数理解析研究所講究録
第 1687 巻 2010 年 50-62

50

On applications of the cellular algebras
Nobuharu Sawada
Department of Mathematics
Tokyo University of Science
ABSTRACT. In this report we explain briefly the results of parts of papers $[SawS]$and [Sa].

1. CELLULAR ALGEBRAS
1.1. Cellular bases. We begin with the definition of a cellular basis.
Let
be a commutative domain with 1 andan associative unital R-algebra
which is free as an R-module. Suppose thatis a (finite) posef, and that for cachthere is a finite indexing set
and elements
for allsuchthat
and 5,$R$$A$$(\Lambda, \geq)$$\lambda\in\Lambda$$c_{s1}^{\lambda}\in A$$\mathcal{T}(\lambda)$$=\{c_{51}^{\lambda}|\lambda\in\Lambda$$\epsilon,$$t\in \mathcal{T}(\lambda)$$t\in \mathcal{T}(\lambda)\}$is a (free) basis of $A$ . For eachletbe the R-submodule ofwith basis.
and $n,$The pairis cellular basis of if(i) the R-linear map $*:Aarrow A$ determined by, for alland alland in, is an algebra anti-isomorphism of ,(ii) for any
and $a\in A$ there exist
such that for all$\{c_{u\mathfrak{v}}^{\mu}|\mu\in\Lambda,$$\mu>\lambda$$A$$\check{A}^{\lambda}$$\lambda\in\Lambda$$\mathfrak{v}\in \mathcal{T}(\mu)\}$$A$$(\varphi, \Lambda)$$\lambda\in\Lambda$$c_{r,\lrcorner\{}^{\lambda^{*}}=c_{ts}^{\lambda}$$t$$\check{\triangleleft}$$A$$\mathcal{T}(\lambda)$$\lambda\in\Lambda,$$r_{\mathfrak{v}}\in R$$t\in \mathcal{T}(\lambda)$(1.1)$mod \check{A}^{\lambda}$$c_{e,\lrcorner\{}^{\lambda}a \equiv\sum_{\mathfrak{o}\in \mathcal{T}(\lambda)}r_{\mathfrak{v}}c_{50}^{\lambda}$$\epsilon\in \mathcal{T}(\lambda)$.

has a cellular basis we say thatis a cellular algebra.
Throughout this section we assume that $(, \Lambda)$ is a fixed cellular basis of thealgebra .
For
letbe the R-module with basis the set ofwhere. Thus,andandhas basis
.
where

If

$A$

$A$

$A$

$\lambda\in\Lambda$$\mathfrak{u},$$\mathfrak{v}\in \mathcal{T}(\mu)$$A^{\lambda}$$\mu\in\Lambda,$$c_{u\mathfrak{v}}^{\mu}$$\check{A}^{\lambda}\subset A^{\lambda}$$A^{\lambda}\check{A}^{\lambda}$$c_{\epsilon t}^{\lambda}+\check{A}^{\lambda}$$\mathfrak{s},$Lemma 1.2 (cf. [Ma, Lemma 2.3]). Let be an element of(i) Suppose thatand $a\in A$ . Then for all$\lambda$$\epsilon\in \mathcal{T}(\lambda)$$\Lambda$$\mu\geq\lambda$$t\in \mathcal{T}(\lambda)$.

$t\in \mathcal{T}(\lambda)$$mod \check{A}^{\lambda}$$a^{*}c_{\epsilon t}^{\lambda} \equiv\sum_{u\in \mathcal{T}(\lambda)}r_{u}c_{ut}^{\lambda}$

whereis the element of $R$ determined by (1.1) for each .
(ii) The R-modules
andare two-sided ideals of .
(iii) Suppose that 5 and are elements of. Then there exists an element$R$such that for any$u$$r_{t1}$$A^{\lambda}$$A$$\check{A}^{\lambda}$$t$$\mathcal{T}(\lambda)$$u,$$v\in \mathcal{T}(\lambda)$$c_{\mathfrak{u}\epsilon}^{\lambda}c_{10}^{\lambda}\equiv r_{\epsilon t}c_{u\mathfrak{v}}^{\lambda}$$mod \check{A}^{\lambda}$.

$r_{s1}$of

51
Fix an element of . If
to be the R-submodule ofdefinewithbasis. Thenis a right A-module by (1.1) and, importantly,
the action ofonis completely independent of . That is,for any. This motivates us to define the right cell module
to be the right
A-module which is free as an R-module with basis
and where for each$\Lambda$$\lambda$$\epsilon\in \mathcal{T}(\lambda)$$\{c_{\mathfrak{s}t}^{\lambda}+\check{A}^{\lambda}|t\in \mathcal{T}(\lambda)\}$

$A$

$s,$

$A^{\lambda}’\check{A}^{\lambda}$$C_{5}^{\lambda}$$C_{\mathfrak{s}}^{\lambda}$$C_{5}^{\lambda}$$C_{\epsilon}^{\lambda}\cong C_{t}^{\lambda}$$\mathfrak{s}$$C^{\lambda}$$t\in \mathcal{T}(\lambda)$$\{c_{t}^{\lambda}|t\in \mathcal{T}(\lambda)\}$$a\in A$(1.2)

$c_{1}^{\lambda}a= \sum_{\mathfrak{v}\in \mathcal{T}(\lambda)}r_{\mathfrak{v}}c_{\mathfrak{v}}^{\lambda}$

where is the element of determined by (1.1). Then
, for any
, via
the canonical R-linear map which sends
for all. In particular,to(1.2) determines a well-defined action of on.
Abusing notation, define the left cell module
to be the free R-module withbasis
and A-action given by$R$$r_{\mathfrak{v}}$$C^{\lambda}\cong C_{5}^{\lambda}$$c_{s1}^{\lambda}+\check{A}^{\lambda}$$c_{t}^{\lambda}$$A$$5\in \mathcal{T}(\lambda)$$t\in \mathcal{T}(\lambda)$$C^{\lambda}$$C^{*\lambda}$$\{c_{t}^{\lambda}|t\in \mathcal{T}(\lambda)\}$$a^{*}c_{t}^{\lambda}= \sum_{\mathfrak{v}\in \mathcal{T}(\lambda)}r_{\mathfrak{v}}c_{\mathfrak{v}}^{\lambda}$

for all $a\in A$ and where, once again,
is given by (1.1). Then
is a left A-module
and.
Moreover, as $(A, A)$ -bimodules,
andare canonically isomorphic
via the R-linear map determined by.
for all and in
Furthermore, as a right A-module,$C^{*\lambda}$$r_{\mathfrak{v}}$$C^{*\lambda}\cong Hom_{R}(C^{\lambda}, R)$$C^{*\lambda}\otimes_{R}C^{\lambda}$$A^{\lambda}/\check{A}^{\lambda}$$c_{51}^{\lambda}+\check{A}^{\lambda}\mapsto c_{5}^{\lambda}\otimes c_{t}^{\lambda}$(1.3)$A^{\lambda}/ \check{A}^{\lambda}\cong C^{*\lambda}\otimes_{R}C^{\lambda}\cong\bigoplus_{s\in \mathcal{T}(\lambda)}C_{5}^{\lambda}$$A^{\lambda}/\check{A}^{\lambda}$$\langle c_{5}^{\lambda},$$\rangle$$\mathcal{T}(\lambda)$.

So, as a right A-module,
is isomorphic to a direct sum of
By Lemma 1.2 (iii) there is a unique bilinear map:, for $s,$
, is given by$\langle,$$t$$\epsilon$$|\mathcal{T}(\lambda)|$copies of$C^{\lambda}\cross C^{\lambda}arrow R$.
such that$C^{\lambda}$$t\in \mathcal{T}(\lambda)$$c_{t}^{\lambda}\rangle$(1.4)$\langle c_{5}^{\lambda},$$c_{1}^{\lambda}\}c_{u\mathfrak{v}}^{\lambda}\equiv c_{\mathfrak{U}6}^{\lambda}c_{t\iota 1}^{\lambda}$$mod \check{A}^{\lambda}$,

where and are any elements of. The bilinear form , is both symmetricand associative.
Let rad $C^{\lambda}=\{x\in C^{\lambda}|\langle x, y\}=0$ for all}. One can see that rad is an
A-submodule of . Accordingly, we define
rad .
$u$

$\mathcal{T}(\lambda)$$\mathfrak{o}$$\langle$$\rangle$$y\in C^{\lambda}$$C^{\lambda}$$C^{\lambda}$$D^{\lambda}=C^{\lambda}/$$C^{\lambda}$

1.2. Simple modules in a cellular algebra. We are almost ready to show thatevery irreducible A-module is isomorphic to, for some
. In tliis section wealso define and describe the decomposition matrix of . Throughout, we assume thatthe poset is finite. Thus is a finite dimensional algebra.
One of the main points of the cellular basis is that it gives rise to many filtrationsin . To formalize this, call a subset
of
a posct ideal if
whenevcr
for some. If
is a poset ideal let
be the R-submodule ofwith basis$D^{\mu}$$l^{4}\in\Lambda$

$A$

$A$

$\Lambda$$A$$\Gamma$$\mu\in\Gamma$$\Gamma$$\Lambda$$A(\Gamma)$$\lambda\in\Gamma$$\lambda>\mu$

$A$

52
. So
is a two-sided ideal. Thenandby Lemma 1.2 (ii).
Lemma 1.3 (cf. [Ma, Lemma 2.14]). Suppose that is finite and let. . .
be any maximal chain of ideals in . Then there exists a total ordering, for all , andof such that
$A( \Gamma)=\sum_{A\in 1^{\neg}}$ $A^{A}$$u,$ $v\in \mathcal{T}(\mu)\}$$\{c_{u\mathfrak{v}}^{\mu}|\mu\in\Gamma$$A(\Gamma)$$\emptyset=\Gamma_{0}\subset\Gamma_{1}\subset$$\Lambda$$\Lambda$$\subset\Gamma_{k}=\Lambda$$i$$\Gamma_{i}=\{\mu_{1}, \ldots, \mu_{i}\}$$\Lambda$$\mu_{1)}\ldots,$$\mu_{k}$$0=A(\Gamma_{0})arrow A(\Gamma_{1})arrow\rangle\cdots-A(\Gamma_{k})=A$.
is a filtration of with composition factors
if and only if the bilinear form ,Let $\Lambda_{0}=\{’\iota\in\Lambda|D^{\mu}\neq 0\}$ . Then
is non-zero. In principle, the next theorem classifies the simple A-modules.
on.
However, in practice, it is often difficult to determine the set$A(\Gamma_{i})/A(\Gamma_{i-1})\cong C^{*r_{i}}\otimes_{R}C^{4}i$$A$$\iota\in\Lambda_{0}$$\langle$$\rangle$$C^{\mu}$$\Lambda_{0}$is flnite.
is a field and that
Theorem 1.4 (Graham-Lehrer). Suppose thatis a complete set of pairwise inequivalent irreducible A-modules.
to be the decomposition. DcfineandSuppose that
. By theinnumber (or composition multiplicity) of the irreducible module
and
,
where
Jordan-H\"older Theorem,
is well-defincd. The matrix, is the so-called decompositi.on matrix of .
is a field. Then the decomCorollary 1.5 (cf. [Ma, Corollary 2.17]). Suppose thatthenandis unitnangular. That is, if
position matrix $D$ of.
only ifand
The last result in this section connects the theory of quasi-hereditary algebras andcellular algebras. Quasi-hereditary algebras are a very important class of algebras
which were introduced by Cline, Parshall and Scott [CPS].
Proposition 1.6 (cf. [Ma, Corollary 2.23]). Suppose that $R$ is a field. Then thefollowing are equivalent.
.
(i)ii) The decomposition matrix $D$ is a square unitnangular matrix.
is quasi-hereditary.
Furthermore, if these conditions are satisfied thenAs this criterion indicates, being quasi-hereditary is a non-degeneracy property on$R$$Tf\iota en$$\Lambda$$\{D^{\mu}|\mu\in\Lambda_{0}\}$$d_{\lambda\mu}=[C^{\lambda} :$\lambda\in\Lambda$$\mu\in\Lambda_{0}$D^{\mu}]$$D^{\mu}$$C^{\lambda}$$\lambda\in\Lambda$$D=(d_{\lambda\mu})$$d_{\lambda\mu}$$A$$\mu\in\Lambda_{0}$

$R$

$A$

$d_{\lambda\mu}\neq 0$$\lambda\in\Lambda$$\mu\in\Lambda_{0}$$d_{\mu\{\iota}=1$$\lambda\geq\mu$$\Lambda=\Lambda_{0}$$\langle$

$A$

$A$

.

2. PRELIMINARIES ON ARIKI-KOIKE ALGEBRAS AND CYCLOTOMIC -SCHUR$q$

ALGEBRAS

be the symmetric group of degree . Let
2.1. Fix positive integers and and let$R$ be an integral domain with 1 andbe elements in $R$ , with invertible .
The Ariki-Koike algebra as sociated to the complex reflection group $W_{n,r}=G(r, 1, n)$ ,subject
over $R$ with generators
is the associative unital algebra
to the following conditions,
$r$

$n$

$n$

$\mathfrak{S}_{n}$$q,$$Q_{1},$$\ldots,$$Q_{r}$$q$$T_{1},$$\mathcal{H}=\mathcal{H}_{n,r}$$(T_{1}-Q_{1})\cdots(T_{1}-Q_{r})$$=0$ ,$(T_{i}-q)(T_{i}+q^{-1})$$=0$$T_{1}T_{2}T_{1}T_{2}$$T\cdot T_{j}$$T_{i}T_{i+1}T_{i}$$=T_{2}T_{1}T_{2}T_{1}$$(i\geq 2)$,,$=T_{j}T_{i}$$(|i-j|\geq 2)$ ,$=T_{i+1}T_{i}T_{i+1}$$(2\leq i\leq n-1)$ .

$T_{n}$$\ldots,$

53
of. The subalgebra
It is known thatis a free R-module of rankof theis isomorphic to the Iwahori-Hecke algebra
generated by.
symmetric group.Thenbe the transposition $(i-1, i)$ in
For $i=2,$
let
is a reduced
where, we set. For
generate
is independent of the choice of a reduced expression. We also
expression. Then
commutes.
. Note that all
put $L_{k}=T_{k}\cdots T_{2}T_{1}T_{2}\cdots T_{k}$ for $k=1,2,$.
Moreover, these elements produce a basis of$n!r^{n}$$\mathcal{H}$$\mathcal{H}$$T_{2},$$\ldots,$$\mathcal{H}(\mathfrak{S}_{n})$$ff_{n}$$T_{n}$$\mathfrak{S}_{n}$$\ldots,$$s_{i}$$T_{w}=T_{i_{1}}\cdots T_{i_{k}}$$w\in \mathfrak{S}_{n}$$\mathfrak{S}_{n}$$\{s_{2}, \ldots, s_{n}\}$$\mathfrak{S}_{n}$$n$$w=s_{i_{1}}\cdots s_{i_{k}}$$T_{w}$$\ldots,$$L_{1},$$n$$\ldots,$$L_{n}$$\mathcal{H}$is free as an RTheorem 2.2 ([AK, Theorem 3.10]). The Ariki-Koike algebra$1\leqi\leq
n\}$$0\leq a_{i}<r$ for.
module with basis$\mathcal{H}$$\{L_{1}^{a_{1}}\cdots L_{n^{n}}^{a}T_{w}|w\in \mathfrak{S}_{n},$of non-negative integers
Recall that a composition of is sequencefor. If. is a partition if in addition
such that.
all $i>k$ then we write
An r-composition (or multicomposition) of is an r-tuple
such that $|\lambda^{(1)}|+\cdots+|\lambda^{(r)}|=n$ . An rof compositions withis a partition. If is an r-partition ofis an r-partition if eachcompositionof the r-composition is the set. The diagram
then we write$\{(i,j, s)|1\leq i\leq\lambda_{j}^{(s)}, 1\leq s\leq r\}$ .
are called nodes. The
The elements ofi.e, if and are twodominance,bypartiallyorderedr-compositions
isof
set of, if
r-compositions then dominates , and we write$\sigma=(\sigma_{1}, \sigma_{2}, \ldots)$$n$$| \sigma|=\sum_{i}\sigma_{i}=n$$\sigma_{l}=0$$\sigma_{1}\geq\sigma_{2}\geq\cdots$$\sigma$$\sigma=(\sigma_{1)}\ldots, \sigma_{k})$$\lambda=(\lambda^{(1)}, \ldots, \lambda^{(r)})$$n$$\lambda^{(i)}=(\lambda_{1}^{(i)}, \lambda_{2}^{(i)}, \ldots)$$\lambda^{(i)}$$\lambda$$\lambda\vdash n$$n$$\lambda$$[\lambda]=$$\lambda$$[\lambda]$$[\lambda]$$\lambda$$n$$1^{r}$$\lambda$$\lambda\underline{\triangleright}\mu$$\mu$$\sum_{c=1}^{s-1}|\lambda^{(c)}|+\sum_{j=1}^{i}|\lambda_{j}^{(s)}|\geq\sum_{c=1}^{s-1}|\mu^{(c)}|+\sum_{j=1}^{i}|\mu_{j}^{(s)}|$for $1\leq s\leq r$ and for all $i\geq 1$ . If
If is an r-composition let. Set
subgroup of$\lambda\underline{\triangleright}\mu$and$\lambda\neq\mu$.
then we write
be the corresponding Young$\lambda\triangleright\mu$$\mathfrak{S}_{\lambda}=\mathfrak{S}_{\lambda(1)}\cross\cdots\cross \mathfrak{S}_{\lambda^{(r)}}$$\lambda$$\mathfrak{S}_{n}$$x_{\lambda}= \sum_{w\in \mathfrak{S}_{\lambda}}q^{l(w)}T_{w}$,$u_{\lambda}^{+}= \prod_{s=2k}^{r}\prod_{=1}^{a_{s}}(L_{k}-Q_{s})$,where $a_{s}=|\lambda^{(1)}|+\cdots+|\lambda^{(s-1)}|$ for $2\leq s\leq r$ . If $s=1$ then we set $a_{s}=0$ . Set.
ofto be the right idealand define$t=(t^{(1)},\ldots,t^{(r)})$$t:[\mu]arrow$bijectionisa
For any r-composition , a -tableau$(t^{(i)})=\mu^{(i)}$$\{1,2, \ldots, n\}$ , where. We write Shape(t) $=\mu$ if
is a tableau of Shapeareis a -tableau. A -tableau is called standard (resp. row standard) if all
be the set of standard -tableaux.
standard (resp. row standard). Let Std
be the -tableau with the numbers 1, 2, . . . ,For each r-composition , let
attached in order from left to right along its rows and from top to bottom, and frombe the unique permutation. If is any row standard -tableau let
tobe the anti-isomorphism given bysuch that $t=t^{\mu}d(t)$ . Furthermore, let.
, and set
for $i=1,2,$$M^{\lambda}=m_{\lambda}\mathcal{H}$$M^{\lambda}$$m_{\lambda}=x_{\lambda}u_{\lambda}^{+}=u_{\lambda}^{+}x_{\lambda}$$\mathcal{H}$

$\mu$

$\mu$

$t^{(i)}$$t$$t^{(i)}$

$t$

$\mu$

$\mu$

$\lambda$$(\lambda)$$n$$t^{\mu}$$/\iota$$l^{t}$$\mu^{(1)}$$\mu^{(r)}$$t$$d(t)\in \mathfrak{S}_{n}$$\mu$$*:\mathcal{H}arrow \mathcal{H}$$\tau_{i}*=T_{i}$$\ldots,$$n$$m_{\epsilon 1}=T_{d(5)}^{*}m_{\lambda}T_{d(t)}$Theorem 2.3 ([DJM, Theorem 3.26]). The Ariki-Koike algebra
Stdmodule with cellular basis {}.
for some$m_{\epsilon t}|\mathfrak{s},$$t\in$$(\lambda)$$\lambda\vdash n$$\mathcal{H}$is free as an R-

54
2.4. We can now give a definition of the cyclotomic q-Schur algebras. A setof
r-compositions of is saturated if is finite and whenever is an r-partition suchthatfor some. If
then
is a saturated set of r-compositions, wedcnotc by
be the set of r-partitioiis in .
$\Lambda$$\Lambda$$n$$\lambda\in\Lambda$$\mu\in\Lambda$$\lambda\underline{\triangleright}\mu$$\lambda$$\Lambda$$\Lambda^{+}$$\Lambda$

Definition 2.5. Suppose thatis a saturated set of multicompositions of
cyclotomic q-Schur algebra with weight poset
is the endomorphism algebra$\Lambda$

$n$

.

The

$\Lambda$$S(\Lambda)=End_{\prime},(M(\Lambda))$be an r-partition and

,

where

$M( \Lambda)=\bigoplus_{\lambda\in\Lambda}M^{\lambda}$.

an r-composition. A -Tableau of typeis a map
such thatfor all$i\geq 1$ and $1\leq s\leq r$ . We regard $T$ as an r-tuple $T=(T^{(1)}, \ldots, T^{(r)})$ , whereis$T^{(s)}(i,j)=T(i,j,s)$$(i,j,s)\in[\lambda]$
the-tableau withfor all. In this way we identify
the standard tableaux above with the Tableaux of type $w=((0), \ldots, (1^{n}))$ . If $T$ is a
Tableau of type then we write Type $(T)=\mu$ .
Given two pairs $(i, s)$ and $(j, t)$ write $(i\}s)\preceq(j, t)$ if either $s<t$ , or $s=t$ and$i\leq j$ .
Let$\lambda$$\lambda$

$\mu$

$\mu$

$\mu_{i}^{(s)}=\#\{x\in[\lambda]|T(x)=(i, s)\}$$T:[\lambda]arrow\{(i, s)|i\geq 1,1\leq s\leq r\}$$T^{(s)}$$\lambda^{(s)}$$\mu$Definition 2.6. A Tableau$T^{(t)}$

$T$

are

is (row) semistandard if, for $1\leq t\leq r$ , the entries in(i) weakly increasing along the rows with respect(ii) stwictly increasing down columns,(iii) $(i, s)$ appears in
only if $s\geq t$ .

$to\preceq$,$T^{(t)}$Letbe the set of semistandard -Tableaux of typeand let. Notice that if
is non-empty, then.
Suppose that is a standard -tableau and let be an r-composition. Let
be
$(i,
k)$
the Tableau obtained from by replacing each entry with
if appears in row. The tableauofis a -Tableau of type . It is not necessarily semistandard.
If and are semistandard -Tableaux of type and respectively, let$\lambda$$\mathcal{T}_{0}(\lambda, \mu)$$\mathcal{T}_{0}(\lambda, \mu)$$\mathcal{T}_{0^{\Lambda}}(\lambda)=\bigcup_{\mu\in\Lambda}\mathcal{T}_{0}(\lambda, \mu)$$t$$\lambda\underline{\triangleright}\mu$$(t^{A})^{(k)}$$\mu(t)$$\lambda$

$\mu$

$t$

$j$

$\mu(t)$$i$$j$$\lambda$

$\mu$

$T$

$S$

$\mathcal{T}_{0}(\lambda)=$$\mu$$\lambda$

$\nu$

$\mu$

.
$m_{ST}= \sum_{s,1\in Std(\lambda)}q^{l(d(s))+l(d(t))}m_{t},\mu(Jr)=S,\nu(1)=T\lrcorner$For
and $T$ as abovc wc dcfiiic a iiial)onby
for all. Hereand allis the Kronecker delta, i.e,
and it is zero otherwise. Thenis well-defined, and it belongs to$S$$\varphi s\tau$$h\in \mathcal{H}$$\alpha\in\Lambda$$M(\Lambda)$$\varphi_{ST}(m_{\alpha}h)=\delta_{\alpha\nu}m_{ST}h$$\delta_{\alpha\nu}$$\varphi_{ST}$$S(\Lambda)$

Theorem 2.7 ([DJM, Theorem 6.6]). The cyclotomic q-Schur algebm
an R-module with cellular basis $C(\Lambda)=\{\varphi_{ST}|S,$for some$T\in \mathcal{T}_{0^{rt}}(\lambda)$

.

,

if
Moreover,$\alpha=\nu$$\delta_{\alpha\nu}=1$$S(\Lambda)$

is free as$\lambda\in\Lambda^{+}\}$.

The basis. Since this basis is cellular,
is called a semistandard basis of
the map $*:S(\Lambda)arrow S(\Lambda)$ which is determined by
is an anti-automorphism.
ofThis involution is closely related to the -involution on. Explicitly,
: M
if
is an
: $M^{\mu}arrow M^{\nu}$ is the-module homomorphism thenhomomorphism given by, for all.
For each r-partition, we define
as the R-span of
such
$S,$
that
, which is a two-sided ideal ofwith. We define the Weyl
$S(\Lambda)$$\{\varphi_{ST}\}$$\varphi_{ST}^{*}=\varphi_{TS}$$S(\Lambda)$$\varphi$$\mathcal{H}$

$*$

Jノ

$arrow M^{\mu}$$\mathcal{H}$$\varphi^{*}$$\varphi^{*}(m_{\mu}h)=(\varphi(m_{l\text{ノ}}))^{*}h$$\lambda\in\Lambda^{+}$$T\in \mathcal{T}_{0^{\Lambda}}(\alpha)$$\alpha\triangleright\lambda$$h\in \mathcal{H}$$S^{\vee\lambda}=S^{\vee}(\Lambda)^{\lambda}$$\varphi_{ST}$$S(\Lambda)$

55
moduleby the rightgenerated by the image-submodule of
, let
where
. For eachbe the image ofin. Then the Weyl module
is R-free with basis. As inthe case of Specht modules there is an inner product onwhich is determined by$W^{\lambda}$$S(\Lambda)$$S(\Lambda)/S^{\vee}(\Lambda)^{\lambda}$$T^{\lambda}=\lambda(t^{\lambda})$$\varphi_{\lambda}=\varphi_{T^{\lambda}T^{\lambda}}\in S(\Lambda)$$W^{\lambda}$$T\in \mathcal{T}_{0^{\Lambda}}(\lambda)$$W^{\lambda}$$\varphi_{T}$$\{\varphi_{T}|T\in \mathcal{T}_{0^{\Lambda}}(\lambda)\}$$\varphi_{T^{\lambda}T}$$W^{\lambda}$$\varphi_{T^{\lambda}S}\varphi_{TT^{\lambda}}\equiv\langle\varphi_{S},$$mod S^{\vee\lambda}$$\varphi_{T}\}\varphi_{T^{\lambda}T^{\lambda}}$Let rad $W^{\lambda}=\{x\in W^{\lambda}|\langle x,$ $y\rangle=0$ for all
is absolutely irreducible and-modules,non-isomorphic irreducible$L^{\lambda}=W^{\lambda}/radW^{\lambda}$$y\in W^{\lambda}\}$.

The quotient module
is a complete set of.

$\{L^{\lambda}|\lambda\in\Lambda^{+}\}$$S(\Lambda)$

2.8. For an r-composition , we define tlie typeof , by $\alpha=(n_{1}, \ldots, n_{r})$with $n_{i}=|\mu^{(i)}|$ , and the size of by $n= \sum_{i=1}^{r}n_{i}$ . We also define a sequence a$\alpha=\alpha(l^{l},)$

$\mu$

$J$

$=$

$\mu$

. (Recall that $a_{i}= \sum_{k=1}^{i-1}|\mu^{(k)}|=\sum_{k=1}^{i-1}n_{k}.$ )We define a partial orderby a $\geq a$’ for $a=(a_{1}, \ldots, a_{r}),$ $a’=$on the set
if
for any . We write $a>a’$ if $a\geq a’$ and $a\neq a’$ . It is clear$a(\mu)=(a_{1}, \ldots, a_{r})$$\geq$$a_{i}\geq a_{i}$$(a_{1}, \ldots, a_{r}’)\in \mathbb{Z}_{\geq 0}^{r}$$\mathbb{Z}_{\geq 0}^{r}$

$i$

that

(2.1)

If

$\lambda\underline{\triangleright}l^{4}$, then $a(\lambda)\geq a(\mu)$ for r-compositions$\lambda,$

$\mu$

.

Hence if
is non-empty, then, and so we have $a(\lambda)\geq a(\mu)$ .
For any r-partition and r-composition , we define a subset$\mathcal{T}_{0}(\lambda, \mu)$$\lambda\underline{\triangleright}\mu$$\lambda$$\mathcal{T}_{0}^{+}(\lambda.\mu)$

$\mu$

by
$\mathcal{T}_{0}^{+}(\lambda, \mu)=\{S\in \mathcal{T}_{0}(\lambda, \mu)|a(\lambda)=a(l^{L})\}$

of$\mathcal{T}_{0}(\lambda, \mu)$.

Note that the condition $a(\lambda)=a(\mu)$ is equivalent to. Take.
Then one can check that
if and only if each entry of
is of theform $(i, k)$ for some . Hence in this casecan be identified with a semistandard-Tableau of typeunder the usual definition of the semistandard Tableaux forl-partitions
and l-compositions. It follows that we have a bijection$\alpha(\lambda)=\alpha(\mu)$$S\in \mathcal{T}_{0}^{+}(\lambda, \mu)$$S^{(k)}$$S\in \mathcal{T}_{0}^{+}(\lambda, \mu)$$S^{(k)}$$i$$\lambda^{(k)}$$\mu^{(k)}$$\lambda^{(k)}$$\mu^{(k)}$$\mathcal{T}_{0}^{+}(\lambda, \mu)\simeq \mathcal{T}_{0}(\lambda^{(1)}, \mu^{(1)})\cross$$\cdot\cdot\cdot$$\cross \mathcal{T}_{0}(\lambda^{(r)}, \mu^{(r)})$via $Srightarrow(S^{(1)}, \ldots, S^{(r)})$ . Moreover, if
is such that $\mu(s)=S$ with $S\in$
Std, then the entries of i-th component of 5 consist of numbers $a_{i}+1,$
for $a(\lambda)=(a_{1}, \ldots, a_{r})$ . In particular,.
for
Fix an r-tuple $m=(m_{1}, \ldots, m_{r})$ of non-negative integers. Then, an r-composition$\mu=(\mu^{(1)}, \ldots, \mu^{(r)})$ withis called an $(r, m)$ -composition,
and $(r, m)$ -partition is defined similarly. We denote by(resp.
the set of $(r, m)$ -compositions (resp. $(r,$ $m)$ -partitions) of size . (Note thatarc naturally idcntified with each othcr for any $m$ such that $m_{i}\geq n$ . However,depends on the choice of $m.$ ) Finally, let$(\lambda)$$\mathfrak{s}\in$$\mathcal{T}_{0}^{+}(\lambda, \mu)$$\ldots,$ $a_{i+1}$$d(5)\in \mathfrak{S}_{y}$$\alpha=\alpha(\lambda)$$\mu^{(i)}=(\mu_{1}^{(i)}, \ldots.\mu_{m}^{(i)},)\in \mathbb{Z}_{\geq 0}^{m_{i}}$$\tilde{\mathcal{P}}_{n_{1}r}=\tilde{\mathcal{P}}_{n,r}(m)$$\mathcal{P}_{n,r}(m))$$\mathcal{P}_{n,r}=$$n$$\underline{\mathcal{P}}_{n,r}(m)$$\mathcal{P}_{n,r}$$S\in \mathcal{T}_{0}(\lambda, /l),$$T\in \mathcal{T}_{0}(\lambda, \nu)$$C^{0}( \Lambda)=\bigcup_{l\mu.\text{ノ}\in\Lambda,\lambda\in\Lambda+}\{\varphi s\tau\in C(\Lambda)|$$a(\lambda)>a(\mu)$and we define$S^{0}(\Lambda)$

as the R-submodule of$S(\Lambda)$ifwith basis,$\alpha(\mu)\neq\alpha(\nu)\}$$C^{0}(\Lambda)$.

56
3. THE STANDARD BASIS FOR$S^{0}(\Lambda)$

3.1. First, we prepare some notation. Let$\Omega=(\Lambda^{+}\cross\{0,1\})\backslash${$1)|\mathcal{T}_{0}(\lambda,$$(\lambda,$

and we define a partial order. For aandor$\lambda_{1}=\lambda_{2}$$\mu)=\emptyset$$(\lambda, \epsilon)\in\Omega_{\dot{J}}$if$\mathcal{T}_{0}^{+}(\lambda)$$I(\lambda, \epsilon)=\{$

if$\bigcup_{\mu\in\Lambda,a(\lambda)>a(\mu)}\mathcal{T}_{0}(\lambda, \mu)$$\Omega$$(\lambda, \epsilon)\in\Omega$$\Omega$$\epsilon=0$

such that $a(\lambda)>a(\mu)$ }if$(\lambda_{1}, \epsilon_{1})>(\lambda_{2}, \epsilon_{2})$,$I(\lambda, \epsilon),$$\lambda_{1}\triangleright\lambda_{2}$$\mathcal{T}_{0}^{+}(\lambda)$,by$J(\lambda, \epsilon)$if$\epsilon=0$,

if$\epsilon=1$,$J(\lambda, \epsilon)=\{$$\epsilon=1,$$\mathcal{T}_{0}(\lambda)$

and. Then. We define a subset$I(\lambda, \epsilon)$$\mathcal{T}_{0}^{+}(\lambda)=\bigcup_{A\in\Lambda}\mathcal{T}_{0}^{+}(\lambda, \mu)$$\mu\in\Lambda$

byon
wc defiiie index scts$(\lambda_{1}, \epsilon_{1})\geq(\lambda_{2}, \epsilon_{2})$$\epsilon_{1}>\epsilon_{2}$where. Assume thatfor any$J(\lambda, \epsilon)$$C^{0}(\lambda, \epsilon)$

are not empty for allof$S^{0}(\Lambda)$$(\lambda, \epsilon)\in$by.

$C^{0}(\lambda, \epsilon)=\{\varphi_{ST}|(S, T)\in I(\lambda, \epsilon)\cross J(\lambda, \epsilon)\}$

It is easy to see thatthe union(3.1)$\bigcup_{(\lambda,\epsilon)\in\Omega}C^{0}(\lambda, \epsilon)$is disjoint and is equal to the set$C^{0}(\Lambda)$.

the R-submodule of, we define by
3.2. For any
with
some
for
where
spanned by. Similarly, we
for every. Note that
. We
with
as the R-submodule spanned bydefinecan now state.
$S_{0}^{\vee(\lambda\epsilon)}=S^{0}(\Lambda)(>(\lambda, \epsilon))$$(\lambda, \epsilon)\in\Omega$$S^{0}(\Lambda)$$(\lambda’, \epsilon’)\in\Omega$$(U, V)\in I(\lambda’, \epsilon’)\cross J(\lambda’, \epsilon’)$$\varphi_{UV}$$S^{0}(\Lambda)\cap S^{\vee\lambda}=S_{0}^{\vee(\lambda,1)}$$(\lambda’, \epsilon’)>(\lambda, \epsilon)$$\lambda\in\Lambda^{+}$$S^{0}(\Lambda)(\geq(\lambda, \epsilon))$$\varphi_{UV}$$(\lambda’, \epsilon’)\geq(\lambda, \epsilon)$is standardly based (in the sense of [DR]) on
Theorem 3.1. The subalgebra, that is,with standard basis.
is disjoint and forms an R-basis for(i) The union$S^{0}(\Lambda)$$C^{0}(\Lambda)$$(\Omega, \geq)$$S^{0}(\Lambda)$$\bigcup_{(\lambda,\epsilon)\in\Omega}C^{0}(\lambda, \epsilon)=C^{0}(\Lambda)$(ii) For any$\varphi\in S^{0}(\Lambda),$$\varphi_{ST}\in C^{0}(\lambda.

\epsilon)$, we have$mod S_{0}^{\vee(\lambda\epsilon)}$$\varphi\cdot\varphi_{ST}\equiv\sum_{S’\in J(\lambda,\epsilon)}f_{S’,(\lambda\epsilon)}(\varphi, S)\cdot\varphi_{S’T}$(3.2)$mod S_{0}^{\vee(\lambda,\epsilon)}$$\varphi s\tau\cdot\varphi\equiv\sum_{T’\in J(\lambda^{\underline{\sigma}})},f_{(\lambda,\epsilon),T’}(T, \varphi)\cdot\varphi_{ST’}$

where
and , respectively.
$\varphi s^{l}\tau,$$T$$\varphi_{ST’}\in C^{0}(\Lambda)$

and$f_{S’,(\lambda,\epsilon)}(\varphi, S),$$f_{(\lambda,\epsilon),T’}(T, \varphi)\in R$,

are independent of$S$

Note that the cellular algebra is a special case of the standardly based.

3.3. Next we introduce the Weyl module forandeasy to see that R-modules. For. Fix asided ideals ofby the R-submodule offor$S^{0}(\Lambda)$$S^{0}(\Lambda)(\geq(\lambda, \epsilon))$$S^{0}(\Lambda)$$Z_{s}^{(\lambda,\epsilon)}$$S^{0}(\Lambda)$$(\lambda, \epsilon)\in\Omega$.

By (3.2) in Theorem 3.1. it isare two, we define the Weyl rnodulewith$S_{0}^{\vee(\lambda,\epsilon)}=S^{0}(\Lambda)(>(\lambda, \epsilon))$$S\in I(\lambda, \epsilon)$$\{S^{0}(\Lambda)(\geq(\lambda, \epsilon))\}/\{S^{0}(\Lambda)(>(\lambda, \epsilon))\}$

57
is the right. Moreover, by (3.2), we see thatis independent of the choice of $S,$ $i.e$ ,-module and the action ofon. However, sincefor all
is not an element infor, one should pay attention that there is no $(canonical$ ”-Weyl module) For the convenience sake let
for the case
. (That is, we can not define.
and put
for anybasis$\{\varphi_{ST}+S_{0}^{\vee(\lambda,\epsilon)}|T\in J(\lambda, \epsilon)\}$$S^{0}(\Lambda)$$Z_{S}^{(\lambda,\epsilon)}$$S^{0}(\Lambda)$$Z_{S_{1}}^{(\lambda,\epsilon)}\simeq Z_{S_{2}}^{(\lambda,\epsilon)}$$S_{1},$$Z_{S}^{(\lambda,\epsilon)}$$T^{\lambda}$$S_{2}\in I(\lambda, \epsilon)$$I(\lambda, 1)$($(\lambda, 1)\in\Omega$$(\lambda, 1)$$Z_{T^{\lambda}}^{(\lambda,1)}.$}$\varphi_{T}^{0}=\varphi_{T^{\lambda}T}+S_{0}^{\vee(\lambda\epsilon)}$$Z^{(\lambda,0)}=Z_{T^{\lambda}}^{(\lambda,0)}$3.4. Suppose that $S,$
any $U,$$T\in \mathcal{T}_{0}^{+}(\lambda)$$T\in J(\lambda, 0)=\mathcal{T}_{0}^{+}(\lambda)$. Then there exists an element $r_{ST}\in R$ such that for$V\in \mathcal{T}_{0}^{+}(\lambda)$

We define a bilinear form
we have
(3.3)

where

$\langle\varphi_{S}^{0},$

$U$

and

,

$\rangle_{0}$$\langle\varphi_{S}^{0},$:$Z^{(\lambda_{1}0)}\cross Z^{(\lambda,0)}arrow R$$\varphi_{T}^{0}\rangle_{0}\cdot\varphi_{UV}\equiv\varphi_{US}\cdot\varphi_{TV}$are any elements of

$V$

(3.4)

Let rad$\langle$$\varphi_{T}^{0}\rangle_{0}=\langle\varphi s,$$\mathcal{T}_{0}^{+}(\lambda)$$Z^{(\lambda,0)}=\{x\in Z^{(\lambda,0)}|\langle x,$ $y\rangle_{0}=0$

Lemma 3.2. rad
We put$Z^{(\lambda.0)}$

is an$L_{0}^{\lambda}=Z^{(\lambda,0)}/$rad$S^{0}(\Lambda)$$Z^{(\lambda,0)}$by$\langle\varphi_{S}^{0},$$mod S_{0}^{\vee(\lambda,0)}$$\varphi_{T}^{0}\rangle_{0}=r_{ST}$. Hence,. It is easy to see thatfor every $S,$$\varphi_{T}\rangle$.

$mod S_{0}^{\vee(\lambda,0)}$$\varphi_{US}\cdot\varphi_{TV}\equiv r_{ST}\cdot\varphi_{UV}$

for all$T\in \mathcal{T}_{0}^{+}(\lambda)$$y\in Z^{(\lambda,0)}\}$-submodule of$Z^{(\lambda,0)}$

.

.

.

. Then we have the following.

Proposition 3.3. Suppose that $R$ is a field, and. Then(i)and(ii) radandis the unique maximal submodule ofducible. Moreover, the Jacobson radical ofis equal to$\lambda\in\Lambda^{+}$$L_{0}^{\lambda}\neq 0$$Z^{(\lambda,0)}$$Z^{(\lambda,0)}$$Z^{(\lambda,0)}$$L_{0}^{\lambda}$

is absolutely irre.

$mdZ^{(\lambda,0)}$4. A RELATIONSHIP BETWEEN $S^{b}(m,$ $n)$ AND$S^{0}(\Lambda)$

First, we recall the definition of modified Ariki-Koike algebras and their cyclotomic
q-Schur algebras $([SawS])$ .

4.1. From now on, throughout this paper, we consider the following condition onparameters
we considor tlie modified Ariki-Koike algebrasin(and their cyclotomic q-Schur algebras).
$Q_{1},$(4.1)$Q_{i}-Q_{j}$$\ldots,$$Q_{r}$$Rwf_{1t^{Y}I1_{\text{ノ}}^{Y}}ver$are invertible in$R$

for any $i\neq j$ .

Let be a square matrix of degree whose i-j entry is given by
for $1\leq i,$ $j\leq$$A=\prod_{i>j}(Q_{i}-Q_{j})$. Thusis invertible
is thc Vandcriiiondc niatrix,dct$A^{-1}=\triangle^{-1}B$$B=(h_{ij})$withby (4.1). We express the inverse of,
anddefine aaspolynomial $F_{i}(X)\in R[X]$ , for $1\leq i\leq r$ , by $F_{i}(X)= \sum_{1\leq j\leq r}h_{ij}X^{j-1}$ .
$A$

$r$

$Q_{j}^{i-1}$

$r$

$A$

$al\iota d\triangle=$

$A$

58
The modified Ariki-Koike algebra
andwith generators(4.2)$T_{2},$$\cdots,$$T_{n}$$\xi_{1},$is an associative algebra overand relations$\mathcal{H}^{b}=\mathcal{H}_{n,r}^{b}$$\xi_{n}$$\ldots,$$T_{i}T_{i+1}T_{i}=T_{i+1}T_{i}T_{i+1}$,,$(2\leq i\leq n)$ ,$T_{i}T_{j}=T_{j}T_{i}$$(|i-j|\geq 2)$ ,$\xi_{t}\xi_{j}=\xi_{j}\xi_{i}$$(1 \leq i,j\leq n)$$(T_{i}-q)(T_{i}+q^{-1})=0$$(2\leq i\leq n)$$(\xi_{i}-Q_{1})\cdots(\xi_{i}-Q_{r})=0$$(1 \leq i\leq n)$$T_{j} \xi_{j}=\xi_{j-1}T_{j}+\triangle-2\sum(Q_{c_{2}}-Q_{c_{1}})(q-q^{-1})F_{C1}(\xi_{j-1})F_{c_{2}}(\xi_{j})$,$T_{j} \xi_{j-1}=\xi_{j}T_{j}-\triangle-2^{<c_{2}}\sum_{c_{1<C2}}^{c_{1}}(Q_{c_{2}}-Q_{c_{1}})(q-q^{-1})F_{c_{1}}(\xi_{j-1})F_{c_{2}}(\xi_{j})$

,

$R$

,

$(k\neq j-1,j)$ .

$T_{j}\xi_{k}=\xi_{j}T_{j}$, the field of rational functions witli
It is known that if, and it gives an alternate presentation
is isomorphic tovariables
apart from 2.1.
of,is isomorphic togenerated byof
The subalgebra.
ofsubalgebra
corresponding
hence it can be naturally identified with the$0\leqc_{i}<r$for
Moreover, it is known by [Sh] that the set$i\leq n\}$ gives rise to a basis of.
Let $V=\oplus_{i=1}^{r}V_{i}$ be a free R-module, with rank $V_{i}=m_{i}$ . We put $m= \sum m_{i}$ .
. We-module structure on
It is known by $[SakS]$ that we can define a right. Note that this construction works
End
denote this representation by :[Sh] that, under the assumption
in
shown
it
is
without the condition (4.1). Also
can be defined. We denote this representation byon(4.1), a right action of.
. By [Sh, Lemma 3.5], we know that:End
We consider the condition$R=\mathbb{Q}(\overline{q},\overline{Q}_{1}, \ldots,\overline{Q}_{r})$$\overline{q},$$\overline{Q}_{r},$$\overline{Q}_{1},$$\mathcal{H}^{b}$$\mathcal{H}$$\ldots,$$\mathcal{H}$$\mathcal{H}^{b}$$\mathcal{H}^{b}(\mathfrak{S}_{n})$$T_{2},$$\ldots,$$\mathcal{H}_{n}$$T_{n}$$\mathcal{H}$$\mathcal{H}(\mathfrak{S}_{n})$$1\leq$$\{\xi_{1}^{c_{1}}\cdots\xi_{n^{n}}^{c}T_{w}|w\in \mathfrak{S}_{n},$$\mathcal{H}^{b}$$V^{\emptyset n}$$\mathcal{H}$$\mathcal{H}arrow$$V^{\otimes n}$$\rho$$V^{\otimes n}$$\mathcal{H}^{b}$$\rho^{b}$$\mathcal{H}^{b}arrow$(4.3)$m_{i}\geq n$${\rm Im}\rho\subset{\rm Im}\rho^{b}$$V^{\otimes n}$

for $i=1,$$\cdots,$ $r$.

Lemma 4.2 ( $[SawS$ , Lemma 1.5]). Under the conditions (4.1), (4.3), there exists an R. (Hereinduces the identity onsuch thatalgebra homomorphism
andidentifications.)
under the previous
we regard
If.
is a field, thenFrom now on, throughout the paper, we fix an r-tuple $m=(m_{1}, \ldots, m_{r})$ of non.
negative integers and always assume the condition (4.3) whenever we consider(i.e, l-composition) of
may be regarded as an element inAnyin orderby arranging the entries of$\rho_{0}:\mathcal{H}arrow \mathcal{H}^{b}$${\rm Im}\rho^{b}={\rm Im}\rho$$\mathcal{H}_{n}\subset \mathcal{H}^{b}$$\mathcal{H}_{n}\subset \mathcal{H},$$R$$\mathcal{H}_{n}$$\rho_{0}$$\mathcal{H}\simeq \mathcal{H}^{b}$$\mathcal{H}^{b}$$\mu\in\tilde{\mathcal{P}}_{n,r}(m)$$n$$\mathcal{P}_{n,1}$$\iota=(’\iota_{j}^{(i)})$,$l^{\iota_{1}^{(1)},\ldots,\mu_{m_{1}}^{(1)},\mu_{1}^{(2)},\ldots,\mu_{m_{2}}^{(2)},\ldots,\mu_{1}^{(r)},\ldots,\mu_{m_{r}}^{(r)}}$which we denote byFor$\{\mu\}$.

$\alpha=(n_{1}, \ldots, n_{r})\in \mathbb{Z}_{\geq 0}$

such that $\sum n_{i}=n_{i}$ we define$c(\alpha)$$c( \alpha)=(r,\ldots,r\tilde{n_{1}- times},\frac{r-1,..,r-1}{n_{2}- times}, \ldots,1,\ldots,1)\tilde{n_{r}- times}$by59
and let $c(\alpha)=(c_{1}, \ldots, c_{n})$ . We defineby.
For any
, put
where
.
We define an R-linear anti-automorphism $harrow h^{*}$ onby the condition that
$T_{i}(2\leq
i\leq
n)$
$\xi_{j}(1\leq
j\leq
n)$
fixes the generators. As discussed in $[SawS, 2.7]$ ,andthis condition induces a well-defined anti-automorphism on. Moreover, by Lemma$[SawS]$, we know that
2.9 in. For
with, we define
Stdan elementby. By the above fact, we have.
$F_{\alpha}\in \mathcal{H}^{b}$$\mu\in\tilde{\mathcal{P}}_{n,r}$$m_{\mu}^{b}=F_{\alpha(\prime 4)}\cdot m_{\{/x\}}$$F_{\alpha}=\triangle-nF_{c_{1}}(\xi_{1})F_{c_{2}}(\xi_{2})\cdots F_{c_{n}}(\xi_{n})$$m_{\{\mu\}}= \sum_{w\in \mathfrak{S}_{\{\}}},q^{l(w)}T_{w}l(=x_{\mu})\in \mathcal{H}_{n}$$\mathcal{H}^{b}$$*$$\mathcal{H}^{b}$$s,$ $t\in$$(m_{\mu}^{b})^{*}=m_{\mu}^{b}$$m_{5}^{b_{t}}\in \mathcal{H}^{b}$$(\lambda)$$\lambda\in \mathcal{P}_{n,r}$$(m_{st}^{b})^{*}=m_{t\epsilon}^{b}$$m_{,\lrcorner t}^{b},=T_{d(\epsilon)}^{*}m_{A}^{b}T_{d(1)}$Theorem 4.3 ( $[SawS$ , Theorem 2.18]). The modified Ariki-Koike algebra
as an R-module with cellular basis {Std}.
for some$m_{\epsilon 1}^{b}|s,$Put
as follows.

$M_{b}^{\mu}=m_{\mu}^{b}\mathcal{H}^{b}$for$\mu\in\tilde{\mathcal{P}}_{n,r}$$t\in$$(\lambda)$$S^{b}(m, n)=End_{\ovalbox{\tt\small REJECT}^{b}}(M^{b}(\tilde{\mathcal{P}}_{nr,)}))$$[SawS]$ , we have$\alpha\in\tilde{\mathcal{P}}_{n,1}$

is free$\lambda\in \mathcal{P}_{n,r}$. We define a cyclotomic q-Schur algebra $S^{b}(m, n)$Definition 4.4. The cyclotomic q-Schur algebra forendomorphism algebra

For an r-tuples$\mathcal{H}^{b}$

where

,

$\mathcal{H}^{b}$with weight poset$M^{b}( \tilde{\mathcal{P}}_{n_{t}r})=\bigoplus_{\mu\in\tilde{\mathcal{P}}_{nr}},M_{b}^{\mu}$$\tilde{\mathcal{P}}_{n_{1}r}$

is the.

, let $M_{b}(\}=\oplus_{A,tJ(’ 4)=(y}M_{b}^{\mu}$ . Then by Proposition 5.2 (i) in
End ノ
as R-algebras.

$S^{b}(m, n)\simeq\oplus_{\alpha\in\overline{\mathcal{P}}_{n,1}}$$\nearrow^{b}M_{b}^{\alpha}$

Theorem 4.5 ( $[SawS$ , Theorem 5.5]). Let $S^{b}(m, n)$ be the cyclotomic q-Schur algebra
associated to the modified Ariki-Koike algebra
and $S(m_{i}, n_{i})$ be the q-Schur algebra
associated to the Iwahori-Hecke algebra. Then there exists an isomorphism of Ralgebras$\mathcal{H}^{b}$$\mathcal{H}_{n_{i}}$.

$S^{b}(m, n)\simeq$$\bigoplus_{(n,\ldots,n_{r}),n=n_{1}^{1}+\cdot\cdot+n_{r}}.S(m_{1}, n_{1})\otimes\cdots\otimes S(m_{r}, n_{r})$Let$\mu,$$\mathcal{T}_{0}^{+}(\lambda, \mu)$$\nu\in\tilde{\mathcal{P}}_{n,r}$

and

and

$\lambda\in \mathcal{P}_{n,r}$$T\in \mathcal{T}_{0}^{+}(\lambda, \nu)$. We assume that $\alpha(\mu)=\alpha(\nu)=\alpha(\lambda)$ . For $S\in$

, put
.

$m_{ST}^{b}= \sum_{\mathfrak{s},t\in Std(\lambda)}q^{l(d(\mathfrak{s}))+l(d(1))}m_{5}^{b}\mu(\lrcorner)=S,\nu(t)=T\iota$

Moreover, for$S\in \mathcal{T}_{0}^{+}(\lambda, \mu)$$\varphi_{ST}^{b}(m_{\alpha}^{b}h)=\delta_{\alpha\nu}m_{ST}^{b}h$, for alland$T\in \mathcal{T}_{0}^{+}(\lambda, \nu)$$h\in \mathcal{H}^{b}$and all,ne can define$\alpha\in\tilde{\mathcal{P}}_{n,r}$$\varphi_{ST}^{b}\in S^{b}(m, n)$by.

Theorem 4.6 ( $[SawS$ , Theorem 5.9]). The cyclotomic q-Schur algebra $S^{b}(m, n)$ is, for somefree as an R-module with cellular basis $C^{b}(m, n)=\{\varphi_{ST}^{b}|S,$$\lambda\in$$T\in \mathcal{T}_{0}^{+}(\lambda)$$\mathcal{P}_{n,r}\}$.

4.2. Letbe as in Section 3. We describe a relationship between the algebra
and fhe cyclotomic q-Schur algebra $S^{b}(m. n)$ in fhe ca.se where. But
in the moment, we shall consider an arbitrary
as in Section 3.
$S^{0}(\Lambda)$$S^{0}(\Lambda)$$\Lambda=\tilde{\mathcal{P}}_{n,r}$$\Lambda$

60First, let $C^{00}(\Lambda)=\{\varphi_{ST}|(S, T)\in I(\lambda, 1)\cross J(\lambda, 1), \lambda\in\Lambda^{+}\}\subset C^{0}(\Lambda)$ and. We note that.
, which is an R-submodule ofbe the R-span ofby the second and fourth formula in [Sa, Lemma
is a two-sided ideal of. We write2.4]. Thus one can define the quotient algebra$\overline{x}=x+S^{00}(\Lambda)(x\in S^{0}(\Lambda))$ .
has a free R-basis
It is easy to see that
$S\in I(\lambda, 0),$ $T\in J(\lambda, 0),$. Note that the condition $(S, T)\in I(\lambda, 0)\cross J(\lambda, 0)$, let
be the R-submodule of. For
is nothing but $S,$
$S^{00}(\Lambda)$$S^{0}(\Lambda)$$\varphi_{ST}\in C^{00}(\Lambda)$$S^{0}(\Lambda)$$S^{00}(\Lambda)$$\overline{S^{0}}(\Lambda)=S^{0}(\Lambda)/S^{00}(\Lambda)$$\overline{S^{0}}(\Lambda)$$\{\overline{\varphi}_{ST}|$$\lambda\in\Lambda^{+}\}$

spanned byshow the following.

$\overline{S^{0}}(\Lambda)$

with$\overline{\varphi}_{ST}$

Theorem 4.7. The algebra$\overline{S_{0}}^{\vee\lambda}=\overline{S_{0}}^{v}(\Lambda)^{\lambda}$$\lambda\in\Lambda^{+}$$T\in \mathcal{T}_{0}^{+}(\lambda)$$S,$$T\in \mathcal{T}_{0}^{+}(\alpha)$$\overline{S^{0}}(\Lambda)$

for various$\alpha\in\Lambda^{+}$

such that$\alpha\triangleright\lambda$. We

has a free basis$\overline{C^{0}}(\Lambda)=\{\overline{\varphi}_{ST}|S, T\in \mathcal{T}_{0}^{+}(\lambda), \lambda\in\Lambda^{+}\}$

satisfying the following properties.
determined by:(i) The R-linear map$S,$.
anti-automorphism,isanand all, and any
. Then for all(ii) Let$r_{V}\in R$ such that$\overline{S^{0}}(\Lambda)arrow\overline{S^{0}}(\Lambda)$$*$$\overline{\varphi}_{ST}^{*}=\overline{\varphi}_{TS}$$of\overline{S^{0}}(\Lambda)$$\lambda\in\Lambda^{+}$$T\in \mathcal{T}_{0}^{+}(\lambda)$, for all$V\in \mathcal{T}_{0}^{+}(\lambda)$$\overline{\varphi}\in\overline{S^{0}}(\Lambda)$$T\in \mathcal{T}_{0}^{+}(\lambda)$$\sum$$\overline{\varphi}_{ST}\cdot\overline{\varphi}\equiv$, there exists$mod S_{\frac{\vee}{0}}^{\lambda}$$r_{V}\overline{\varphi}_{SV}$$V\in \mathcal{T}_{0}^{+}(\lambda)$

for any$S\in \mathcal{T}_{0}^{+}(\lambda)$

In particular,, where$\overline{C^{0}}(\Lambda)$$r_{V}$

is independent of the choice of $T$ .

is a cellular basis of

In the case where $S^{b}(m, n)$ is defined,
we have the following proposition.
Proposition 4.8. Let
exists an algebm isomorphism
such that $S,$$\Lambda=\tilde{\mathcal{P}}_{n,r}$$b$$\overline{S^{0}}(\Lambda)$.

can be identified witli $S^{b}(m, n)$ , i.e,

and assume that (4.1) and (4.3) holds. Then there
satisfying the following. For:.
, we have
and$\overline{S^{0}}(\Lambda)arrow S^{b}(m, n)$$\lambda\in\Lambda^{+}$$T\in \mathcal{T}_{0}^{+}(\lambda)$$\overline{\varphi}_{ST}\in\overline{C^{0}}(\Lambda)$$\overline{S^{0}}(\Lambda)$$(\overline{\varphi}_{ST})^{b}=\varphi_{ST}^{b}$

for arbitrary . The
We now return to the general setting, and consider(cover”$S^{b}(m, n)$ .
theof
is a natural
above proposition says that the. Hence, by the cellular theory
is an element inFor-submodule ofas the rightof[GL], one can define a Weyl modulein. We denote by
the image ofspanned by the image ofDefine a bilinear
is a free R-basis ofThen the set
by requiring thatonform ,$\Lambda$$\overline{S^{0}}(\Lambda)$$\overline{S^{0}}(\Lambda)$$\lambda\in\Lambda^{+},$$\overline{S^{0}}(\Lambda)$$\overline{\varphi}_{\lambda}=\overline{\varphi}_{T^{\lambda}T^{\lambda}}$$\overline{Z}^{\lambda}$$\overline{S^{0}}(\Lambda)$$\overline{S^{0}}(\Lambda)$$\overline{S^{0}}(\Lambda)/\overline{S_{0}}^{\vee\lambda}$$\overline{\varphi}_{7^{\lambda}T}$$\overline{\varphi}_{T}$$\overline{\varphi}_{\lambda}$$\overline{Z}^{\lambda}$$\overline{S^{0}}(\Lambda)/\overline{S_{0}}^{\vee\lambda}$$\{\overline{\varphi}_{T}|T\in \mathcal{T}_{0}^{+}(\lambda)\}$$\overline{Z}^{\lambda}$$\langle$$\}_{\overline{0}}$$\overline{\varphi}_{T^{\lambda}S}\overline{\varphi}_{TT^{\lambda}}\equiv\langle\overline{\varphi}_{S},$$\overline{\varphi}_{T}\rangle_{\overline{0}}\cdot\overline{\varphi}_{\lambda}$

mod$\overline{S_{0}}^{\vee\lambda}$

where
. Let
for all $S,$
for all}. In the case where is a ficld, by a gciicral tlicory of ccllular
gives a complete set of non-isomorphic
algebras. the set-modules. Furtherinore, we have the following result.
irreducible$T\in \mathcal{T}_{0}^{+}(\lambda)$$0$$y\in\overline{Z}^{\lambda}$$\overline{L}^{\lambda}=\overline{Z}^{\lambda}/rad\overline{Z}^{\lambda}$$R$$\{\overline{L}^{\lambda}|\lambda\in\Lambda^{+}, \overline{L}^{\lambda}\neq 0\}$$\overline{S^{0}}(\Lambda)$$rad\overline{Z}^{\lambda}=\{x\in\overline{Z}^{\lambda}|\langle x,$$y\rangle_{\overline{0}}=$61. Hence,
Proposition 4.9. Suppose that $R$ is a field. Thenfor any-modules. Thereis a complete set of non-isomorphic irreducible
fore,
is quasi-hereditary.
$\overline{L}^{\lambda}\neq 0$$\lambda\in\Lambda^{+}$$\{\overline{L}^{\lambda}|\lambda\in\Lambda^{+}\}$$\overline{S^{0}}(\Lambda)$$\overline{S^{0}}(\Lambda)$

The following result connects the decomposition numbers in

and in$\overline{Z}^{\lambda}$$Z^{(\lambda,0)}$.

Theorem 4.10. Suppose that $R$ is a field. Then(i)
is a complete set of pairwise inequivalent irreducible.
-module-modules occumng in the composition factors of the$\{L_{0}^{\alpha}|\alpha\in\Lambda^{+}, \lambda\underline{\triangleright}\alpha\}$(ii) For$\lambda,$$\mu\in\Lambda_{f}^{+}$

we have.

$[\overline{Z}^{\lambda}:\overline{L}^{\mu}]=[Z^{(\lambda 0)}):L_{0}^{\mu}]$(iii) For$\lambda,$$Z^{(\lambda_{1}0)}$$S^{0}(\Lambda)$$S^{0}(\Lambda)$such that$\mu\in\Lambda^{+}$$\alpha(\lambda)\cdot\neq\alpha(\mu)$, we have$[\overline{Z}^{\lambda}:\overline{L}^{\mu}]=0$.

5. AN ESTIMATE FOR DECOMPOSITION NUMBERS

We are now ready to estimate the decomposition numbers for the cyclotomic qSchur algebras.
5.1. We keep the notation in Section 4, and consider the general

Theorem 5.1. Suppose that $R$ is a field. Then, for all$[\overline{Z}^{\lambda} :\overline{L}^{\mu}]=[Z^{(\lambda,0)} :$\lambda,$

L_{0}^{\mu}]=[W^{\lambda} :$\Lambda$

with$\mu\in\Lambda^{+}$

L^{\mu}]$.
$\alpha(\lambda)=\alpha(\mu)$,.

5.8. We return to the setting in 4.1. Let
under the condition (4.1) and, we denote by(4.3). For an r-partition
the R-submodule of $S^{b}(m, n)$. Moreover, for an r-partition
spanned bywithsuch that $S,$
, and in fact
is the unique semistandard -Tableau of. Thus,Moreover,.
type
such thatis the unique element in Std. We define the Weyl
, andis the identity map on
spanned by the image
moduleas the right-submodule of. For each.
in
, we denote byof
the image of.
Then we know that the Weyl module
is R-free with basis
The Weyl module
enjoys an associative symmetric bilinear form, defined by the$\Lambda=\tilde{\mathcal{P}}_{n,r}$$S_{b}^{\vee\lambda}$$\lambda\in \mathcal{P}_{n,r}$$T\in \mathcal{T}_{0}^{+}(\alpha)$$\varphi_{ST}^{b}$$\lambda\in \mathcal{P}_{n,r},$$\alpha\triangleright\lambda$$T^{\lambda}$$T^{\lambda}\in \mathcal{T}_{0}^{+}(\lambda,\cdot\lambda)$$\lambda$$t=t^{\lambda}$$\lambda$$m_{T^{\lambda}T^{\lambda}}^{b}=m_{t^{\lambda}t^{\lambda}}^{b}=m_{\lambda}^{b}$$M_{b}^{\lambda}$$S^{b}$ $(m_{\dot{l}} n)$$W_{b}^{\lambda}$$S^{b}(m, n)/S_{b}^{\vee\lambda}$$T\in \mathcal{T}_{0}^{+}(\lambda, \mu)$$\varphi_{\lambda}^{b}$$\lambda(t)=T^{\lambda}$$(\lambda)$$\varphi_{\lambda}^{b}=\varphi_{T^{\lambda}T^{\lambda}}^{b}$$\varphi_{T}^{b}$$S^{b}(m, n)/S_{b}^{\vee\lambda}$$\varphi_{T^{\lambda}T}^{b}$$\{\varphi_{T}^{b}|T\in \mathcal{T}_{0}^{+}(\lambda)\}$$W_{b}^{\lambda}$$W_{b}^{\lambda}$

equation$\varphi_{T^{\lambda}S}^{b}\varphi_{TT^{\lambda}}^{b}\equiv\langle\varphi_{S}^{b},$for all $S,$
for all$\varphi_{T}^{b}\rangle_{b}\cdot\varphi_{\lambda}^{b}$$mod S_{b}^{\vee\lambda}$

where radLet}. By [ $SawS$ , Proposition 5.11], we know that, for all r-partition
is a complete set
is an absolutely irreducible and$(m, n)$ -niodules. Furthermore, for, weof non-isomorphic irreducible. Note that the above
indenote by
the composition multiplicity ofdefinition of the Weyl module
coincides with the definition of the Weyl module
when $S^{b}(m, n)$ is isomorphic tounder the isomorphism in Proposition 4.8.
$0$$T\in \mathcal{T}_{0}^{+}(\lambda)$.

$W_{b}^{\lambda}=\{x\in W_{b}^{\lambda}|\langle x,$ $y\rangle_{b}=$$L_{b}^{\lambda}=W_{b}/radW_{b}^{\lambda_{i}}$$y\in W_{b}^{\lambda}$$\lambda\in \mathcal{P}_{n,r},$$\{L_{b}^{\lambda}|\lambda\in \mathcal{P}_{n,r}\}$$L_{b}^{\lambda}$$S^{b}$$[\dagger W_{b}^{\lambda} :$\lambda,$

L_{b}^{\mu}]$$L_{b}^{\mu}$$W_{b}^{\lambda}$$W_{b}^{\lambda}$$\overline{Z}^{\lambda}$$\overline{S^{0}}(\Lambda)$$b$$\mu\in \mathcal{P}_{n,r}$62Consequently, under the isomorphism , we havefor every. On the other hand, note that in the case where $r=1$ , the notation
for $S^{b}(m, n)$ coincides with the standard notation for q-Schur algebras discussed as in[Ma, Chapter 4]. So, we use freely such a notation. For, we denote byinis defined as the composition multiplicity of
for
.
and
$[W_{b}^{\lambda} :$b$$\lambda,$

L_{b}^{\mu}]=[\overline{Z}^{\lambda} :\overline{L}^{\mu}]$$\mu\in \mathcal{P}_{n,r}$$\lambda,$$[W^{\lambda^{(\iota)}} :$\mu\in \mathcal{P}_{n,r}$L^{4}(t)](1\leq i\leq r)$$L^{\mu^{(t)}}$$W^{\lambda^{(i)}}$$l^{l}=(’ 4(\iota), \ldots, ’\iota^{(r)})$$\lambda=(\lambda^{(1)}, \ldots, \lambda^{(r)})$

Proposition 5.2 ( $[SawS$ , Proposition 5.14]). Let
and that (4.1) and (4.3) are satisfied. Let
in Theorem 4.5, we have$\lambda,$$\Lambda=\tilde{\mathcal{P}}_{n_{1}r}$$\mu\in \mathcal{P}_{n_{1}r}$. Suppose that $R$ is a field,. Then under the isomorphism$[W^{\lambda}:L^{\mu}]=\{\begin{array}{ll}\prod_{i=1}^{r}[W^{\lambda^{(\cdot)}}:L^{\mu^{(i)}}] if \alpha(\lambda)=\alpha(\mu),0 otherwise.\end{array}$Corollary 5.3. Let
satisfied. Then, for all$\Lambda=\tilde{\mathcal{P}}_{n_{2}r}$$\lambda,$. Suppose that $R$ is a field, and that (4.1) and (4.3) are$\mu\in \mathcal{P}_{n_{1}r}$with$\alpha(\lambda)=\alpha(\mu)$, we have$[W^{\lambda}:L^{\mu}]= \prod_{i=1}^{r}[W^{\lambda^{(\cdot)}}:L^{\mu^{(\cdot)}}]$.

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$(\mathbb{Z}r\mathbb{Z})l\mathfrak{S}_{r\iota}$

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