assume that
8=
A. Let its corresponding Young diagram be
λ=
{(i,j):j≤ −
1
,
1
≤i≤λ−j}
, which is a corner justified set of boxes. A Young tableau of shape
λ
(or
λ-tableau for short) is a bijection
T:λ→ [d].
For example, when
λ=(m,m)
or
(m,m,m)
, a
λ
-tableau represents a 2
×m
or 3
×m
grid, respectively,
whose boxes are filled in by every number, exactly once, from 1 to
d=
2
m
or
d=
3
m
, respectively. We
also write
Sh(T)=λ
to mean the composition corresponding to the function
T
, and
Sh(λ)
to mean the
set of all Young tableaux having composition
λ
. The set of all Young tableaux admits a (right) action of
the symmetric group WA
d−1=6dby permuting the letters that are filled in the boxes.
A Young tableau is called row standard if the entries in each row are increasing; it is called standard
if the entries in each row and each column are increasing. Denote by
rStdA(λ)
and
StdA(λ)
as the sets of
row standard and standard Young tableaux of shape
λ
, respectively. We remark that the size of
StdA(λ)
is counted by the hook formula, and hence
(2-22) #StdA((m,m)) =1
m+12m
mand #StdA((m,m,m)) =2·(3m)!
m! · (m+1)! · (m+2)!.

1676 MEE SEONG IM
We say
S∈rStdA(λ)
is in standard form if integers from 1 to
d
are written in each box, increasing
along each row from left to right, and then moving down the consecutive rows from top to bottom.
Write
rStdA
J(λ)
to be those Young tableaux in
rStdA(λ)
containing standard form Young tableau
S
,
together with those of the form S·w, where w∈DA
J.
It is well-known that the following assignment is a bijection:
(2-23) rStdA
J(λ) →DA
J,T7→ wT,
where DA
J= hsλ1,sλ1+λ2,...,sλ1+···+λn−1i.
Example 2.5.
Let
λ=(
3
,
2
)∈3A(
2
,
5
)
and
J= {
1
,
2
,
4
}
. Then
WA
J= hs1,s2,s4i
and
DA
J= hs3i
. Let
T=123
4 5 ∈rStdA
J(λ)
, which is in standard form. Then
wT=e
. Furthermore,
T·s3=124
3 5
, and
wT·s3=(3,4).
We now generalize this bijection to type
8=
B by defining the Young tableaux of classical type. For
λ∈3B(n,d), denote the corresponding Young diagram by
(2-24) λ=(i,0): −λ\
0≤i≤λ\
0t {±(i,j):j≤ −1,0≤i≤λ−j−1}if n=2r+1,
{±(i,j):j≤ −1,1≤i≤λ−j}if n=2r.
Note that when
n
is even, the corresponding Young diagram is just a type A Young diagram, with a copy
obtained by rotating it 180 degrees.
Example 2.6.
For
d=
2,
5B(
2
,
2
)= {(
2
,
1
,
2
)}
and
5B(
4
,
2
)= {(
1
,
1
,
1
,
1
,
1
)}
, and hence the Young
diagrams are
and ,
respectively. For d=3, we have
λ∈5B(3) (7) (1,5,1) (2,3,2) (3,1,3) (1,1,3,1,1) (1,2,1,2,1) (1,1,1,1,1,1,1)
λ
A
λ
-tableau (Young tableau) of type
8=
B is again a bijection
T:λ→[±d]
for
n
odd, and
T:λ→[±d]\{
0
}
for
n
even. We also write
Sh(T)=λ
by a slight abuse of notation, and we write
Sh(λ)
to be the set of
all Young tableaux with composition
λ
. Such a
λ
-tableau is called row standard if the entries in each
row of
λ
-tableau are increasing, while it is called standard if the entries in each row and each column of
λ
-tableau are increasing. Denote by
rStdB(λ)
and
StdB(λ)
the sets of type B row standard and standard
λ-tableaux, respectively.
For
n
even, we say
S∈rStdB(λ)
is in standard form if integers
{−r,...,r}\{
0
}
are written in each
box, increasing along each row from left to right, and moving down the rows from top to bottom. For
n
odd, we say
S∈rStdB(λ)
is in standard form if integers from
−r
to
r
are written in each box, increasing
along each row from left to right, and moving down the rows from top to bottom.
Write
rStdB
J(λ) ⊆rStdB(λ)
to be those Young tableaux containing standard form Young tableau
S
,
together with those of the form S·w, where w∈DB
J.

THE TRANSITION MATRIX FROM THE SPECHT BASIS TO THE KAZHDAN–LUSZTIG BASIS 1677
The following assignment is an obvious bijection:
(2-25) rStdB
J(λ) →DB
J,T7→ wT,
where
DB
J=hsλ1,sλ1+λ2,...,sλ1+···+λr−1iif nis even,
hsλ\
0
,sλ\
0+λ1,...,sλ\
0+λ1+···+λr−1iif nis odd.
Example 2.7. Given tableau T=−3−2−10123in standard form, we have wT=e.
Example 2.8.
Given tableau
T=−3−2
−1 0 1
2 3
in standard form, we have
wT=e
. As for
T·s1=−3−1
−2 0 2
1 3
,
wT·s1=s1=(1,2)(−1,−2).
We define R≥rTfor R,T∈rStd8(λ) if there exists transpositions sr1,...,srmsuch that
R=S·sra1· · · srab,T=S·sr1· · · srm,
where 1
≤a1<· · · <ab≤m
,
S
is in standard form, and
sra1· · · srab
and
sr1· · · srm
are reduced expressions.
2.4. Specht module.
We refer to [10;5;6] for an extensive background on Specht modules. For now
we assume that
8=
A. Let
Sh(λ)
be the set of all Young tableaux of composition
λ
. Let
{T}
be the
equivalence class (called a tabloid) of the
λ
-tableau
T
under the (right) action of
WA
λ⊂WA
d
. That is,
two tableaux are row equivalent if one may be obtained from the other from a permutation within rows,
and
{T}
is the row equivalence class of the tableau
T
. It is clear that each class contains a unique row
standard λ-tableau.
Example 2.9. All tabloids of composition (3,2)of type A are of the form
n123
4 5 o,n124
3 5 o,n125
3 4 o,n134
2 5 o,n135
2 4 o,
n145
2 3 o,n234
1 5 o,n235
1 4 o,n245
1 3 o,n345
1 2 o.
We define the Specht vector corresponding to T∈Sh(λ) by
(2-26) vA
T=X
w∈col(T)
(−1)`A(w){T·w},
where
col(T)
is the subset of the symmetric group
6d
consisting of the permutations which reorder the
columns of T. For T∈Std8(λ),R∈rStd8(λ), and λ∈38(d), define cA
R,T∈Zby
(2-27) vA
T=X
R∈rStdA(λ)
cA
R,T{R}.
Note that vA
Tform a standard basis for the Specht module indexed by standard λ-tableaux.
We now generalize this to type B by defining
(2-28) vB
T=X
R∈rStdB(λ)
cB
R,T{R}for T∈Sh(λ), λ ∈3B(n,d), and Tin standard form.

1678 MEE SEONG IM
Example 2.10.
Let
λ=(
3
,
2
)
and
J= {
1
,
2
,
4
}
. Then for
8=
A, the standard Specht vectors are of the
form
vA
123
4 5
=n123
4 5 o−n423
1 5 o−n153
4 2 o+n453
1 2 o,
vA
124
3 5
=n124
3 5 o−n324
1 5 o−n154
3 2 o+n354
1 2 o.
In type B, for
T=−5−4
−3−2−1
123
4 5
and T·s3=−5−3
−4−2−1
124
3 5
the standard Specht vectors are of the form
vB
T=




−5−4
−3−2−1
1 2 3
4 5





−




−5−1
−3−2−4
4 2 3
1 5





−




−2−4
−3−5−1
1 5 3
4 2





+




−2−1
−3−5−4
4 5 3
1 2





,
vB
T·s3=




−5−3
−4−2−1
1 2 4
3 5





−




−5−1
−4−2−3
3 2 4
1 5





−




−2−3
−4−5−1
1 5 4
3 2





+




−2−1
−4−5−3
3 5 4
1 2





.
Lemma 2.11. If c8
R,T6= 0, then wR≤wTwith respect to the Bruhat order.
Proof. Let
c8
R,T6=
0. Recall the bijection
rStd8
J(λ) →D8
J
in (2
-
23) and (2
-
25), where
T7→ wT
. Then
there exists
w∈col(T)
such that
R=T·w
. If
w=Id
, then we are done. So assume
w6= Id
. Since
T∈Std8(λ)
, permute the entries within each row of
R
such that it is in
rStd8(λ)
. Without loss of
generality, rename this representative by
R
. Note that
R
may not be in
Std8(λ)
since
T
is in
Std8(λ)
.
So for
T7→ wT
,
R=T·w7→ wT·w=wR
,
`8(wTw) ≥`8(w)
since
w
is not the identity permutation
and wT∈D8
Jis a reduced word while w∈W8
d\D8
J. This shows R≤rTimplies wR≤wT.
Example 2.12. Let r=2, n=5. So d=3,
T=
−3
−2
−1 0 1
2
3
,T·s1=
−3
−1
−2 0 2
1
3
,T·s1s2=
−2
−1
−3 0 3
1
2
,
and
vB
T= {T} − {T·s2}, v B
T·s1= {T·s1} − {T·s1(s2s1s2)}, v B
T·s1s2= {T·s1s2} − {T·s1s2s1}.
We denote the (classical) Specht module over Ccorresponding to λ∈58(d)by
(2-29) S8
λ=M
T∈Std8(λ)
Cv8
T,
where the (right) action of
W8
d
is given by
v8
T·si=v8
T·si
. We call
{v8
T:T∈Std8(λ)}
the tableaux basis
for S8
λ.

THE TRANSITION MATRIX FROM THE SPECHT BASIS TO THE KAZHDAN–LUSZTIG BASIS 1679
Note that for
T∈Std8(λ)
, it is not guaranteed that
T·si∈Std8(λ)
, and hence
v8
T·si
is, in general, not
a single Specht vector corresponding to a standard
λ
-tableau, but a linear combination of Specht vectors.
3. The change-of-basis matrix
Now we embed the Specht module
S8
J
into the induced trivial module
M8
J
by identifying the tabloid
{R}
for R∈rStd8(d), with the standard basis element M8
wRin M8
J, where wR∈D8
J.
Under the embedding, we denote by
a8
x,T
(for
T∈rStd8(λ)
and
x∈D8
J
) the change-of-basis coeffi-
cients from the Specht basis to the Kazhdan–Lusztig basis, i.e.,
(3-1) v8
T=X
x∈D8
J
a8
x,TMx.
Recall that ≤is the Bruhat order on D8
J⊂W8
d.
Theorem 3.1. For J ⊆I8
d,fix total orders ≤Don D8
Jand ≤ron rStd8
J(λ) satisfying
x≤y∈D8
J⇒x≤Dy and wR≤wT∈D8
J⇒R≤rT for R,T∈rStd8
J(λ),
respectively. The matrix (a8
x,T)x,Tis upper unitriangular in the sense that
a8
wT,T=1 for T∈rStd8
J(λ), a8
x,T=0 if x6≤DwT.
Moreover,a8
x,T=0if x 6≤ wT.
Note that if a partial ordering does not exist, then the appropriate change-of-basis coefficient is zero.
Proof. Combining Lemmas 2.3 and 2.11, we obtain the following:
(3-2) v8
T=X
R∈rStd8(λ)
wR≤wT
c8
R,TX
x≤wR
m8
x,wR(1)Mx=X
x≤wTX
R∈rStd8(λ)
wR≤wT
c8
R,Tm8
x,wR(1)Mx.
It follows from the unitriangularity of (c8
R,T)and (m8
wT,wR(1)) that
a8
wT,T=X
R∈rStd8(λ)
wR≤wT
c8
R,Tm8
wT,wR(1)=1.
Also, if x6≤DwTthen x6≤ wT, and so a8
x,T=0. 
Remark 3.2.
For type A, Theorem 3.1 recovers [7, Theorem 4.1]for arbitrary
J
. If
J= [
2
n] − {n}
, then
it also recovers [8, Theorems 5.5 and 5.7].
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[1]
H. Bao, J. Kujawa, Y. Li, and W. Wang, “Geometric Schur duality of classical type”,Transform. Groups
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[3]
R. Dipper and G. James, “Representations of Hecke algebras of general linear groups”,Proc. London Math. Soc.
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(1986), 20–52.

1680 MEE SEONG IM
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MEE SE ON G IM:meeseongim@gmail.com
Department of Mathematics, United States Naval Academy, Annapolis, MD, United States
RMJ — prepared by msp for the
Rocky Mountain Mathematics Consortium