Cell algebra structures on monoid and twisted monoid algebras

0.2 MB·English

by  Robert D. May

Cell algebra stuctures on monoid and twisted
5
1
monoid algebras
0
2

y
Robert May
a
M

5 1 Introduction
2

] In [11] a class of algebrascalled cell algebrasis defined generalizingthe cellular
A
algebras of Graham and Lehrer in [7]. (These algebras had previously been
R introduced and studied as “standardly based algebras” by Du and Rui in [4].)
. It was shown in [11] that if R is any commutative domain with unit and M
h
is either a transformation semigroup T or a partial transformation semigroup
t r
a PT then the semigroup algebra R[M] is a cell algebra with a standard cell
m r
basis. In this paper we show that the algebra R[M] is a cell algebra for any
[ finitemonoidM withtheproperty(calledthe“R-C.A.”property)thatforevery

2 D-class D the group algebra R[GD] for the Schutzenberger group of D has a
v cell algebra structure. The choice of a cell basis for each R[GD] determines a
4 standardcell basis for R[M]. If allthe groupsG for a givenM aresymmetric
D
3
groups (as for T or PT ) then M satisfies the R-C.A. property for any R. If
r r
3
k is analgebraicallyclosedfield of characteristic0 or p where p does not divide
4
0 the order of any GD, then any finite monoid M satisfies the k-C.A. property.
. We use the standard cell basis obtained for M and the properties of cell
1
algebrasgivenin[11]toderivepropertiesofthealgebrasR[M]forsuchmonoids.
0
5 For example, if M is a regular semigroup and each cell algebra R[GD] has the
1 property that (Λ ) = Λ then R[M] is quasi-hereditary. As a special case,
D 0 D
: this yields the theorem of Putcha [13] that the complex monoid algebra C[M]
v
i is quasi-hereditary for any finite regular monoid. If M is an inverse semigroup
X
satisfying the R-C.A. property, then we show that R[M] is semi-simple if and
r onlyifeachcellalgebraR[G ]issemi-simple. WhenRisanalgebraicallyclosed
a D
fieldk ofgoodcharacteristic,thisyieldsthatk[M]issemi-simpleforanyinverse
semi-group(aresultlongknownevenwithoutthealgebraicallyclosedcondition;
see [1] which cites [12]).
We also show that if π : M ×M → R is any twisting satisfying a certain
simple compatibility condition then the twisted monoid algebra Rπ[M] is also
a cell algebra (with the same sets Λ,L,R and cell basis C as the cell algebra
R[M]).
In [5], [14], and [9], East, Wilcox, and Guo and Xi have explored analogous
results for cellular algebra structures on semigroups and twisted semigroups.
Many of the complications in their work arise from the need to construct the
1
involution anti-automorphism∗ required by the definition of a cellular algebra.
Cellalgebrasrequirenosuchmapping,andtheresultsobtainedforsuchalgebras
appear to be both considerably simpler and of more general applicability than
the corresponding result for cellular algebras. Yet questions such as whether
an algebra is quasi-hereditary or semi-simple appear to be not much harder to
address when given a cell basis than when given a cellular basis.
2 Properties of finite monoids
In this section we review some facts about finite monoids and the standard
Green’s relations on a finite monoid M. Most of these results are well-known;
see, for example, [1], [6], or [10] for references.
Green’s relations L, R, J, H and D are defined, for x,y ∈M, by
xLy ⇔Mx=My, xRy ⇔xM =yM , xJy ⇔MxM =MyM
xHy ⇔xLy and xRy
xDy ⇔xLz and zRy for some z ∈M ⇔xRz and zLy for some z ∈M.
IfKisoneofGreen’srelationsandm∈M,denotebyK theK-classcontaining
m
m. Write L, R, J, H, or D for the set of all L, R, J, H, or D classes in M.
According to “Green’s lemma”, if aRb and s,t ∈ M are such that as =
b, bt=a, then the map x→xs is a bijective, R-class preserving map from L
a
onto L with inverse the map y →yt. There is a dual result when aLb
b
For a finite monoid M, one has D = J (see chapter 5 of [8]) and the set D
of D-classes inherits a partial order defined, for x,y ∈ M, by D 6 D ⇔ x ∈
x y
MyM. Evidently D 6D and D 6D for any x,y ∈M.
xy x xy y
Let R be a commutative domain with unit. For any subset S ⊆ M, write
R[S]forthefreeR-modulewithbasisS. LetA=R[M],themonoidalgebrafor
M, and define AD = ⊕D′6DR[D′], AˆD = ⊕D′<DR[D′]. Then both AD and
AˆD are two sided ideals in A.
For a givenD ∈D, choosea “baseelement” γ ∈D andconsider the H-class
H = H ⊆ D. Let RT(H) = {m∈M :Hm⊆H} and for m ∈ RT(H) define
γ
the right translation r : H → H by (h)r = hm, h ∈ H. Each map r is
m m m
a bijection from H to H and GR = {r :m∈RT(H)} is a group, the (right)
H m
Schutzenberger group for H. (Up to isomorphism, GR depends only on D, not
H
theparticularH-classH containedinD.) Evidentlyr r =r foranym,n∈
m n mn
RT(H). The map φR : GR → H defined by φR : r 7→ γ m is bijective and
H H H m D
extendslinearlytoabijectivemapφR :R GR →R[H]offreeR-modules. For
H (cid:2) H(cid:3)
anym,n∈RT(H),φR(r )·n=(γ m)·n=γ (mn)=φR (r )=φR (r r ),
H m D D H mn H m n
so in general for any x∈R GR we have by linearity φR (x)·n=φR (xr ).
(cid:2) H(cid:3) H H n
In parallel fashion, let LT(H) = {m∈M :mH ⊆H} and for m ∈ LT(H)
define the left translation l : H → H by l (h) = mh, h ∈ H. Then again
m m
each map l is a bijection from H to H and GL = {l :m∈LT(H)} is a
m H m
group,the (left, ordual)SchutzenbergergroupforH. Thenl l =l for any
m n mn
m,n∈LT(H), the map φL :GL →H defined by φL :l 7→mγ is bijective,
H H H m D
2
and the linearly extended map φL : R GL → R[H] is a bijection of free R-
H (cid:2) H(cid:3)
modules. The mapψ = φR −1◦φL :GL →GR is thenalsobijective. Choose
(cid:0) H(cid:1) H H H
an (injective) map ϑ: GR →RT(H) such that r = g for all g ∈GR. Then
H ϑ(g) H
foranym∈LT(H)definem¯ ∈RT(H)bym¯ =ϑ◦ψ(l ),sothatmγ =γ m¯.
m D D
For any m∈LT(H), n∈RT(H) we have m·φR (r )=m·γ n=mγ ·n=
H n D D
γ m¯ ·n = φR (r r ). Then for any x ∈ R GR , m ∈ LT(H) we have by
D H m¯ n (cid:2) H(cid:3)
linearity mφR (x)=φR (r x).
H H m¯
Lemma 2.1. Take any m∈M.
i. If there exists an h ∈ H such that hm ∈ H then m ∈ RT(H) and r
m
gives a bijective map of H onto H.
ii. If there exists an h∈H such that mh∈Hthen m∈LT(H) and l gives
m
a bijective map of H onto H.
Proof. For i., write h = m ·γ , hm = m ·γ for some m ,m ∈ LT(H).
1 D 2 D 1 2
Then l ∈ GL and we can choose m¯ ∈ LT(H) such that l = (l )−1.
m1 H 1 m¯1 m1
Then γ =m¯ ·h. Any element of H has the form n·γ for some n∈LT(H),
D 1 D
andwehavenγ ·m=n·(m¯ h)·m=nm¯ (hm)=nm¯ (m γ )=(nm¯ m )·γ .
D 1 1 1 2 D 1 2 D
But nm¯ m ∈ LT(H) (since n,m¯ ,m are), so nγ ·m = (nm¯ m )·γ ∈ H.
1 2 1 2 D 1 2 D
Then m∈RT(H) as desired. The proof of ii. is parallel.
Wenowinvestigategeneralproductsam, mawherea∈D ∈Dandm∈M.
As mentioned above, we have D 6 D and D 6 D . We need to study
am a ma a
the cases when D = D or D = D . The first tool is the following result
am a ma a
(see e.g chapter 5 of [8]):
Lemma 2.2. For any finite monoid M and a,m∈M,
i If D =D , then R =R .
a am a am
ii If D =D , then L =L .
a ma a ma
Proof. We use the facts (again, see chapter 5 of [8]) that in any finite monoid
D = J and that for any element x in a finite monoid some power xn is idem-
potent. For i., assume D = D for some a,m ∈ M. To prove R = R , it
a am a am
suffices to check that a ∈amM and am∈ aM. Since am∈ aM is obvious, we
need show only a ∈ amM. Now D = D ⇒ J = J ⇒ a ∈ MamM ⇒
a am a am
a = x(am)y = xa(my) for some x,y ∈ M. Then by repeated substitutions we
have a = xka(my)k for any positive integer k. Choose k = n so that xn is an
idempotent e. Then a = ea(my)n and ea = e·ea(my)n = ea(my)n = a. But
then a = a(my)n = am· y(my)n−1 ∈ amM as desired. The proof of ii. is
similar.
We also use the “egg-box” picture of a given fixed class D ∈D:
Picture the H-classes contained in D in a rectangular array where the H-
classes in a given column are all in the same L-class and the H-classes in a
givenroware allin the same R-class. So the number of columns in the arrayis
3
the number n(D,L) of distinct L-classes in D, while the number of rows is the
number n(D,R) of distinct R-classes in D. Write H for the H-class in row i,
i j
column j. We can assume that H is our “base class” H .
1 1 γ
The following result follows from Green’s lemma and its dual:
Lemma 2.3. For each i ∈ {1,2,··· ,n(D,R)} , j ∈ {1,2,··· ,n(D,L)} there
exist elements a ,a¯ ,b ,¯b ∈M such that
i i j j
i. For each column j, the left translation h 7→ a h defines a bijective map
i
l : H → H with inverse l−1 : H → H defined by left translation
i 1 j i j i i j 1 j
l−1 :h7→a¯ h.
i i
ii. For each row i, the right translation h 7→ hb defines a bijective map
j
r : H → H with inverse r−1 : H → H defined by right translation
j i 1 i j j i j i 1
r−1 :h7→h¯b .
j j
Then for any row i, column j, and h ∈ H , h 7→ a hb gives a bijective
γ i j
map ψ : H = H → H of H-classes which extends linearly to a bijection
i j γ 1 1 i j
ψ :R[H ]→R[ H ].
i j γ i j
We can now state the main result of this section.
Proposition 2.1. Suppose that for some d∈D,m∈M we have dm∈D. For
some i,j, this d ∈ H = ψ (H ) = a H b and so d = a φR(g )b for some
i j i j γ i γ j i H d j
g ∈GR. Then:
d H
i. dm∈ H for some k. Define m∗ =b m¯b .
i k j k
ii. m∗ ≡b m¯b ∈RT(H).
j k
iii. The right translation r :h7→hm gives a bijection r : H → H .
m m i j i k
iv. If h=aiφRH(g)bj ∈iHj, then hm=(h)rm =aiφRH(grm∗)bk.
v. For any x ∈ R[ H ], write x = a φR (y)b for some y ∈ R GR . Then
i j i H j (cid:2) H(cid:3)
xm=aiφRH(yrm∗)bk.
Proof. SinceD =D =D,wehaveR =R bylemma2.2. SoH and H
d dm d dm d dm
lie in the same R-class and are in the same row of the “egg-box” for D. So if
d∈ H , thendm∈ H forsome k, provingi. Write d=a φR(g )b anddm=
i j i k i H d j
a φR(g )b forsomeg ,g ∈GR. Thendm=a φR(g )b m=a φR(g )b .
i H dm k d dm H i H d j i H dm k
Multiplying by a¯ on the left and by ¯b on the left gives φR(g )b m¯b =
i k H d j k
φR(g ) ∈ H. So by lemma 2.1, m∗ ≡ b m¯b ∈ RT (H) proving ii. Then
H dm j k
rm∗ : H →H is a bijection φRH(g)m∗ = φRH(grm∗) for any g ∈ GRH. Now take
any h=a φR(g)b ∈ H and compute hm¯b =a φR(g)b m¯b =a φR(g)m∗ =
i H j i j k i H j k i H
aiφRH(grm∗) Multiplying on the right by bk then proves iv., and v. then follows
by linearity. For iii., observe that r can be written as a composition of bijec-
m
tions: rm = iψk ◦φRH ◦r¯m∗ ◦(cid:0)φRH(cid:1)−1 ◦(iψj)−1 where r¯m∗ : GRH → GRH is the
bijection g 7→grm∗.
As an immediate corollary we have
4
Corollary 2.1. For any H-class H ⊆D and any m∈M either
i j
i. R[iHj]·m⊆AˆD =⊕D′<DR[D′] or
ii. R[ H ]·m = R[ H ] for some k and if x = a φR (y)b for some y ∈
i j i k i H j
R(cid:2)GRH(cid:3), then xm=aiφRH(yrm∗)bk, where m∗ =bjm¯bk ∈RT(H).
Weobtainthedualversionsofproposition2.1andcorollary2.1givennextby
usingpartii. oflemmas2.1and2.2alongwiththeformulamφR(g)=φR (r g)
H H m¯
for g ∈GR, where m¯ =ϑ◦ψ(l )∈RT(H).
H m
Proposition 2.2. Suppose that for some d∈D,m∈M we have md∈D. For
some i,j, this d ∈ H = ψ (H ) = a H b and so d = a φR(g )b for some
i j i j γ i γ j i H d j
g ∈GR. Then:
d H
i. md∈ H for some k. Define m∗ =a¯ ma .
k j k j
ii. m∗ ≡a¯kmaj ∈LT(H), so m¯∗ =ϑ◦ψ(lm∗)∈RT(H).
iii. The left translation l :h7→mh gives a bijection l : H → H .
m m i j k j
iv. If h=aiφRH(g)bj ∈iHj, then mh=lm(h)=akφRH(rm¯∗g)bj.
v. For any x ∈ R[ H ], write x = a φR (y)b for some y ∈ R GR . Then
i j i H j (cid:2) H(cid:3)
mx=akφRH(rm¯∗ y)bj.
Corollary 2.2. For any H-class H ⊆D and any m∈M either
i j
i. m·R[iHj]⊆AˆD =⊕D′<DR[D′] or
ii. m·R[ H ] = R[ H ] for some k and if x = a φR (y)b for some y ∈
i j k j i H j
R(cid:2)GRH(cid:3), then mx = akφRH(rm¯∗ y)bj, where m∗ ≡ a¯kmaj ∈ LT (H) and
m¯∗ =ϑ◦ψ(lm∗)∈RT(H).
3 Cell algebra structures on monoid algebras
In [11], a class of algebras called cell algebras is defined which generalize the
cellularalgebrasofGrahamandLehrer[7]. Thesealgebrashadpreviouslybeen
introduced and studied as “standardly based algebras” by Du and Rui in [4].
Such algebras share many of the nice properties of cellular algebras. We will
give conditions on a monoid M and domain R such that the monoid algebra
R[M] will be a cell algebra and will construct a standard cell basis for such
algebras. We first review the definition.
Let R be a commutative integral domain with unit 1 and let A be an
associative, unital R-algebra. Let Λ be a finite set with a partial order >
and for each λ ∈ Λ let L(λ),R(λ) be finite sets of “left indices” and “right
indices”. Assume that for each λ ∈ Λ,s ∈ L(λ), and t ∈ R(λ) there is
an element Cλ ∈ A such that the map (λ,s,t) 7→ Cλ is injective and
s t s t
C = Cλ :λ∈Λ,s∈L(λ),t∈R(λ) is a free R-basis for A. Define R-
submo(cid:8)dsuletsofAbyAλ =R - span of {(cid:9)Cµ :µ∈Λ,µ>λ,s∈L(µ),t∈R(µ)}
s t
and Aˆλ =R - span of { Cµ :µ∈Λ,µ>λ,s∈L(µ),t∈R(µ)}.
s t
5
Definition 3.1. Given (A,Λ,C), A is a cell algebra with poset Λ and cell basis
C if
i For any a ∈ A,λ ∈ Λ, and s,s′ ∈ L(λ), there exists r = r (a,λ,s,s′) ∈ R
L L
such that, for any t∈R(λ), a· sCtλ = rL· s′Ctλ modAˆλ, and
s′∈PL(λ)
ii For any a ∈ A,λ ∈ Λ, and t,t′ ∈ R(λ), there exists r = r (a,λ,t,t′) ∈ R
R R
such that, for any s∈L(λ), Cλ·a= r · Cλ modAˆλ .
s t t′∈PR(λ) R s t′
NowforeachD-classD inafinite monoidM,chooseabaseelementγ and
D
base H-class H = H as above. Then define the Schutzenberger group G of
γ D
D to be G = GR. (Up to isomorphism, this is independent of the choice of
D H
base class H.) Let R[G ] be the group algebra of G over R.
D D
Definition 3.2. A monoid M satifies the R-C.A. condition for a given domain
R if R[G ] has a cell algebra structure for every D-class D in M.
D
IfanH-classH containsanidempotent,thenH isactuallyasubgroupofM.
In fact, the maximal subgroups of M are just the H-classes which contain an
idempotent. Inthis casethe groupH isisomophic to the Schutzenbergergroup
GR. If M is a regular semigroup,then everyD-class D containsanidempotent
H
e, so G is isomorphic to a maximal subgroup H of M. Then for regular
D e
semigroupsM the R-C.A.conditionis equivalent to requiringthat R[G]havea
cell algebra structure for every maximal subgroup G of M.
If M is a monoid(such as a transformationsemigroupT or a partialtrans-
r
formation semigroup PT ) for which every group G is a symmetric group,
r D
then the usual Murphy basis gives a cellular (and hence cell) algebra structure
to R[G ] for any domain R. Thus M satisfies the R-C.A. condition for any R.
D
For any finite monoid M, if k is a field of characteristic 0 or characteristic
p where p does not divide the order of any G , then by Maschke’s theorem,
D
every k[G ] is semisimple. If, in addition, k is algebraically closed, then each
D
k[G ] is split semisimple. Such algebrasare products ofmatrix algebrasoverk
D
and have a natural cellular (hence cell) algebra basis. Thus if k is algebraically
closedandofgoodcharacteristicrelativeto M,thenanyM satisfiesthe k-C.A.
condition.
Our main result is the following theorem.
Theorem 3.1. Let M be a finite monoid satifying the R-C.A. condition for a
domain R. Then A=R[M] is a cell algebra. The choice of a cell basis for each
algebra R[G ] gives rise to a standard cell basis for A.
D
Proof. For a given D ∈ D, put A = R[G ] and assume Λ ,L ,R define a
D D D D D
cell algebra structure on A with cell basis
D
C = Cλ :λ∈Λ ,s∈L (λ),t∈R (λ) .
D (cid:8)s t D D D (cid:9)
Define a poset Λ to consist of all pairs (D,λ) where D is a D-class in M and
λ ∈ Λ . Define the partial order by (D ,λ ) > (D ,λ ) if D < D or D =
D 1 1 2 2 1 2 1
6
D and λ > λ in Λ . For (D,λ) ∈ Λ, define L(D,λ) to be all pairs (R,s)
2 1 2 D1
whereRisanR-classcontainedinD ands∈L (λ). Similarly,defineR(D,λ)
D
tobeallpairs(L,t)whereLisanL-classcontainedinDandt∈R (λ). Finally,
D
given (D,λ) ∈Λ, (R,s)∈ L(D,λ), (L,t) ∈ R(D,λ), assume R corresponds to
row i and L corresponds to column j in the “egg-box” for D. Then define
C(D,λ) = ψ φR Cλ =a φR Cλ b .
(R,s) (L,t) i j(cid:0) H(cid:0)s t(cid:1)(cid:1) i H(cid:0)s t(cid:1) j
For fixed D,R,L, since Cλ :λ∈Λ ,s∈L (λ),t∈R (λ) is a basis for
(cid:8)s t D D D (cid:9)
R[G ] by assumption and ψ and φR are bijective,
D i j H
C(D,λ) :λ∈Λ ,s∈L (λ),t∈R (λ)
n(R,s) (L,t) D D D o
will give a basis for R[ H ]. Then since A=R[M] is the direct sum of the free
i j
submodules R[H] as H varies over all H-classes in M,
C = C(D,λ) :(D,λ)∈Λ,(R,s)∈L(D,λ),(L,t)∈R(D,λ)
n(R,s) (L,t) o
is a basis for A. To show C is a cell basis, we must check the cell conditions (i)
and (ii). Note first that any element m in a D-class D will be in the span of
m
C(Dm,µ) :µ∈Λ ,(R,s)∈L(D ,µ),(L,t)∈R(D ,µ) . ThenifD <
n(R,s) (L,t) Dm m m o m
D for some D-classD, we have(D ,µ)>(D,λ) for any µ∈Λ andλ∈Λ ,
m Dm D
som∈Aˆ(D,λ). So⊕D′<DR[D′]=AˆD ⊆Aˆ(D,λ). Toprove(i),wecanassumea
is a basis element m∈M. Take (D,λ)∈Λ, (R,s)∈L(D,λ), (L,t)∈R(D,λ)
andassumeR correspondsto rowi andL correspondsto columnj inthe “egg-
box” for D. Then C(D,λ) = ψ φR Cλ =a φR Cλ b ∈R[ H ]. By
(R,s) (L,t) i j(cid:0) H(cid:0)s t(cid:1)(cid:1) i H(cid:0)s t(cid:1) j i j
corollary 2.2 there are two cases to consider.
Case i: m·R[iHj] ⊆ ⊕D′<DR[D′] . Then m·(R,s) C((LD,,tλ)) ∈ m·R[iHj] ⊆
⊕D<D′R[D′]⊆Aˆ(D,λ) and we can satisfy (i) by taking all coefficients rL to be
zero.
Caseii: m·R[ H ]=R[ H ]forsomek. Bycorollary2.2,since C(D,λ) =
i j k j (R,s) (L,t)
aiφRH(cid:0)sCtλ(cid:1)bj ∈ R[iHj], then m (R,s)C((LD,,tλ)) = akφRH(cid:0)rm¯∗sCtλ(cid:1)bj, where m∗ ≡
a¯kmaj ∈ LT(H) and m¯∗ = ϑ◦ψ(lm∗) ∈ RT(H). Since g = rm¯∗ ∈ GD, the
cell algebra property (i) for R[GD] gives g· sCtλ = s′∈LPD(λ)rL· s′Ctλ modAˆλD,
where r depends on s,s′,λ, and m, but is independent of t. Then
L
m C(D,λ) =a φR g Cλ b
(R,s) (L,t) k H(cid:0) s t(cid:1) j
= X rL·akφRH(cid:0)s′Ctλ(cid:1)bj modulo akφRH(cid:16)AˆλD(cid:17)bj
s′∈LD(λ)
where
a φR Aˆλ b ⊆span C(D,µ) :µ>λ,s∈L (µ),t∈R (µ) ⊆Aˆ(D,λ).
k H(cid:16) D(cid:17) j n(R,s) (L,t) D D o
7
But akφRH(cid:0)s′Ctλ(cid:1)bj = (R′,s′)C((LD,,tλ)) where R′ is the R-class corresponding to
row k of the “egg-box”. Then m· (R,s)C((LD,,tλ)) = s′∈LPD(λ)rL·akφRH(cid:0)s′Ctλ(cid:1)bj =
s′∈LPD(λ)rL· (R′,s′)C((LD,,tλ)) modAˆ(D,λ). Since rL is independent of L and t, this
yields property (i) for this case ii.
The proof of condition (ii) is parallel. Thus C is a cell basis and A=R[M]
is a cell algebra.
If M is a finite monoid satisfying the R-C.A. condition, we will assume a
fixed cell algebra structure is given to each R[G ]. We will then call the cell
D
algebrastructure obtained in the proof of theorem 3.1 the standard cell algebra
structure on R[M].
In [5], [14], and [9], East, Wilcox, and Guo and Xi worked with finite regu-
lar semigroups with cellular algebra (hence cell algebra) structure on R[G] for
maximal subgroups G of M. Their examples therefore satisfy the R-C.A. con-
dition and can be seen to be cell algebras without considering the complicated
involutionrequirements involvedin showing a cellular algebra structure. In [5],
Eastgives examples of inversesemigroups with cellular R[G] for all maximal G
which lack an appropriate involution and therefore do not have cellular R[M].
These examples would be cell algebras by theorem 3.1.
More typical examples of cell algebras that are not cellular are the algebras
R[M]forM atransformationsemigroupT orpartialtransformationsemigroup
r
PT . Thesewereshowntobecellalgebrasin[11]. Fortheseexamples,aD-class
r
D consists of mappings of a given rank i and the group G is the symmetric
D
groupG . Since, as remarkedabove,the symmetric groupshavecellular (hence
i
cell) structures on their groupalgebras,theorem 3.1 applies and provides a cell
algebra structure on M.
4 Properties of the cell algebra A = R[M]
In this section we assume that M is a finite monoid satisfying the R-C.A. con-
dition, that is, such that for every D ∈ D the group algebra R[G ] of the
D
Schutzenberger group for D is a cell algebra. Then by Theorem3.1, A=R[M]
is acell algebraandwe canapply the resultsin [11]to A withthe standardcell
algebra structure.
For (D,λ) ∈ Λ, the left cell module C(D,λ) is a left A-module which is a
L
free R-module with basis C(D,λ) :(R,s)∈L(D,λ) . Similarly, the right
(cid:8)(R,s) (cid:9)
cell module C(D,λ) for (D,λ) is a right A-module and a free R-module with
R
(D,λ)
basis C :(L,t)∈R(D,λ) . For each (D,λ) ∈ Λ there is an R-bilinear
n (L,t) o
map h−,−i : (cid:16)CR(D,λ), LC(D,λ)(cid:17)→R defined by the property (cid:16)(R′,s′)C((LD,,tλ))(cid:17)·
(cid:16)(R,s)C((LD′,,λt′))(cid:17)=DC((LD,,tλ)), (R,s)C(D,λ)E (R′,s′)C((LD′,,λt′)) mod Aˆ(D,λ) foranychoice
of R′,s′,L′,t′.
8
Right and left radicals are defined by
rad C(D,λ) = x∈C(D,λ) :hx,yi=0 for all y ∈ C(D,λ)
(cid:16) R (cid:17) n R L o
rad C(D,λ) = y ∈ C(D,λ) :hx,yi=0 for all x∈C(D,λ) .
(cid:16)L (cid:17) n L R o
Then define D(D,λ) = CR(D,λ) and D(D,λ) = LC(D,λ) . Finally, define
R rad(cid:16)CR(D,λ)(cid:17) L rad(LC(D,λ))
Λ = (D,λ)∈Λ:hx,yi6=0 for some x∈C(D,λ),y ∈ C(D,λ) . Evidently,
0 n R L o
λ∈Λ ⇔D(D,λ) 6=0⇔ D(D,λ) 6=0. A major result of [11] is
0 R L
Theorem 4.1. Assume R=k is a field. Then
(a) D(D,µ) :(D,µ)∈Λ is a complete set of pairwise inequivalent irreducible
n R 0o
right A-modules and
(b) D(D,µ) :(D,µ)∈Λ is a complete set of pairwise inequivalent irre-
L 0
(cid:8) (cid:9)
ducible left A-modules.
To find Λ for A we need to relate the bracket h−,−i on A to the brackets
0
h−,−i on the cell algebras A = R[G ]. For a given D-class D, write R
D D D i
for the R-class corresponding to row i of the “egg-box” and L for the L-class
j
corresponding to column j.
Definition 4.1. R , L are matched if there exist x ∈ L , y ∈ R such that
i j j i
xy ∈D. R , L are unmatched if L R ∩D =∅.
i j j i
Lemma 4.1. Assume Ri, Lj are matched. Write x=ai′φRH(ξ)bj ∈R[Lj] and
y = aiφRH(η)bj′ ∈ R[Ri] for any ξ,η ∈ R[GD]. Then xy = ai′φRH(cid:0)ξrm(i,j)η(cid:1)bj′
where m(i,j)≡b a γ ∈RT(H).
j i
Proof. If R , L are matched, then for some choice of i′,j′ and some elements
i j
x ∈ i′Hj ⊆ Lj, y ∈ iHj′ ⊆ Ri we have xy ∈ i′Hj′ ⊆ D. But then (for the
given i′,j′) we have xy ∈i′Hj′ ⊆D for any x∈i′Hj, y ∈iHj′ by propositions
2.1 and 2.2. In particular, for x = ai′γbj ∈ i′Hj, y = aiγbj′ ∈ iHj′ we get
xy = ai′γbjaiγbj′ ∈ i′Hj′. Then multiplying by a¯i′ on the left and ¯bj′ on the
right gives γb a γ ∈H. But then m(i,j)≡b a γ ∈RT(H) by lemma 2.1.
j i j i
Now consider any i′,j′ and any g =r ,h=r ∈G and let
m n D
x=ai′φRH(g)bj ∈R[Lj], y =aiφRH(h)bj′ ∈R[Ri].
Then
xy =(ai′φRH(g)bj)(aiφRH(h)bj′)=ai′γmbjaiγnbj′
=ai′γm·m(i,j)·nbj′ =ai′φRH(cid:0)rmrm(i,j)rn(cid:1)bj′
=ai′φRH(cid:0)grm(i,j)h(cid:1)bj′.
Then by linearity of φRH, xy = ai′φRH(cid:0)ξrm(i,j)η(cid:1)bj′ for arbitrary ξ,η ∈ R[GD]
when x=ai′φRH(ξ)bj ∈R[Lj], y =aiφRH(η)bj′ ∈R[Ri], provingthe lemma.
9
For i∈{1,2,··· ,n(D,R)}, define C(D,λ) to be the free R-module with
(cid:0)L (cid:1)i
basis C(D,λ) :λ∈Λ ,s∈L (λ) , so C(D,λ) = ⊕ C(D,λ) . Simi-
(cid:8)(Ri,s) D D (cid:9) L i (cid:0)L (cid:1)i
(D,λ)
larly, for j ∈ {1,2,··· ,n(D,L)}, define C to be the free R-module
(cid:16) R (cid:17)
j
with basis C(D,λ) :λ∈Λ ,t∈R (λ) , so C(D,λ) = ⊕ C(D,λ) . Notice
n (Lj,t) D D o R j (cid:16) R (cid:17)j
that for each i, φ : C(D,λ) 7→ Cλ gives an isomorphism (of R-modules)
i (Ri,s) s
φ : C(D,λ) → Cλ. Similarly, φ : C(D,λ) 7→ Cλ gives an isomorphism
i (cid:0)L (cid:1)i L j (Lj,t) t
φ : C(D,λ) →Cλ.
j (cid:16) R (cid:17) R
j
Proposition 4.1. Take X ∈ C(D,λ) , Y ∈ C(D,λ) . Then
(cid:16) R (cid:17)j (cid:0)L (cid:1)i
(a) If R , L are not matched, then hX,Yi=0,
i j
(b) If R , L are matched, then
i j
hX,Yi= φ (X),r φ (Y) = r φ (X),φ (Y)
(cid:10) j m(i,j) i (cid:11)D (cid:10) m(i,j) j i (cid:11)D
where m(i,j)≡b a γ ∈RT(H).
j i
Proof. It suffices to check (a) and (b) for basis elements X,Y, so assume X =
C((LDj,,λt)), Y = (Ri,s)C(D,λ) andputx=(Ri′,s′) C((LDj,,λt)) =ai′φRH(cid:0)s′Ctλ(cid:1)bj ∈Lj and
y =(Ri,s) C((LDj,′λ,t)′) =aiφRH(cid:0)sCtλ′(cid:1)bj′ ∈Ri. Thenxy =hX,Yi (Ri′,s′)C((LDj,′λ,t)′) mod
Aˆ(D,λ).
If R , L are not matched, then xy ∈AˆD ⊆ Aˆ(D,λ), so hX,Yi= 0, proving
i j
(a).
If R , L are matched, then, by lemma 4.1,
i j
xy =ai′φRH(cid:0)s′Ctλ·rm(i,j)· sCtλ′(cid:1)bj′
=ai′φRH(cid:0)(cid:10)Ctλ,rm(i,j)· sCλ(cid:11)D s′Ctλ′(cid:1)bj′ modAˆ(D,λ)
=(cid:10)Ctλ,rm(i,j)· sCλ(cid:11)D·ai′φRH(cid:0)s′Ctλ′(cid:1)bj′ modAˆ(D,λ)
=(cid:10)φj(X),rm(i,j)·φi(Y)(cid:11)D·(Ri′,s′)C((LDj,′λ,t)′) modAˆ(D,λ).
ThisgiveshX,Yi= φ (X),r φ (Y) = r φ (X),φ (Y) , proving
(cid:10) j m(i,j) i (cid:11)D (cid:10) m(i,j) j i (cid:11)D
part (b).
We note the following corollary for future use.
Corollary 4.1. Let M be a finite monoid satisfying the R-C.A. condition and
placethestandardcellalgebrastructureonR[M]. Thenforanyλ∈Λ , D ∈D:
D
1. If rad Cλ 6=0, then rad C(D,λ) 6=0,
D L L
(cid:0) (cid:1) (cid:0) (cid:1)
2. If rad Dλ 6=0, then rad D(D,λ) 6=0.
D(cid:0) R(cid:1) (cid:16) R (cid:17)
10