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ROW AND COLUMN REMOVAL THEOREMS FOR HOMOMORPHISMS OF SPECHT

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					  ROW AND COLUMN REMOVAL THEOREMS FOR HOMOMORPHISMS
          OF SPECHT MODULES AND WEYL MODULES

                                      ´
                                   SINEAD LYLE AND ANDREW MATHAS


       Abstract. We prove a q-analogue of the row and column removal theorems for homomor-
       phisms between Specht modules proved by Fayers and the first author [6]. The original paper
       considered a complement to (Donkin’s generalisations of) James’s row and column removal
       theorems regarding Specht modules for the symmetric groups. Here, we consider the Specht
       modules of the Iwahori–Hecke algebra HR,q (Sn ) when q = −1.
          Similar theorems are then shown to hold in the context of homomorphisms between the Weyl
       modules of the q-Schur algebra. In fact, the latter results are shown to hold in more generality:
       specifically we can allow q = −1.




                                               1. Introduction
   Fix a positive integer n ≥ 1 and a commutative ring R. Let q be an invertible element
of R and let H = Hn = HR,q (Sn ) be the Iwahori–Hecke algebra of the symmetric group S n of
degree n.
   For each partition λ of n Dipper and James [2] defined a right H R,q (Sn )-module S λ called a
Specht module. These modules are important because when H R,q (Sn ) is semisimple the Specht
modules form a complete set of pairwise non–isomorphic irreducible H R,q (Sn )-modules. More
generally, every irreducible HR,q (Sn )-module arises as the head of some Specht module.
   In this paper we prove the following two theorems, which are q-analogues of the row and
column removal theorems for homomorphisms from [6].
Theorem 1.1. Let R be a ring and q an invertible element of R with q = −1. Let λ and µ be
partitions of n, and suppose that for some s we have
                                        λ1 + · · · + λ s = µ 1 + · · · + µ s .
Define partitions
                   µT = (µ1 , . . . , µs ),                           λT = (λ1 , . . . , λs ),
                   µB = (µs+1 , µs+2 , . . . ),                      λB = (λs+1 , λs+2 , . . . ).
Then, putting m = |λB | = |µB |, we have
                                                    T     T              B     B
                   HomHn (S µ , S λ ) ∼ HomHn−m (S µ , S λ ) ⊗ HomHm (S µ , S λ ).
                                      =
  For any partition ν, let ν be the partition which is conjugate to ν; that is, ν is the partition
obtained by swapping the rows and columns of ν.
Theorem 1.2. Let R be a ring and q an invertible element of R with q = −1. Let λ and µ be
partitions of n, and suppose that for some r we have
                                        λ1 + · · · + λ r = µ 1 + · · · + µ r .
  1991 Mathematics Subject Classification. 20C08, 05E10.
  Key words and phrases. Hecke algebras, Schur algebras.
  This research was supported by ARC grant DP0343023.
                                                          1
2                                       ´
                                     SINEAD LYLE AND ANDREW MATHAS


Define
    λL = min(λ1 , r), min(λ2 , r), . . . ,                      µL = min(µ1 , r), min(µ2 , r), . . . ,
    λR = max(λ1 − r, 0), max(λ2 − r, 0), . . . ,                µR = max(µ1 − r, 0), max(µ2 − r, 0), . . . .
Then, putting l = |λR | = |µR |, we have
                                                                L        L                 R     R
                      HomHn (S λ , S µ ) ∼ HomHn−l (S λ , S µ ) ⊗ HomHl (S λ , S µ ).
                                         =
   The partitions λ, λL , λR , λT , λB appearing in these theorems may be viewed pictorially as
follows:
                                                λT                   λR
                         λ                                      λ L

                                                     λB
                                                                                .
   One of the reasons why these results are interesting is that Donkin [4] and James [8] have
proved analogous results for the decomposition numbers of H. To state these results precisely,
define e > 1 to be minimal such that 1 + q + · · · + q e−1 = 0 (setting e = ∞ if no such integer
exists) and if S is an H-module and D is a simple H-module let [S : D] be the decomposition
multiplicity of D in S. Finally, if λ is an e-restricted partition (that is, λ i − λi+1 < e for all i),
let D λ = S λ / rad S λ be the corresponding simple H-module (see, for example, [9]). Then, under
the assumptions of Theorem 1.1 we have
                                                         T          T         B      B
                                   [S µ : D λ ] = [S µ : D λ ] · [S µ : D λ ].
There is an analogous statement under the assumptions of Theorem 1.2.
   This paper is structured as follows. In the next section we recall the necessary machinery from
the literature and also prove some preliminary combinatorial results. Section 3 gives the proof
of Theorem 1.1 and Theorem 1.2. Finally, in section 4, we establish analogous results relating
homomorphisms between Weyl modules of the q-Schur algebras. For precise statements, see
Theorem 4.1 and Theorem 4.2.

                                               2. Definitions
    We begin by establishing some definitions. The notation used will be that of [9].
Definition 2.1. Let n be a positive integer and S n the symmetric group acting on n letters.
Suppose R is a ring and let q be an invertible element of R. The Iwahori–Hecke algebra H =
HR,q (Sn ) of Sn is the unital associative R-algebra with generators T 1 , T2 , . . . , Tn−1 and relations
                  (Ti − q)(Ti + 1) = 0,                                      for i = 1, 2, . . . , n − 1,
                               Ti Tj = T j Ti ,                              for 1 ≤ i < j − 1 ≤ n − 2,
                       Ti+1 Ti Ti+1 = Ti Ti+1 Ti ,                           for i = 1, 2, . . . , n − 2.
Set si to be the transposition (i, i + 1) ∈ Sn . Let ω ∈ Sn and suppose si1 si2 . . . sil is a reduced
expression for ω; that is, if ω = sj1 sj2 . . . sjk then k ≥ l. We say that ω has length l and write
 (ω) = l. Define Tω = Ti1 Ti2 . . . Tik . Then H is free as an R-module with basis {T ω | ω ∈ Sn }.
Definition 2.2. Let λ and µ be compositions of n. Say that λ dominates µ and write λ                         µ if
                                                     k              k
                                                         λi ≥           µi
                                                 i=1            i=1
for all k. If λ     µ and λ = µ, we write λ µ.
                              ROW AND COLUMN REMOVAL THEOREMS                                        3


Definition 2.3. Let λ = (λ1 , λ2 , . . .) be a composition of some integer n. The diagram of λ is
the set
                                    {(i, j) | 1 ≤ i and 1 ≤ j ≤ λi }
regarded as an array of nodes in the plane. A λ-tableau is obtained from the diagram of λ by
replacing each node with one of the integers 1, 2, . . . , n, allowing no repeats. If t is a λ-tableau
we write Shape(t) = λ. A λ-tableau t is row standard if its entries increase along the rows; and
is standard if λ is a partition and the entries of t increase across the rows and down the columns.
If λ is a partition, define Std(λ) to be the set of standard λ-tableaux. The symmetric group S n
acts from the right in a natural way on the set of tableaux.
Definition 2.4. Let λ and µ be compositions of n. A λ-tableau of type µ is obtained by
replacing each node in the diagram of λ with one of the integers 1, 2, . . . such that the number
i occurs µi times. A λ-tableau T (of type µ) is semistandard if λ is a partition, and the entries
of T are non-decreasing across the rows and strictly increasing down the columns. Let T 0 (λ, µ)
denote the set of semistandard λ-tableaux of type µ, and let T λ denote the unique semistandard
λ-tableau of type λ.
Definition 2.5. Let λ be a composition of n. Define t λ to be the standard λ-tableau in which
the integers 1, 2, . . . , n are entered in increasing order along its rows, and S λ to be the row
stabilizer of tλ . Set
                                              mλ =       Tω .
                                                   ω∈Sλ
We define the permutation module          Mλ   to be the right H-module M λ = mλ H.
Definition 2.6. Suppose that λ is a composition of n and that t is a row standard λ-tableau.
Define d(t) ∈ Sn to be such that t = tλ d(t). Now define ∗ to be the R-linear anti–automorphism
of H determined by Ti∗ = Ti . If s, t are row standard λ-tableaux, define m st ∈ H by
                                                 ∗
                                          mst = Td(s) mλ Td(t) .
Then Murphy [10] has shown that
                          {mst | s, t ∈ Std(λ) for some partition λ of n}
is a cellular basis of H in the sense of Graham and Lehrer [7]. Further, if µ is any composition
of n and s, t are row standard µ-tableaux then m st can be written as a R-linear combination of
muv where u, v ∈ Std(λ), and λ is a partition of n such that λ µ.
Definition 2.7. Let λ, µ be compositions of n and let t be a λ-tableau. Define µ(t) to be the
λ-tableau of type µ obtained by replacing each entry in t with its row index in t µ .
  Suppose that S is a semistandard λ-tableau of type µ and t is a standard λ-tableau. Define
                           mSt =          mst s ∈ Std(λ) and µ(s) = S .
                                     s
As proved in [10], M µ is free as an R-module with basis
                    {mSt | S ∈ T0 (λ, µ), t ∈ Std(λ) for some partition λ of n}.
Definition 2.8. Let λ be a partition of n. Define H λ to be the two–sided ideal of H with basis
                                 {mst | s, t ∈ Std(ν) where ν      λ}
    ˇ
and Hλ to be the two–sided ideal of H with basis
                                 {mst | s, t ∈ Std(ν) where ν  λ}.
We define the Specht module      Sλ                             ˇ
                                     to be the right H-module (Hλ + mλ )H.
4                                      ´
                                    SINEAD LYLE AND ANDREW MATHAS

                                                            ˇ
   It follows that S λ is free as an R-module with basis {Hλ + mtλ t | t ∈ Std(λ)}.
   We now look at the q-Schur algebra. Fix a non–negative integer d and let Λ(d, n) be the set of
compositions of n with at most d non–zero parts. Let Λ + (d, n) be the set of partitions contained
in Λ(d, n) and for λ ∈ Λ+ (d, n), set T0 (λ) = µ∈Λ(d,n) T0 (λ, µ).

Definition 2.9. The q-Schur algebra S = S(d, n) is the endomorphism algebra of H defined by
                                                       

                                          S = EndH                M µ .
                                                        µ∈Λ(d,n)

Definition 2.10. Let λ, µ ∈ Λ(d, n), ν ∈ Λ + (d, n) and suppose S ∈ T0 (ν, λ), T ∈ T0 (ν, µ). If
τ ∈ Λ(d, n), define ϕST ∈ S by

            ϕST (mτ h) = δτ µ         mst h s, t ∈ Std(ν) and λ(s) = S, µ(t) = T      ∈ Mλ
                                s,t

for all h ∈ H. In fact [9, 4.13],
                                {ϕST | ν ∈ Λ+ (d, n) and S, T ∈ T0 (ν)}
is a cellular basis of S.
Definition 2.11. Let λ ∈ Λ+ (d, n), µ ∈ Λ(d, n) and T ∈ T0 (λ, µ). Define the map
                                           ϕT :               M τ → Sλ
                                                  τ ∈Λ(d,n)

by
                 ϕT (mτ h) = δτ µ           ˇ
                                            Hλ + mλ Td(t) h t ∈ Std(λ) and µ(t) = T
                                      t
for all h ∈ H.
   By restriction, we will consider ϕT to be an element of HomH (M µ , S λ ).
   Dipper and James [3, Cor. 8.7] have proved a stronger version of the next result for the
Specht modules which are dual to the S λ . This lemma can be deduced from the Dipper–James
result; however, we give an independent proof of it because the lemma is central to what follows
(and the Dipper–James result is more difficult to prove).
   If s is a standard tableau let s↓m be the subtableau of s which contains the numbers 1, . . . , m.
If s and t are standard λ-tableaux write s t if Shape(s↓m) Shape(t↓m), for m = 1, . . . , n. If
s t and s = t we write s t.
   The following Lemma should be well–known, however, we do not know of a reference in the
literature. (After much work, Dipper and James [3, Cor. 8.7] prove the corresponding result for
Specht modules which are dual to the ones considered here.)
Lemma 2.12. Let λ and µ be partitions of n and suppose that q = −1. Then Hom H (M µ , S λ )
is free as an R-module with basis {ϕT | T ∈ T0 (λ, µ)}.
Proof. First observe that if φ ∈ HomH (M µ , S λ ) then φ(mµ h) = φ(mµ )h, for all h ∈ H, so φ is
completely determined by the value of φ(m µ ). In particular, the maps ϕT are linearly indepen-
dent because {ϕT (mµ ) | T ∈ T0 (λ, µ)} is linearly independent by the remark after Definition 2.8.
  It remains to show that the set {ϕT (mµ ) | T ∈ T0 (λ, µ)} spans HomH (M µ , S λ ). Let φ ∈
HomH (M µ , S λ ) and write φ(mµ ) = {Hλ +ˇ        rt mtλ t | t ∈ Std(t)}, for some rt ∈ R. Fix a
tableau s such that rs = 0 and rt = 0 whenever s t. We claim that µ(s) is a semistandard
                             ROW AND COLUMN REMOVAL THEOREMS                                              5


tableau. If not, then there exist integers i < j which are in the same column of s such that
(i, j) ∈ Sµ . Now, (j − 1, j) ∈ Sµ so mµ Tj = qmµ . Therefore
(†)   ˇ
      Hλ +                                                              ˇ
               rt mtλ t = φ(mµ ) = q −1 φ(mµ Tj−1 ) = q −1 φ(mµ )Tj−1 = Hλ +       q −1 rt mtλ t Tj−1 .
Let s = s(j − 1, j). If i < j − 1 then s is a standard tableau and s dominates s , so mtλ s Tj−1 =
mtλ s , and rs = 0 by the choice of s. However, now (†) and [9, Cor. 3.9] show that m tλ s
appears with non–zero coefficient in φ(m µ ), so this is a contradiction and we must have i = j −1.
Therefore, j − 1 and j are in the same column of s and so there exist a v ∈ R such that
                      ˇ                 ˇ
                      Hλ + mtλ s Tj−1 = Hλ + −mtλ s , +       {av mtλ v | v s},
by [9, Cor. 3.21]. Hence, comparing the coefficient of m tλ s on both sides of (†) shows that
rs = −q −1 rs . Since q = −1 we again have a contradiction. Hence, µ(s) is semistandard as
claimed.
   To complete the proof it is enough to show that r t = rt whenever µ(t) = µ(t ). The proof of
this is similar to [9, Prop. 3.18] and is left as an exercise for the reader.
Definition 2.13. Let µ ∈ Λ(d, n). Define
                                         ϕµ = ϕT µ T µ ∈ S.
Then ϕµ acts as the identity on M µ and zero on M τ for τ = µ. Set Mµ to be the right S-module
                                        Mµ = ϕµ S.
Then Mµ has a basis given by
                      {ϕST | S ∈ T0 (λ, µ), T ∈ T0 (λ) where λ ∈ Λ+ (d, n)}.
Definition 2.14. For λ ∈ Λ+ (d, n), we define S λ to be the two-sided ideal of S given by
                    S λ = {ϕST | S, T ∈ T0 (ν) where ν ∈ Λ+ (d, n) and ν      λ}
    ˇ
and S λ to be the two-sided ideal of S given by
                    ˇ
                   S λ = {ϕST | S, T ∈ T0 (ν) where ν ∈ Λ+ (d, n) and ν      λ}.
We define the Weyl module     Wλ   to be the right S-module
                                                ˇ
                                       W λ = (S λ + ϕλ )S.
Then W λ is free as an R-module with basis
(2.15)                                ˇ
                                     {S λ + ϕT λ T | T ∈ T0 (λ)}.
                           ˇ
In fact, we may identify S λ + ϕT λ T with the map ϕT ∈ HomH ( τ M τ , S λ ) of Definition 2.11.
Then W   λ is isomorphic to the S-submodule of Hom (              τ λ
                                                       H     τ M , S ) with basis {ϕT | T ∈ T0 (λ)}.
(Note that if q = −1 this is actually a basis of Hom H ( τ M τ , S λ ) by Lemma 2.12.)
Definition 2.16. Let λ ∈ Λ+ (d, n), µ ∈ Λ(d, n) and T ∈ T0 (λ, µ). Define θT ∈ HomS (Mµ , W λ )
by
                                      θT (ϕµ f ) = ϕT f
for all f ∈ S.
  Using (2.15) is now straightforward to prove the following.
Lemma 2.17. Suppose that λ ∈ Λ+ (d, n) and µ ∈ Λ(d, n). Then {θT | T ∈ T0 (λ, µ)} is a basis
of HomS (Mµ , W λ ).
6                                         ´
                                       SINEAD LYLE AND ANDREW MATHAS


                                3. The Hecke algebras of type A
   In this section we prove Theorem 1.1 and Theorem 1.2. We start by showing that these two
results are equivalent.
Lemma 3.1. For any partitions λ and µ of n, we have
                          HomHn (S µ , S λ ) ∼ HomHn (S λ , S µ ).
                                             =
Proof. For any Hn -module M , let M = HomR (M, R) be the dual of M . Then there exists an
                                                                              #
Hn -automorphism # such that (S ν )# ∼ (S ν ) [10, Theorem 5.2]; explicitly, Tω = (−q) (ω) Tω−1
                                     =                                                      −1

for all ω ∈ Sn . Hence
                         HomH (S µ , S λ ) ∼ HomH ((S µ )# , (S λ )# )
                                         n =                    n

                                                   ∼ HomH ((S µ ) , (S λ ) )
                                                   =     n

                                                   ∼ HomH (S λ , S µ ).
                                                   =            n




Proposition 3.2. Theorem 1.1 holds if and only if Theorem 1.2 holds.
Proof. If λ and µ satisfy the conditions of Theorem 1.1, then λ and µ satisfy the conditions of
Theorem 1.2, with r = s, l = m and with
                        (λ )L = (λT ) ,                                  (λ )R = (λB ) ,
                        (µ )L = (µT ) ,                                  (µ )R = (µB ) .
By Lemma 3.1 we have HomHn (S µ , S λ ) ∼ HomHn (S λ , S µ ) and
                                        =
                                   T     T                  B       B
                 HomHn−m (S µ ,S λ ) ⊗ HomHm (S µ , S λ )
                                        ∼ HomH        λL      L              R     R
                                        =      n−m (S    , S µ ) ⊗ HomHm (S λ , S µ ).
The proof of the converse is similar.
 It is therefore sufficient to prove only Theorem 1.1.
 Henceforth in this section, we fix a ring R and an invertible element q ∈ R such that q = −1.
We also fix integers m and n with 0 ≤ m ≤ n and partitions λ and µ of n which satisfy
                                 λ1 + · · · + λ s = µ 1 + · · · + µ s = n − m
for some s. For any composition ν of n which satisfies ν 1 + · · · + νs = n − m write
                        ν T = (ν1 , . . . , νs )   and          ν B = (νs+1 , νs+2 , . . .).
Definition 3.3. Let H = Hn be the algebra generated by T1 , T2 , . . . , Tn−1 subject to the re-
lations of Definition 2.1 The subalgebra H T ∼ Hn−m is defined to be the subalgebra of H
                                                  =
generated by
                                    T1 , T2 , . . . , Tn−m−1
                     B ∼ H is defined to be the subalgebra of H generated by
and the subalgebra H = m
                                         Tn−m+1 , Tn−m+2 , . . . , Tn−1 .
Note that ifhT  ∈  HT and     hB∈ HB then hT hB = hB hT .
   Define Sn−m to be the permutation group acting on the set {1, 2, . . . , n − m} and S B the
                                                                                        m
permutation group acting on {n − m + 1, n − m + 2, . . . , n}. Suppose that ν is any partition
                                                     T         B
of n such that ν1 + ν2 + . . . + νs = n − m. Define tν and tν to be the standard ν-tableaux
                               ROW AND COLUMN REMOVAL THEOREMS                                         7


with entries 1, 2, . . . , n − m and n − m + 1, n − m + 2, . . . , n respectively which increase along
the rows, and define Sν T ⊆ Sn−m and SBB ⊆ SB accordingly. Now define
                                            ν       m

                                           mν T =             Tω ,
                                                    ω∈Sν T

                                           mν B =             Tω .
                                                    ω∈SBB
                                                          ν

Note that
                                           mν = m ν T mν B .
Then we have
                                               T
                                           M ν = mν T H T ,
                                               B
                                           M ν = mν B H B .
Hence we have cellular bases of H T and HB using Definition 2.6 in the obvious way. We may
                ˇ T ˇ B      T       B
similarly define Hν , Hν , S ν and S ν .
Definition 3.4. Suppose that AT ∈ T0 (λT , µT ) has entry xi in position i of row j and AB ∈
                                                               j
                         i
T0 (λB , µB ) has entry yj in position i of row j. Define (AT , AB ) to be the λ-tableau of type µ
               i
with entry zj in position i of row j, where

                                        i      xi
                                                j     (j ≤ s)
                                       zj =     i
                                                              .
                                               yj + s (j > s)
  For example, if
                                         111112                        112
                                  AT =   2223   ,             AB =     23 ,
                                         333                           3
then
                                                      1   1   1112
                                                      2   2   23
                                           T   B      3   3   3
                                        (A , A ) =    4   4   5    .
                                                      5   6
                                                      6

Lemma 3.5. If AT and AB are semistandard, then so is (AT , AB ), and
                          {(AT , AB ) | AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB )}
is precisely the set of semistandard λ-tableaux of type µ.
Proof. The first statement is clear. Suppose A is a semistandard λ-tableau of type µ with entry
 i
zj in position i of row j. Then the entries of A strictly increase down the columns, so that for
                       i
j > s we have that zj > s. Since the number, n − m, of entries less than or equal to s equals the
                                                                      i
number of positions in rows 1, . . . , s, it also is also true that z j ≤ s for j s. So the top s rows
of A constitute a λ T -tableau of type µT (which is clearly semistandard), and a similar statement

holds for the lower rows.
Definition 3.6. Suppose that a is a ν-tableau which contains the entries 1, 2, . . . , n − m in rows
1, 2, . . . , s. Define aT to be the ν T -tableau consisting of rows 1 to s of a, and a B the ν B -tableau
consisting of rows s + 1, s + 2, . . . Similarly, if we have such an a T , aB , define a = (aT , aB )
in the obvious manner. Say that aT ∈ Std(ν T ) if it is a standard ν T -tableau containing the
entries 1, 2, . . . n − m and aB ∈ Std(ν B ) if it is a standard ν B -tableau containing the entries
n − m + 1, n − m + 2, . . . n.
8                                            ´
                                          SINEAD LYLE AND ANDREW MATHAS


Lemma 3.7. Let A ∈ T0 (ν, µ). Then
       a | a ∈Std(ν) and µ(a) = A =
               (aT , aB ) | aT ∈ Std(ν T ) and µT (aT ) = AT , aB ∈ Std(ν B ) and µB (aB ) = AB .
Proof. This follows from Lemma 3.5.
                                                                      T        T                  B    B
Definition 3.8. Now suppose that ΦT ∈ HomHT (M µ , S λ ), and ΦB ∈ HomHB (M µ , S λ ).
Then
                       ΦT =       ΓAT ϕAT AT ∈ T0 (λT , µT )
                                                AT
and
                                  ΦB =                 ΓAB ϕAB AB ∈ T0 (λB , µB )
                                               AB
for some ΓAT , ΓAB ∈ R. Define Φ = (ΦT , ΦB ) ∈ HomH (M µ , S λ ) by

              Φ=                  ΓAT ΓAB ϕ(AT ,AB ) AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB ) .
                      AT AB

    By Lemma 2.12 and Lemma 3.5 this defines an R-linear bijection
                                      T        T                  B    B
                     HomHT (M µ , S λ ) ⊗ HomHB (M µ , S λ ) → HomH (M µ , S λ ).
Lemma 3.9. Suppose that λ                 ν µ. Then ν 1 + . . . + νs = n − m and either ν T µT or ν B µB .
Proof. The proof follows directly from the definition of                    .
Lemma 3.10. Suppose that ν                    µ and that S ∈ T 0 (ν, µ). Let
                                                    ∗
                              m=                   Td(s) mν s ∈ Std(ν) and µ(s) = S .
                                          s

                                                                                     ˇ T
Then m = mµ hT hB for some hT , hB such that hT ∈ HT and hB ∈ HB and either mµT hT ∈ Hµ
            ˇ B
or mµB hB ∈ Hµ .
Proof. Define
                      mT =                Td(sT ) mν T | sT ∈ Std(ν T ) and µT (sT ) = S T .
                                           ∗

                                 sT

Then mT ∈ M     µT   and so mT = mµT hT for some hT ∈ HT . Similarly, we can define

                      mB =                Td(sB ) mν B sB ∈ Std(ν B ) and µB (sB ) = S B .
                                           ∗

                                 sB

so that mB = mµB hB for some hB ∈ HB . If ν T                                    ˇ T
                                                                µT then mµT hT ∈ Hµ ; otherwise, by Lemma
                          ˇ B
3.9, ν B µB and mµB hB ∈ Hµ . Then
                             ∗
               m=           Td(s) mν s ∈ Std(ν) and µ(s) = S
                       s

                 =                Td(sT ) Td(sB ) mν T mν B sT ∈ Std(ν T ) and µT (sT ) = S T ,
                                   ∗       ∗

                       sT   sB

                                               sB ∈ Std(ν B ) and µB (sB ) = S B
                                ROW AND COLUMN REMOVAL THEOREMS                                    9


by Lemma 3.7

                = m T mB
                = m µT h T m µB h B
                = m µ hT hB .


Proposition 3.11. Suppose that λ, µ, ν are partitions of n such that λ, ν           µ and λ   ν. Let
S ∈ T0 (ν, µ), t ∈ Std(ν) and A ∈ T0 (λ, µ). Then
                                             ϕA (mSt ) = 0.

Proof. Consider the map ϕT λ A of Definition 2.10. Then ϕA = πϕT λ A where π : M λ → S λ is the
natural projection, and ϕT λ A consists of left multiplication by certain elements of H. Now, since
mSt ∈ Hν , we have that ϕT λ A (mSt ) ∈ Hν , and hence can be written as an R-linear combination
of elements of the form muv where u, v ∈ Std(τ ) for some τ ν.
   However, ϕT λ A (mSt ) ∈ M λ , and hence can be written as an R-linear combination of elements
of the form muv where u, v ∈ Std(τ ) for some τ λ. Since λ ν, this shows that ϕ T λ A (mSt ) ∈ Hλˇ
and so ϕA (mSt ) = 0 as required.
Proposition 3.12. Suppose that λ, µ, ν are partitions of n such that λ         ν µ and that
                                                         T    T
                                       ΦT ∈ HomHT (M µ , S λ )
                                                         B    B
                                      ΦB ∈ HomHB (M µ , S λ )
are such that
                                                                                T
                                                                      ˇ
                  ΦT (mµT h) = 0 whenever h ∈ HT is such that mµT h ∈ Hµ

and

                                                                      ˇ B
                  ΦB (mµB h) = 0 whenever h ∈ HB is such that mµB h ∈ Hµ .
Suppose that

                            ΦT =           ΓAT ϕAT AT ∈ T0 (λT , µT )
                                      AT

                            ΦB =           ΓAB ϕAB AB ∈ T0 (λB , µB ) .
                                      AB

Define the map Φ = (ΦT , ΦB ) as in Definition 3.8. Let S ∈ T0 (ν, µ), t ∈ Std(ν). Then
                                             Φ(mSt ) = 0.
Proof. Assume that we have the conditions of Proposition 3.12. Using the notation of Lemma
3.10
                                       ∗
                      mSt =           Td(s) mν Td(t) s ∈ Std(ν) and µ(s) = S
                                 s
                           = mµ hT hB Td(t)
10                                           ´
                                          SINEAD LYLE AND ANDREW MATHAS


for suitable hT ∈ HT , hB ∈ HB . Then

 Φ(mSt ) =            ΓAT ΓAB ϕ(AT ,AB ) (mµ )hT hB Td(t)
              AT AB

          =           Γ AT Γ AB             ˇ
                                            Hλ + mλ Td(a) hT hB Td(t) a ∈ Std(λ) and µ(a) = (AT , AB )
              AT AB                   a

          =           Γ AT Γ AB                   ˇ
                                                  Hλ + mλT Td(aT ) hT mλB Td(aB ) hB Td(t)
              AT AB                  aT    aB

                                    aT ∈ Std(λT ) and µT (aT ) = AT , aB ∈ Std(λB ) and µB (aB ) = AB

            ˇ
          = Hλ +            Γ AT          mλT Td(aT ) hT                       Γ AB        mλB Td(aB ) hB   Td(t) .
                       AT            aT                                   AB          aB

Note that
                                           ˇ T
                            ΦT (mµT hT ) = Hλ +                           Γ AT        mλT Td(aT ) hT
                                                                     AT          aT
                                B
                            Φ (mµB h ) = H  B          ˇ λB      +        Γ AB        mλB Td(aB ) hB .
                                                                     AB          aB

                    ˇ        T
                                    ˇ                        B
Now either mµT hT ∈ Hµ or mµB hB ∈ Hµ . Without loss of generality, suppose that m µT hT ∈
ˇ
H µT . Then

                               Γ AT                    ˇ T
                                      mλT Td(aT ) hT ∈ Hλ
                                           AT           aT

and
                                                                                             B
                                                Γ AB         mλB Td(aB ) hB ∈ M λ .
                                          AB           aB

Hence we can write
                                                                                                                
                 ˇ
       Φ(mSt ) = Hλ +                    ∆xT yT mxT yT xT , y T ∈ Std(τ T ) where τ T                      λT 
                                xT ,y T
                                                                                                                     
                                          ∆xB yB mxB yB xB , y B ∈ Std(τ B ) where τ B                       λB  Td(t)
                                xB ,y B

for some ∆xT yT , ∆xB yB ∈ R. Now for any such mxT yT , mxB yB we may write
                                             ∗                    ∗
                            mxT yT mxB yB = Td(xT ) mτ T Td(yT ) Td(xB ) mτ B Td(yB )
                                                     ∗       ∗
                                                  = Td(xT ) Td(xB ) mτ T mτ B Td(yT ) Td(yB )
                                                     ∗
                                                  = Td(xT ,xB ) mτ Td(yT ,yB )
               T   T          B B
where τ = (τ1 , τ2 , . . . , τ1 , τ2 , . . .) is a composition of n such that τ λ and (x T , xB ), (y T , y B )
are row standard τ -tableaux. By Definition 2.6, this can be written as an R-linear combination
of terms muv where u, v ∈ Std(τ ) and τ                                           ˇ           ˇ
                                                       τ λ. Hence mxT yT mxB yB ∈ Hλ . Since Hλ is a right
ideal, it follows that mxT yT mxB yB Td(t) ∈ H       ˇ λ and so Φ(m ) = 0 as required.
                                                                    St
                            ROW AND COLUMN REMOVAL THEOREMS                                    11

                                                      T    T                         B    B
Proposition 3.13. Suppose that ΦT ∈ HomHT (M µ , S λ ), and ΦB ∈ HomHB (M µ , S λ ) are
such that
                                                                   ˇ T
               ΦT (mµT h) = 0 whenever h ∈ HT is such that mµT h ∈ Hµ
and
                                                                              B
                                                                     ˇ
                 ΦB (mµB h) = 0 whenever h ∈ HB is such that mµB h ∈ Hµ .
                                                                    ˆ
Define the map Φ = (ΦT , ΦB ) as in Definition 3.8, and define the map Φ : S µ → S λ by
                           ˆ ˇ
                           Φ[(Hµ + mµ )h] = [Φ(mµ )]h for all h ∈ H.
     ˆ
Then Φ is a well defined H-homomorphism.
                                                                 ˇ
Proof. It is sufficient to show that if h ∈ H is such that m µ h ∈ Hµ then [Φ(mµ )]h = Φ(mµ h) = 0.
Suppose h satisfies these conditions. Then m µ h is an R-linear combination of terms of the form
mSt where S ∈ T0 (ν, µ) and t ∈ Std(ν) for some ν µ. Choose such an m St . If λ                ν
then Φ(mSt ) = 0 by Proposition 3.11. If λ ν then Φ(m St ) = 0 by Proposition 3.12. Hence
Φ(mµ h) = 0 as required.
Definition 3.14. Suppose Φ ∈ Hom H (M µ , S λ ) is such that Φ(mµ h) = 0 for all h ∈ H such
            ˇ          ˆ
that mµ h ∈ Hµ . Define Φ ∈ HomH (S µ , S λ ) by setting
                           ˆ ˇ
                           Φ[(Hµ + mµ )h] = [Φ(mµ )]h for all h ∈ H.
              ˆ                                                         ˆ
Similarly, if Φ ∈ HomH (S µ , S λ ), define Φ ∈ HomH (M µ , S λ ) by Φ = Φπ where π : M µ →
S µ is the natural projection. We use analogous notation the for corresponding H T and HB

homomorphisms.
Definition 3.15. Suppose that
                                                  T     T
                                   ˆ
                                   ΦT ∈ HomHT (S µ , S λ ),
                                                  B     B
                                   ˆ
                                   ΦB ∈ HomHB (S µ , S λ ).
Define
                                                      T    T
                                   ΦT ∈ HomHT (M µ , S λ ),
                                                      B    B
                                   ΦB ∈ HomHB (M µ , S λ )


as in Definition 3.14. Note that ΦT , ΦB satisfy the conditions of Proposition 3.13. Hence define
                                ˆ     ˆ ˆ
                                Φ = (ΦT , ΦB ) ∈ HomH (S µ , S λ )
as in Proposition 3.13.
  We can now prove the main result of this section.
Proof of Theorem 1.1. We have to show that
                                                 T     T              B     B
                   HomH (S µ , S λ ) ∼ HomHT (S µ , S λ ) ⊗ HomHB (S µ , S λ ).
                                     =
                     T   T                        B   B
    ˆ                           ˆ
Let ΦT ∈ HomHT (S µ , S λ ) and ΦB ∈ HomHB (S µ , S λ ). Then there is an R-linear mapping
               T     T              B     B
from HomHT (S µ , S λ ) ⊗ HomHB (S µ , S λ ) into HomH (S µ , S λ ) given by sending
                                     ˆ    ˆ      ˆ ˆ
                                     ΦT ⊗ ΦB → ( ΦT , ΦB )
12                                       ´
                                      SINEAD LYLE AND ANDREW MATHAS


and extending linearly. By construction this mapping is clearly injective. It remains only to
show that it is surjective.
        ˆ
  Take Φ ∈ HomH (S µ , S λ ) and form the map Φ ∈ Hom H (M µ , S λ ). Then there exist ΓAT AB ∈ R
such that
                                    Φ=       ΓAT AB ϕ(AT ,AB ) .
                                                 AT ,AB

                                      ˇ T                                ˇ
Choose hT ∈ HT such that mµT hT ∈ Hµ . It therefore follows that mµ hT ∈ Hµ , and so, by
construction of the map Φ, we have that Φ(m µ hT ) = 0. Thus

                  Γ AT AB            mλT Td(aT ) hT mλB Td(aB ) aT ∈ Std(λT ) and µT (aT ) = AT ,
         AT ,AB             aT ,aB

                                                 aB ∈ Std(λB ) and µB (aB ) = AB            ˇ
                                                                                          ∈ Hλ .

Hence for fixed AB , it is necessary that

                                                Γ AT AB          mλT Td(aT ) hT
                                           AT               aT

can be expressed in terms of muv where u, v ∈ Std(ν T ) and ν T                   λT . Hence

                                           Γ AT AB                          ˇ T
                                                           mλT Td(aT ) hT ∈ Hλ .
                                      AT             aT

So for fixed AB , the map
                                                ΦT =            Γ AT AB ϕ AT
                                                           AT

                                   ˆ       T     T
                                                                                ˆ
is such that the corresponding map ΦT : S µ → S λ is well defined. Suppose that {φi i ∈ I}
                                 T     T
form a basis of HomHT (S µ , S λ ). So we can write
                                            ˆ
                                            ΦT =                  ˆ
                                                            βi,AB φi
                                                      i

and so

                                             Φ=             βi,AB (φi , ϕAB ).
                                                     i,AB

A similar argument for fixed φi shows that in fact

                                             Φ=             αij (φT , φB )
                                                                  i    j
                                                     i,j

      ˆ               T    T
                                  ˆ              B     B
where φT ∈ HomHT (S µ , S λ ) and φB ∈ HomHB (S µ , S λ ) for all i, j; thus under our mapping,
       i                           j
ˆ
Φ appears as the image of
                                                                 T     T              B    B
                                ˆ    ˆ
                           αij (φT ⊗ φB ) ∈ HomHT (S λ , S µ ) ⊗ HomHB (S λ , S µ ).
                                 i    j
                     i,j
                                  ROW AND COLUMN REMOVAL THEOREMS                                               13


                                        4. The q-Schur algebra
  Let R be a ring and q an invertible element of R; we allow q = −1. In this section we prove
the following analogues of Theorems 1.1 and 1.2 for homomorphisms between Weyl modules.
Theorem 4.1. Fix integers n, m with 0 ≤ m ≤ n and positive integers d, d , d . Let λ, µ ∈
Λ+ (d, n) and suppose that for some s we have
                                  λ1 + · · · + λs = µ1 + · · · + µs = n − m.
Define µT , µB , λT , λB as in Theorem 1.1 and suppose that
                      µT ∈ Λ+ (d , n − m),                             λT ∈ Λ+ (d , n − m),
                      µB ∈ Λ+ (d , m),                                λB ∈ Λ+ (d , m).
Then we have
                                                    T     T                                    µB         B
          HomS(d,n) (W µ , W λ ) ∼ HomS(d ,n−m) (W µ , W λ ) ⊗ HomS(d
                                 =                                                    ,m) (W         , W λ ).
Theorem 4.2. Fix integers n, l with 0 ≤ l ≤ n and positive integers d, d , d . Let λ, µ ∈ Λ+ (d, n)
and suppose that for some r we have
                                   λ1 + · · · + λr = µ1 + · · · + µr = n − l.
Define   λ L , λR , µL , µR   as in Theorem 1.2 and suppose that
                       µL ∈ Λ+ (d , n − l),                            λL ∈ Λ+ (d , n − l),
                       µR ∈ Λ+ (d , l),                                λR ∈ Λ+ (d , l).
Then we have
                                                      L     L                                  λR       R
            HomS(d,n) (W λ , W µ ) ∼ HomS(d ,n−l) (W λ , W µ ) ⊗ HomS(d
                                   =                                                  ,l) (W        , W µ ).
   Before we begin the proof, we note that in almost all cases these two results follow directly
from Theorem 1.1 and Theorem 1.2 because Dipper and James [3, Cor. 8,6] have shown that
                            HomH (S λ , S µ ) ∼ HomS(d,n) (W λ , W µ )
                                              =
whenever q = −1 or µ is 2-restricted. We give a direct argument so as to cover these cases as
well.
  The next lemma shows that we can reduce to the case where d ≥ n, d ≥ n − l and d ≥ l.
Lemma 4.3. Suppose that λ, µ ∈ Λ+ (d, n). Then HomS(d,n) (W λ , W µ ) ∼ HomS(d+1,n) (W λ , W µ ).
                                                                      =
Proof. This follows from the general theory of Schur functors, as can be found in [1, §3.1]. To see
this let ϕd = {ϕα | α ∈ Λ(d, n)} ∈ S(d + 1, n). Then ϕd is an idempotent so P = ϕd S(d + 1, n)
is a projective S(d + 1, n)-module. Since S(d, n) ∼ ϕd S(d + 1, n)ϕd we can define a functor
                                                  =
            α : mod- S(d + 1, n) −→ mod- S(d, n)                by           α = Hom S(d+1,n) (P,        ).
If λ, µ ∈ Λ+ (d, n) then every composition factor of the S(d + 1, n)-modules W λ and W µ are
contained in the head of P , so Hom S(d+1,n) W λ , W µ ∼ HomS(d,n) α(W λ ), α(W µ ) by [1,
                                                           =
Cor. 3.1c]. It is easy to see that if ν ∈ Λ(d, n) then α(W ν ) ∼ W ν , as S(d, n)-modules, so this
                                                               =
completes the proof.
  To show that Theorem 4.1 and Theorem 4.2 are equivalent we need some more notation. Fix
an integer d ≥ n and for λ ∈ Λ(d, n) define
                                                                (w)
                                          nλ =          (−q)−         Tw .
                                                 w∈Sλ
14                                  ´
                                 SINEAD LYLE AND ANDREW MATHAS


Set N λ = nλ H and let SR = SR (d, n) be the endomorphism algebra
                                                       

                                    SR = EndH                  M µ .
                                                     µ∈Λ(d,n)

Now N λ = (M λ )# , where # is the automorphism used in the proof of Lemma 3.1. Therefore,
HomH (M µ , M λ ) ∼ HomH (N µ , N λ ) and, consequently, # induces a canonical algebra isomor-
                  =
                                                                  #
phism β : S(d, n) − SR (d, n); explicitly, β φ (h) = φ(h# ) , for all φ ∈ S(d, n) and all
                       →
h ∈ H. For λ ∈ Λ+ (d, n) define WR = β(W λ ). Then WR is an analogue of a Weyl module for
                                    λ                     λ

the algebra SR (d, n).
   Donkin has shown that if d ≥ n then SR (d, n) is the Ringel dual of the q-Schur algebra
S(d, n) [5]. Consequently, we have a contravariant functor γ which sends S(d, n)-modules to
SR (d, n)-modules. Further, if ν ∈ Λ(d, n) then γ(W ν ) ∼ (WR ) , where (WR ) is the contragre-
                                                        =   ν               ν

dient dual of WRν . Proofs and further explanations of these results can be found in [5].


Lemma 4.4. Suppose that d ≥ n and λ, µ ∈ Λ + (d, n) Then
                            HomS(d,n) (W µ , W λ ) ∼ HomS(d,n) (W λ , W µ ).
                                                   =
Proof. Since d ≥ n, SR (d, n) is the Ringel dual of S(d, n). Therefore, using the functors above,
we have
          HomS(d,n) (W µ , W λ ) ∼ HomSR (d,n) WR , WR ,
                                 =
                                                 µ    λ
                                                                         by applying β,
                                 ∼ HomS (d,n) (W λ ) , (W ) ,
                                 =
                                                          µ
                                                                         by duality,
                                          R             R       R
                                 ∼ HomS(d,n) W , W µ ,
                                 =                  λ

where the last line follows by applying γ.
Proposition 4.5. Theorem 4.1 holds if and only if Theorem 4.2 holds.
Proof. By Lemma 4.3 we may assume that d, d , d are all large enough so that we can apply
Lemma 4.4. The result now follows by the argument of Proposition 3.2
     We now prove Theorem 4.1. The proof is closely related to the proof of Theorem 1.1.
     Fix integers n, m with 0 ≤ m ≤ n, and positive integers d, d , d . Fix λ, µ ∈ Λ+ (d, n) where
                               λ1 + . . . + λ s = µ 1 + . . . + µ s = n − m
for some s and
                                       λT , µT ∈ Λ+ (d , n − m)
                                       λB , µB ∈ Λ+ (d , m)
We use the definitions of H T ∼ Hn−m and HB ∼ Hm from Definition 3.3 and define
                             =             =
                                                  

                                    S = EndH                 Mτ
                                                  τ ∈Λ(d,n)

to be an endomorphism algebra of H,
                                                                    

                                  S T = EndHT                      Mτ
                                                     τ ∈Λ(d ,n−m)
                              ROW AND COLUMN REMOVAL THEOREMS                                                          15


to be an endomorphism algebra of H T and
                                                                           

                                  S B = EndHB                      Mτ
                                                       τ ∈Λ(d ,m)

                                                                                          T         T
to be an endomorphism algebra of H B . Then modules such as Mτ , W τ                                    are defined in the
obvious way.
Definition 4.6. Let
                                                                                   T            T
                θT =        ΓAT θAT AT ∈ T0 (λT , µT ) ∈ HomS T (Mµ , W λ )
                       AT
                                                                                      B          B
                θB =        ΓAB θAB AB ∈ T0 (λB , µB ) ∈ HomS B (Mµ , W λ )
                       AB

and define

                 θ = (θ T , θ B ) : Mµ → W λ        by         θ=               ΓAT ΓAB θ(AT ,AB ) .
                                                                    AT AB

  Note that by Lemma 2.17, this defines a bijection
                              T     T                      B        B
                 HomS T (Mµ , W λ ) ⊗ HomS B (Mµ , W λ ) → HomS (Mµ , W λ ).
Proposition 4.7. Let ν ∈ Λ+ (d, n), τ ∈ Λ(d, n) be such that λ                     ν          µ. Let U ∈ T 0 (ν, µ), V ∈
T0 (ν, τ ), A ∈ T0 (λ, µ). Then
                                               θA (ϕU V ) = 0.
Proof. Define θT λ A ∈ HomS (Mµ , Mλ ) by θT λ A (ϕµ f ) = ϕT λ A f , for all f ∈ S. Then θT λ A is a
well defined S-homomorphism and θA = πθT λ A where π : Mλ → W λ is the natural projection.
Note that ϕU V = ϕµ ϕU V . Then
                                   θT λ A (ϕU V ) = (θT λ A (ϕµ ))ϕU V
                                                  = ϕ T λ A ϕU V .

Now ϕT λ A ∈ S λ and ϕU V ∈ S ν . So, since λ                                                ˇ
                                                               ν, we have that ϕT λ A ϕU V ∈ S λ . Hence
θA (ϕU V ) = π(ϕT λ A ϕU V ) = 0.
Proposition 4.8. Suppose that ν ∈ Λ+ (d, n) is such that λ                        ν           µ and τ ∈ Λ(d, n). Let
U ∈ T0 (ν, µ), V ∈ T0 (ν, τ ). Suppose that
                                                               T        T
                                        θ T ∈ HomS T (Mµ , W λ )
                                                               B        B
                                        θ B ∈ HomS B (Mµ , W λ )
are such that
                                                                                                    T
                                                                           ˇ
                    θ T (ϕµT f ) = 0 whenever f ∈ S T is such that ϕµT f ∈ S µ

and
                                                                                                    B
                                                                           ˇ
                    θ B (ϕµB f ) = 0 whenever f ∈ S B is such that ϕµB f ∈ S µ .
16                                        ´
                                       SINEAD LYLE AND ANDREW MATHAS


Suppose that

                                   θT =             ΓAT θAT AT ∈ T0 (λT , µT )
                                             AT

                                   θB =             ΓAB θAB AB ∈ T0 (λB , µB ) .
                                             AB

Define the map θ = (θ T , θ B ) as in Definition 4.6. Then
                                                         θ(ϕU V ) = 0.
Proof. Note that ϕU V = ϕµ ϕU V . Then

                    θ(ϕU V ) =                    ΓAT ΓAB θ(AT ,AB ) (ϕµ )ϕU V
                                      AT    AB

                                  =               ΓAT ΓAB ϕ(AT ,AB ) ϕU V ∈ HomH (M τ , S λ ).
                                      AT AB

Then, using the notation of Lemma 3.10, there exist h T ∈ HT , hB ∈ HB such that
     [θ(ϕU V )](mτ ) =             ΓAT ΓAB ϕ(AT ,AB ) (ϕU V (mτ ))
                         AT AB

                    =              ΓAT ΓAB ϕ(AT ,AB )
                         AT   AB

                                                     ∗
                                                    Td(u) mν Td(v) u, v ∈ Std(ν), µ(u) = U, τ (v) = V
                                       u     v


                    =              ΓAT ΓAB ϕ(AT ,AB )          mµ hT hB           Td(v)
                         AT   AB                                              v

                      ˇ
                    = Hλ +                                ΓAT ΓAB mλT mλB Td(aT ) Td(aB ) hT hB       Td(v)
                                  AT   AB     aT    aB                                            v

                                            aT ∈ Std(λT ), aB ∈ Std(λB ), µT (aT ) = AT , µB (aB ) = AT

                                 ˇ T                       ˇ B
where either ν T µT and mµT hT ∈ Hµ or ν B µB and mµB hB ∈ Hµ . Without loss of generality,
                    ˇ T                               T    T
assume mµT hT ∈ Hµ . Consider ϕU T T ν T ∈ HomH (M ν , M µ ). Then θ T (ϕU T T ν T ) = 0 by
assumption and hence
                                                                                          T
                                                                                    ˇ
                                   ΓAT mλT Td(aT ) hT = [θ T (ϕAT T ν T )](mν T ) ∈ Hλ .
                         A T aT

Also,
                                                                                    B
                                                    ΓAT mλB Td(aB ) hB ∈ M λ .
                                           A T aB
Proposition 4.8 now follows from the proof of Proposition 3.13.
                                                                  T       T                            B      B
Proposition 4.9. Suppose that θ T ∈ HomS T (Mµ , W λ ), and θ B ∈ HomS B (Mµ , W λ ) are
such that
                                                                                              T
                                                                            ˇ
                     θ T (ϕµT f ) = 0 whenever f ∈ S T is such that ϕµT f ∈ S µ
                                ROW AND COLUMN REMOVAL THEOREMS                                  17


and
                                                                          ˇ B
                   θ B (ϕµB f ) = 0 whenever f ∈ S B is such that ϕµB f ∈ S µ .
                                                                      ˆ
Define the map θ = (θ T , θ B ) as in Definition 4.6, and define the map θ : W µ → W λ by
                               ˆ ˇ
                               θ[(S µ + ϕµ )f ] = [θ(ϕµ )]f for all f ∈ S.
     ˆ
Then θ is a well defined S-homomorphism.
                                                                 ˇ
Proof. It is sufficient to show that if f ∈ S is such that ϕ µ f ∈ S µ then [θ(ϕµ )]f = θ(ϕµ f ) = 0.
Suppose f satisfies these conditions. Then ϕ µ f is an R-linear combination of terms of the form
ϕU V where U ∈ T0 (ν, µ) and V ∈ T0 (ν, τ ) for some ν ∈ Λ+ (d, n), τ ∈ Λ(d, n) such that ν µ.
Choose such an ϕU V . If λ ν then θ(ϕU V ) = 0 by Proposition 4.7. If λ ν then θ(ϕ U V ) = 0
by Proposition 4.8. Hence θ(ϕµ f ) = 0 as required.
Definition 4.10. Suppose that
                                                          T     T
                                         ˆ
                                         θ T ∈ HomS T (W µ , W λ ),
                                                          B     B
                                         ˆ
                                         θ B ∈ HomS B (W µ , W λ ).
Define
                                                          T     T
                                         θ T ∈ HomS T (Mµ , W λ ),
                                                          B     B
                                         θ B ∈ HomS B (Mµ , W λ ).


                                      ˆ                 ˆ
in the obvious manner; that is, θ T = θ T π T and θ B = θ B π B where πT and πB are the projections
onto W  µT and W µB respectively. Note that θ T and θ B satisfy the conditions of Proposition 4.9.

Hence define
                                  ˆ    ˆ ˆ
                                 θ = (θ T , θ B ) ∈ HomS (W µ , W λ )
as in Proposition 4.9.
Proposition 4.11. Let θ ∈ HomS (Mµ , W λ ) be such that
                                                                      ˇ
                          θ(ϕµ f ) = 0 for any f ∈ S such that ϕµ f ∈ S µ .
Then there exist ΓAT AB ∈ R such that

               θ=             ΓAT AB θ(AT ,AB ) AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB ) .
                    AT ,AB

Fix AB and consider
                                                                      T   T
                             θT =        ΓAT AB θAT ∈ HomS T (Mµ , W λ ).
                                    AT
Then
                                                                         ˇ T
                      θ T (ϕµT f ) = 0 for any f ∈ S T such that ϕµT f ∈ S µ .
Proof. Suppose we have the conditions of Proposition 4.11. By Proposition 4.7 and Proposition
                                                                                    T
4.9, it is sufficient to show that θ T (ϕU V ) = 0 for any U ∈ T0 (ν T , µT ), V = T ν such that
ν T ∈ Λ+ (d, n) and λT ν T µT . We have

                             Td(u) mν T u ∈ Std(ν T ) and µT (u) = U
                              ∗
                                                                          = m µT h T
                      u
18                                            ´
                                           SINEAD LYLE AND ANDREW MATHAS


for some hT ∈ HT . Then θ T (ϕU V ) = 0 if and only if
                      Γ AT AB           mλT Td(aT ) hT aT ∈ Std(λT ) and µT (at ) = AT         ˇ T
                                                                                             ∈ Hλ .
                 AT               aT
                                                 B
Now fix ν = (ν T , ν B ), X = (U T , T ν ), Y = T ν . By assumption, θ(ϕXY ) = 0. Hence
               Γ AT AB                 mλT mλB Td(aT ) Td(aB ) hT aB ∈ Std(λB ) and µB (aB ) = AB       ˇ
                                                                                                      ∈ Hλ
      AT ,AB             aT    aB

and so
                                                Γ AT AB                           ˇ T
                                                                 mλT Td(aT ) hT ∈ Hλ
                                           AT             aT
as required.
Definition 4.12. Suppose θ ∈ HomH (Mµ , W λ ) is such that θ(ϕµ f ) = 0 for all f ∈ S such that
       ˇ           ˆ
ϕµ f ∈ S µ . Define θ ∈ HomS (W µ , W λ ) by setting
                                         ˆ ˇ
                                         θ[(S µ + ϕµ )f ] = [θ(ϕµ )]f for all f ∈ S.
              ˆ                                                        ˆ
Similarly, if θ ∈ HomS (W µ , W λ ), define θ ∈ HomS (Mµ , W λ ) by θ = θπ where π : Mµ →
W µ is the natural projection. We use analogous notation for the corresponding S T and S B

homomorphisms.
     We can now prove our second main result.
Proof of Theorem 4.1. We have to show that
                                                     T     T               B     B
                      HomS (W µ , W λ ) ∼ HomS T (W µ , W λ ) ⊗ HomS B (W µ , W λ ).
                                        =
    ˆ               T     T
                                 ˆ                  B         B
Let θ T ∈ HomHT (W µ , W λ ) and θ B ∈ HomS B (W µ , W λ ). Then there is an R-linear mapping
                T     T                 B     B
from HomS T (W µ , W λ ) ⊗ HomS B (W µ , W λ ) into HomS (W µ , W λ ) given by sending
                                     ˆ     ˆ      ˆ ˆ
                                     θ T ⊗ θ B → (θ T , θ B )
and extending linearly. By construction, this mapping is injective. It remains only to show that
it is surjective.
        ˆ                               ˆ
   Let θ ∈ HomS (W µ , W λ ). Form θ = θπ ∈ HomS (Mµ , W λ ) so that θ satisfies the conditions
of Proposition 4.11. Suppose
                      θ=                ΓAT AB θ(AT ,AB ) AT ∈ T0 (λT , µT ), AB ∈ T0 (λB , µB )
                              AT ,AB

for some ΓAT AB ∈ R. Fix AB . By Proposition 4.11, the map
                                                     θT =           Γ AT AB θ AT
                                                               AT
                                       ˆ      T      T
                                                                                 ˆ
is such that the corresponding map θ T : W µ → W λ is well defined. Suppose that {ψi | i ∈ I}
is a basis of HomS T (W µT , W λT ). So we can write

                                                 ˆ
                                                 θT =                  ˆ
                                                                 βi,AB ψi
                                                           i

and so
                                                     θ=          βi,AB (ψi , θAB ).
                                                          i,AB
                                  ROW AND COLUMN REMOVAL THEOREMS                                         19


A similar argument for fixed ψi shows that in fact
                                                                T    B
                                               θ=         αij (ψi , ψj )
                                                    i,j

       ˆT
where ψi ∈ HomS T (W        µT   ,W   λT         ˆB              B     B
                                                                                                   ˆ
                                           ) and ψj ∈ HomS B (W µ , W λ ); thus under our mapping, θ
appears as the image of
                                                    T     T               B     B
                             ˆT   ˆB
                        αij (ψi ⊗ ψj ) ∈ HomS T (W λ , W µ ) ⊗ HomS B (W λ , W µ ).
                  i,j



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 [2] R. Dipper and G. James, Representations of Hecke algebras of general linear groups, Proc. L.M.S. (3), 52
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 [4] S. Donkin, A note of decomposition numbers for general linear groups and symmetric groups, Math. Proc.
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 [5]       , On tilting modules for algebraic groups, Math. Z., 212 (1993), 39–60.
 [6] M. Fayers and S. Lyle, Row and column removal theorems for homomorphisms between Specht modules,
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 [7] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math., 123 (1996), 1–34.
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  School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia

  School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Australia

				
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