Identities of graded simple algebras

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by  Dušan D. Repovš

IDENTITIES OF GRADED SIMPLE ALGEBRAS
DUSˇANREPOVSˇ ANDMIKHAILZAICEV
7
1
0 Abstract. We study identities of finite dimensional algebras over a field of
2 characteristiczero,gradedbyanarbitrarygroupoidΓ. Firstweprovethatits
n graded colength has a polynomially bounded growth. For any graded simple
a algebraA weprove theexistence ofthe gradedPI-exponent, providedthatΓ
J isacommutativesemigroup. IfAissimpleinanon-gradedsensetheexistence
ofthegradedPI-exponent isprovedwithoutanyrestrictionsonΓ.
6

]
A
R
1. Introduction
.
h
Westudynumericalcharacteristicsofidentitiesoffinitedimensionalgradedsim-
t
a ple algebras over a field of characteristics zero. The main object of our investiga-
m
tions is the asymptotic behaviour of sequences of graded codimensions and graded
[ colengths of such algebras (all necessary definitions and notions will be given in

the next section). Given a graded algebra A, one can associate the sequence of so-
1
v called graded codimensions {cgnr(A),n = 1,2,...}. This sequence is an important
7 numericalinvariantofgradedidentitiesofA. Itisknownthatthissequenceisexpo-
7 nentially bounded, that is cgr(A) ≤an for some real a, provided that dimA<∞.
n
5
In this case the following natural question arises: does the limit
1
0 (1) expgr(A)= lim n cgr(A)
1. n→∞ q n
0 exist and what are its possible values? If the limit (1) exists then it is called the
7 graded PI-exponent of A.
1
In the non-graded case, codimension growth is well understood. Existence and
:
v integrality of the (non-graded) PI-exponent was conjectured by Amitsur in 1980’s
Xi for associative PI-algebras. Amitsur’s conjecture was confirmed in [1, 2]. Later
the same result was provedfor finite dimensional Lie algebras[3, 4, 5], Jordanand
r
a alternative algebras [6, 7, 8] and many other algebraic systems. In the general
nonassociative case, for any real α > 1, examples of algebras with PI-exponent
equal to α were constructed in [9]. Recently, the first example of algebra A such
that the PI-exponent of A does not exist, was constructed [10]. Nevertheless, for
any finite dimensional simple algebra the PI-exponent does exist [11].
For graded algebras there are only partial results of this kind. For example, if
A is an associative graded PI-algebra then its graded PI-exponent always exists
Date:January9,2017.
2010 Mathematics Subject Classification. Primary17B01,16P90; Secondary16R10.
Key words and phrases. Polynomial identities, graded algebras, codimensions, exponential
growth.
Thefirstauthor wassupportedbytheSRAgrantsP1-0292-0101, J1-7025-0101, J1-6721-0101
andJ1-5435-0101. Thesecondauthor waspartiallysupportedbytheRFBRgrant16-01-00113a.
Wethanktherefereeforcomments andsuggestions.
1
2 D.D.REPOVSˇANDM.V.ZAICEV
and it is an integer [12]. An existence of the Z -graded PI-exponent for any finite
2
dimensionalsimpleLiesuperalgebrahasrecentlybeenprovedin[13]. Notethatfor
finite dimensional Lie superalgebras, both graded and ordinary PI-exponents can
befractional[11,14,15]. Themainpurposeofthispaperistoprovetheexistenceof
gradedPI-exponentsforanyfinitedimensionalgradedsimplealgebra(seeTheorem
2).
AnotherimportantnumericalcharacteristicofidentitiesofanalgebraAistheso-
calledcolength sequence l (A). Except for its independent interest, asymptotic be-
n
haviourof{l (A)}playsanimportantroleinthestudiesofasymptoticsof{c (A)}.
n n
The polynomial type of growth of {l (A)} is very convenient for investigations of
n
codimension growth.
Polynomialupper boundsofthe colengthforanyassociativePI-algebrawerees-
tablished in [16]. For an arbitrary (non-associative) finite dimensional algebra the
same restriction was obtained in [17]. In the case of finite dimensional Lie super-
algebras, polynomial growth of Z -graded colength has recently been confirmed in
2
[18]. Inordertogetthe mainresultofthe paper wewillfind the polynomialupper
bound for graded colength of a finite dimensional graded algebra (see Theorem 1).
2. Preliminaries
Let Γ be a groupoid. An F-algebra A is said to be Γ-graded if there is a vector
space decomposition
A= A
g
Mg∈Γ
andA A ⊆A forallg,h∈Γ. Anelementa∈Aiscalledhomogeneous of degree
g h gh
gifa∈A andinthiscasewewritedeg a=g. AsubspaceV ⊆Aishomogeneous
g Γ
iff V = (V ∩A ). We call A graded simple if it has no homogeneous ideals.
g∈Γ g
For instaLnce, if Γ is a group and A=F[Γ] is its group algebra then A is Γ-graded
simple but is not simple in the usual sense. On the other hand, any simple algebra
with an arbitrary grading is graded simple.
We recall some key notions from the theory of graded and ordinary identities
andtheir numericalinvariants. We referthe readerto [19,20]for details. Consider
an absolutely free algebra F{X} with a free generating set
X = X , |X |=∞ for any g ∈Γ.
g g
g[∈Γ
One can define a Γ-gradingon F{X} by setting deg x=g, when x∈X , and ex-
Γ g
tendthisgradingtotheentireF{X}inthenaturalway. Apolynomialf(x ,...,x )
1 n
in homogeneous variables x ∈ X ,...,x ∈ X is called a graded identity of a
1 g1 n gn
Γ-graded algebra A if f(a ,...,a ) = 0 for any a ∈ A ,...,a ∈ A . The set
1 n 1 g1 n gn
Idgr(A) of all graded identities of A forms an ideal of F{X} which is stable under
graded homomorphisms F{X}→F{X}.
First, let Γ be finite, Γ = {g ,...,g } and X = X ... X . Denote by
1 t g1 gt
Pn1,...,nt the subspace of F{X} of multilinear polynomiaSls ofStotal degree n =
n +···+n in variables x(1),...,x(1) ∈ X ,...,x(t),...,x(t) ∈ X . Then the
1 t 1 n1 g1 1 nt gt
value
P
c (A)=dim n1,...,nt
n1,...,nt P ∩Idgr(A)
n1,...,nt
IDENTITIES OF GRADED SIMPLE ALGEBRAS 3
is called a partial codimension of A while
n
(2) cgr(A)= c (A)
n (cid:18)n ,...,n (cid:19) n1,...,nt
n1+·X··+nt=n 1 t
is called a graded codimension of A. Recall that the support of the grading is the
set
Supp A={g ∈Γ|A 6=0}.
g
Note that if Supp A6=Γ, say, Supp A={g ,...,g }, k <t, then the value
1 k
n P
(3) dim n1,...,nk
(cid:18)n ,...,n (cid:19) P ∩Idgr(A)
n1+·X··+nk=n 1 k n1,...,nk
coincides with (2). This allows us to consider (3) as the definition of the graded
codimension of A even if Γ is infinite, provided that Supp A={g ,...,g }.
1 k
For convenience, denote
P
(4) P (A)= n1,...,nk .
n1,...,nk P ∩Idgr(A)
n1,...,nk
Given 1 ≤ j ≤ k, consider the action of the symmetric group S on P
nj n1,...,nk
defined by
σf(...,x(j),...,x(j),...)=f(...,x(j) ,...,x(j) ,...).
1 nj σ(1) σ(nj)
ThenthespacesP andP (A)becomeF[H]-modules,whereH =S ×
n1,...,nk n1,...,nk n1
···×S . AnyF[H]-moduleP (A)isdecomposedintothesumofirreducible
nk n1,...,nk
F[H]-submodules and in the languages of group characters it can be written as
(5) χ (P (A))= m χ .
H n1,...,nk λ(1),...,λ(k) λ(1),...,λ(k)
λ(1)⊢n1X,...,λ(k)⊢nk
Here,χ isthecharacteroftheirreducibleH-representationdefinedbythe
λ(1),...,λ(k)
k-tuple (λ(1),...,λ(k)) of partitions λ(1) ⊢n ,...,λ(k) ⊢n and m is the
1 k λ(1),...,λ(k)
multiplicity of the corresponding F[H]-module in P (A). The integer
n1,...,nk
(6) l (A)= m
λ(1),...,λ(k) λ(1),...,λ(k)
λ(1)⊢n1X,...,λ(k)⊢nk
is called the partial colength, whereas the integer
(7) lgr(A)= l (A)
n n1,...,nk
n1+·X··+nk=n
is called the graded colength of A.
As it was mentionedin the introduction, gradedcodimensions are exponentially
bounded if A is finite dimensional. Namely,
(8) cgr(A)≤dn+1
n
where d = dimA (see [21] and also [7, Proposition 2]). This result was proved in
[7, 21] under the assumption that Γ is a finite group. The same argument is valid
foranarbitrarygroupoid. Relation(8)allowsustoconsiderupperandlowerlimits
of n cgr(A) and we can define the lower and the upper graded PI-exponents as
n
follopws:
expgr(A)=liminf n cgr(A), expgr(A)=limsup n cgr(A).
n n
n→∞ q n→∞ q
4 D.D.REPOVSˇANDM.V.ZAICEV
IfthelowerandtheupperlimitscoincidethenwealsodefinethegradedPI-exponent
by
expgr(A)=expgr(A)=expgr(A).
Representation theory of symmetric groups is a useful tool for studying asymp-
totics of codimension growth. Basic notions of S -representations can be found in
n
[22] and its application to PI-theory in [19, 20].
Recall that, given a partition λ ⊢ n, there is exactly one (up to isomorphism)
irreducibleS -representationdefinedbyλ. Itscharacteranddimensionaredenoted
n
by χ and χ (1) = d , respectively. For the group H = S × ··· × S any
λ λ λ n1 nk
irreduciblerepresentationisdefinedbythek-tupleofpartitionsλ(1) ⊢n ,...,λ(k) ⊢
1
n and its character and dimension are χ . Moreover,
k
λ(1),...,λ(k)
(9) χ (1)=d ···d ,
λ(1),...,λ(k) λ(1) λ(k)
respectively. In particular, (5) and (9) imply the equality
(10) c (A)=χ (P (A))(1)=
n1,...,nk H n1,...,nk
m d ···d .
λ(1),...,λ(k) λ(1) λ(k)
λ(1)⊢n1X,...,λ(k)⊢nk
Let d ≥ 1 be a fixed integer and let ν = (ν ,...,ν ) ⊢ m be a partition of m
1 q
with q ≤ d. Dimension of an irreducible F[S ]-module with the character χ is
m ν
closely connected with the following function:
1
Φ(ν)= .
(νm1)νm1 ···(νmd)νmd
Here we assume that ν =...=ν =0 in the case q <d and 00 =1. The values
q+1 d
Φ(ν)m and d are close in the following sense.
ν
Lemma 1. (see [11, Lemma 1]) Let m≥100. Then
Φ(ν)m
≤d ≤mΦ(ν)m.
md2+d ν
(cid:3)
We willusethe followingpropertyofΦ. Letν andρbe anytwopartitionsofm,
suchthatν =(ν ,...,ν ),ρ=(ρ ,...,ρ ),p,q≤dandq =porq =p+1,ρ =1.
1 p 1 q p+1
As before, we consider ρ and ν as partitions with d components. We say that the
Young diagramD is obtained fromdiagramD by pushing down one box if there
ρ ν
exist 1≤i<j ≤d such that ρ =ν −1, ρ =ν +1 and ρ =ν for all remaining
i i j j t t
1≤t≤d.
Lemma 2. (see [11, Lemma 3], [23, Lemma 2]) Let D be obtained from D by
ρ ν
pushing down one box. Then Φ(ρ)≥Φ(ν).
(cid:3)
3. Polynomial growth of graded colength
Consider a finite dimensional Γ-graded algebra A with the support Supp A =
{g ,...,g }, A=A ⊕···⊕A . Let
1 k g1 gk
d =dimA ,...,d =dimA
1 g1 k gk
IDENTITIES OF GRADED SIMPLE ALGEBRAS 5
be dimensions of the homogeneous components. Recall that an irreducible F[S ]-
t
modulecorrespondingtothepartitionµ⊢tcanberealizedasaminimalleftF[S ]-
t
ideal generated by an essential idempotent e where T is some Young tableaux
Tλ λ
with Young diagram D . For H = S ×···×S , any irreducible F[H]-module
λ n1 nk
is isomorphic to the tensor product of F[S ],...,F[S ]-modules with characters
n1 nk
χ ,...,χ ,respectively. Thefollowingremarkeasilyfollowsfromtheconstruc-
λ(1) λ(k)
tion of essential idempotents and therefore we omit the proof.
Lemma 3. Let λ(1) = (λ(1),...,λ(1)),...,λ(k) = (λ(k),...,λ(k)) be partitions of
1 q1 1 qk
n ,...,n , respectively. Suppose that the multiplicity m on the right hand
1 k λ(1),...,λ(k)
side of (5) is nonzero. Then q ≤d ,...,q ≤d .
1 1 k k
(cid:3)
For convenience we shall assume as before that q =d ,...,q =d even if q is
1 1 k k i
strictly less than d for some i.
i
Denote by R = R{X ∪...∪X } the relatively free algebra of the variety
var A of Γ-graded algebrga1s generatedgkby A. Denote by Rn1,...,nk the subspace of
d1,...,dk
polynomials in R of degree n in the set of variables {X(1),...,X(1)} ⊆ X , of
1 1 d1 g1
degree n in {X(2),...,X(2)}⊆X , etc.
2 1 d2 g2
Lemma 4. Multiplicities on the right hand side of (5) satisfy the inequalities
m ≤dimRn1,...,nk.
λ(1),...,λ(k) d1,...,dk
Proof. Let P be the subspace of multilinear polynomials of degree n =
n1,...,nk
n + ··· + n on x(1),...,x(1),...,x(k),...,x(k) in R. Here x(i) ∈ X for all
1 ek 1 n1 1 nk j gi
1 ≤ i ≤ k,1 ≤ j ≤ n . Then P is isomorphic to P (A) as an F[H]-
i n1,...,nk n1,...,nk
module. Denote for brevity q =m and consider the F[H]-submodule
e λ(1),...,λ(k)
M =M ⊕···⊕M
1 q
of P , where M ,...,M are isomorphic irreducible F[H]-modules with H-
n1,...,nk 1 q
character χ . Any M is generated as an F[H]-module by a multilinear
j
e λ(1),...,λ(k)
polynomial of the type
f (x(1),...,x(1),...,x(k),...,x(k))=
j 1 n1 1 nk
e ···e h (x(1),...,x(1),...,x(k),...,x(k))
Tλ(1) Tλ(k) j 1 n1 1 nk
with a multilinear polynomial h ∈P .
j n1,...,nk
One can split the set of indeterminates x(1),...,x(k) into a disjoint union of
e 1 nk
subsets
P(1) ={x(1),...,x(1) },
1 1 λ(1)
1
P(1) ={x(1) ,...,x(1) },
2 λ(1)+1 λ(1)+λ(1)
1 1 2
···
P(1) ={x(1) ,...,x(1)},
d1 n1−λ.(d.11.)+1 n1
P(k) ={x(k),...,x(k) },
1 1 λ(k)
1
6 D.D.REPOVSˇANDM.V.ZAICEV
P(k) ={x(k) ,...,x(k) },
2 λ(k)+1 λ(k)+λ(k)
1 1 2
···
P(k) ={x(k) ,...,x(k)}
dk nk−λ(d1k)+1 nk
so that all f ,...,f are symmetric on any subset P(1),...,P(k).
1 q 1 dk
Now we identify all variables in each symmetric subset, that is, we apply a
homomorphism ϕ:R→R such that
• ϕ(x(j))=x(j) if 1≤α≤λ(j),
α 1 1
···
• ϕ(x(j))=x(j) if λ(j)+···+λ(j) <α≤n
α dj 1 dj−1 j
for all j = 1,...,k. Then all ϕ(f ),...,ϕ(f ) lie in Rn1,...,nk. Note that the total
1 q d1,...,dk
linearization of each of ϕ(f ) is equal to f with a nonzero coefficient independent
j j
of j. Hence a nontrivial linear relation α ϕ(f )+···+α ϕ(f ) = 0 implies the
1 1 q q
same relation α f +···+α f = 0. But f ,...,f belong to distinct irreducible
1 1 q q 1 q
summands M ,...,M , respectively. In particular, they are linearly independent.
1 q
Hence q does not exceed dimRn1,...,nk, and the proof is completed. (cid:3)
Now we restrict the dimensiodn1,.o..f,dRk n1,...,nk.
d1,...,dk
Lemma 5. Let A = A ⊕ ··· ⊕ A be a Γ-graded algebra with the support
g1 gk
{g ,...,g } and let d =dimA ,...,d =dimA . Then
1 k 1 g1 k gk
(11) dimRdn11,,......,,dnkk ≤(d1+···+dk)(n1+1)d21···(nk+1)d2k.
Proof. Let {a(gi),...,a(gi)} be a basis of the subspace A ,1 ≤ i ≤ k. Consider a
1 di gi
polynomial ring F[Y], where Y =Y ∪...∪Y and
g1 gk
Y ={ygi | 1≤m≤d ,j =1,2,...}.
gi m,j i
Then algebra F[Y] can be naturally endowed by a Γ-grading with Supp F[Y] =
{g ,...,g } if we set deg y = g when y ∈ Y . Denote A = A⊗F[Y] and fix
1 k Γ i gi
elements
di e
zgi = a(gi)⊗ygi , j =1,2,...,
j m m,j
mX=1
inA. Then alg{zgi} is alsoa Γ-gradedalgebra,where degzgi =g . Moreover,A≃
j j i
RandRn1,...,nk isasubspaceofA⊗T,whereT isthesubspaceofF[Y]spannedby
moneomiadl1s,..o.,fddkegree at most n in the set of indeterminates {ygt | 1≤m,j ≤de},
t m,j t
t=1,...,k. Clearly,
(12) dimT ≤(n1+1)d21···(nk+1)d2k
hence (11) follows from (12). (cid:3)
Now we are ready to get an upper bound for graded colength of A.
Theorem 1. Let A= A be a finite dimensional algebra graded by groupoid
g∈Γ g
Γ with Supp A = {g1,L...,gk}. Let also dimAgi = di,1 ≤ i ≤ k. Then the nth
graded colength of A satisfies the inequality
lngr ≤d(n+1)k+d21···+d2k+d1+···+dk
where d=dimA=d +···+d .
1 k
IDENTITIES OF GRADED SIMPLE ALGEBRAS 7
Proof. ByLemma3,thetotalnumberofpartitionsλi ⊢nidoesnotexceed(ni+1)di
for any i=1,...,k. Hence, by (6) and Lemmas 4 and 5, we have
ln1,...,nk(A)≤d(n1+1)d21+d1···(nk+1)d2k+dk
and
lngr ≤d(n+1)k+d21···+d2k+d1+···+dk
as follows from (7). (cid:3)
4. Existence of graded PI-exponents
Webeginthissectionwithatechnicalresultconnectingdimensionsofirreducible
representationsofsymmetricgroupsandmultinomialcoefficients. Givenapartition
µ = (µ ,...,µ ) ⊢ m, we denote by qµ(qµ ,...,qµ ) the partition of qm, where
1 t 1 t
q ≥1 is an arbitrary integer. We also define the height ht(µ) as ht(µ)=t. Recall
that d =χ is the dimension of the correspondingirreducible representationof
µ µ(1)
S .
m
Lemma 6. Let n ,...,n be positive integers, n +···+n = n ≥ 100. Let also
1 k 1 k
λ(1),...,λ(k) bepartitionsofn ,...,n ,respectively,suchthatht(λ(1)),...,ht(λ(k))≤
1 k
d. If q ≥100 then
qn 1 k(d2+d+1) 1 n q
d ···d ≥ d ···d .
(cid:18)qn ,...,qn (cid:19) qλ(1) qλ(k) (cid:18)qn(cid:19) (cid:20)n2k(cid:18)n ,...,n (cid:19) λ(1) λ(k)(cid:21)
1 k 1 k
Proof. Given nonnegative real α ,...α with α +···+α =1, we denote
1 k 1 k
1
Φ(α ,...α )= .
1 k (α1)α1···(αk)αk
From the Stirling formula for factorials it easily follows that
1 m m m m m m m
(13) Φ 1,··· , k ≤ ≤mΦ 1,··· , k ,
mk (cid:16) m m (cid:17) (cid:18)m1,...,mk(cid:19) (cid:16) m m (cid:17)
where m ,...,m arenonnegative integersandm +···+m =m. Applying (13)
1 k 1 k
to
qn
P =
(cid:18)qn ,...,qn (cid:19)
1 k
we obtain
1 k qn qn qn 1 k n n n q
P > Φ 1,..., k = Φ 1,..., k .
(cid:18)qn(cid:19) (cid:18) qn qn (cid:19) (cid:18)qn(cid:19) h (cid:16) n n (cid:17) i
Applying again (13) we get
1 k 1 n q
(14) P > .
(cid:18)qn(cid:19) (cid:20)n(cid:18)n ,...,n (cid:19)(cid:21)
1 k
It follows from Lemma 1 and (13) that
1 1 k(d2+d) q
d ···d > ··· Φ(λ(1))n1···Φ(λ(k))nk >
qλ(1) qλ(k) (cid:18)qn1 qnk(cid:19) h i
k(d2+d)
1
(15) [d ···d ]q.
(cid:18)qn(cid:19) λ(1) λ(k)
8 D.D.REPOVSˇANDM.V.ZAICEV
Now our lemma is a consequence of (14) and (15). (cid:3)
Recall that A is a d-dimensional Γ-graded algebra. Now let
(16) a=expgr(A)=limsup n cgr(A).
n
n→∞ q
The next lemma is the main step of the proof of Theorem 2.
Lemma 7. Let Γ be a commutative semigroup and let a in (16) be strictly greater
than 1. If A is graded simple then for any ε > 0 and any δ > 0 there exists an
increasing sequence of positive integers n(1),n(2),... such that
(i) n cgr(A)>(1+δ)(1−ε) for all n=n(1),n(2),...; and
n
(ii) np(q+1),n(q) ≤d, for all q =1,2,... .
Proof. Clearly, there exists an integer n(1) such that
cgr (A)>(a−ε)n(1)
n(1)
and n(1) can be chosen arbitrary large. There are also n ,...,n ≥ 0 such that
1 k
n +···+n =n(1) and
1 k
n(1) 1
c (A)> (a−ε)n(1).
(cid:18)n ,...,n (cid:19) n1,...,nk (n(1)+1)k
1 k
(see (2)). Without loss of generality, we can suppose that k =|Supp A|. Consider
theH =S ×···×S -actiononP . Itfollowsfrom(6),(7),(10)thatthere
n1 nk n1,...,nk
exist partitions λ(1) ⊢n ,...,λ(k) ⊢n such that
1 k
1
d ···d > c (A).
λ(1) λ(k) lgr (A) n1,...,nk
n(1)
By Theorem 1 we have
lgr (A)<d(n(1)+1)k(d+1)2
n(1)
hence
1
d ···d > c (A).
λ(1) λ(k) d(n(1)+1)k(d+1)2 n1,...,nk
and
n(1) 1
(17) d ···d > (a−ε)n(1).
(cid:18)n ,...,n (cid:19) λ(1) λ(k) d(n(1)+1)k(d+1)2+k
1 k
There exists a multilinear polynomial
f =f(x(1),...,x(1),...,x(k),...,x(k))6∈Idgr(A)
1 n1 1 nk
where x(i) ∈X for all 1≤i≤k,1≤j ≤n such that f generates an irreducible
j gi k
F[H]-module with the character
χ (F[H]f)=χ .
H λ(1),...,λ(k)
There are homogeneousin Γ-gradinga1,...,a1 ,...,ak,...,ak with deg ai =
1 n1 1 nk Γ j
g such that
i
Q=f(a1,...,ak )6=0.
1 nk
IDENTITIES OF GRADED SIMPLE ALGEBRAS 9
Since Γ is associative and commutative it follows that Q is homogeneous in Γ-
grading of A. Therefore one can find d′ ≤ d and homogeneous c1,...,cd′ ∈ A
satisfying the inequality
(18) (Q∗c1∗...∗cd′)∗Q6=0
wherea∗bdenotes the rightorthe left multiplicationby b (otherwiseSpan<Q>
is a graded ideal of A). Denote
g =f =f(x(1),...,x(1) ,...,x(k),...,x(k) ),
1 1 1,1 1,n1 1,1 1,nk
where x(i) are new homogeneous variables, deg x(i) =g , and take
α,β Γ α,β i
g2 =(f1∗z1∗···∗zd′)∗f2,
where degΓz1 =degΓc1,...,degΓzd′ =degΓcd′,
f =f(x(1),...,x(1) ,...,x(k),...,x(k) ),
2 2,1 2,n1 2,1 2,nk
with x(i) ∈ X . Then g ∈ P is not an identity of A, as follows
α,β gi 2 2n1+q1,...,2nk+qk
from (18), and q ,...,q ≥0,q +···+q =d′.
1 k 1 k
The square H×H of group H acts on {x(i) } where the first copy of H acts on
α,β
{x(i)}, while the second copy acts on {x(i)} and
1,β 2,β
χ (F[H ×H]g )=χ ⊗χ .
H×H 2 λ(1),...,λ(k) λ(1),...,λ(k)
Denote n(2) = 2n(1) +d′. Repeating this procedure we construct for all q =
3,4,... a multilinear polynomial
g =g (x(1),...,x(1) ,...,x(k),...,x(k) ,z ,z ,...)
q q 1,1 1,n1 q,1 q,nk 1 2
of degre n(q) such that:
(i) all x(i) ,z are homogeneous and x(i) ∈X ;
α,β j α,β gi
(ii) g is not an identity of A;
q
(iii) n(q) =qn(1)+d(q), d(q) ≤(q−1)d≤dq, n(q)−n(q−1) ≤d; and
(iv) q copiesHq =H×···×H ofH actsong permutingx(i) andg generates
q α,β q
an irreducible F[Hq]-module M with
χ(M)=(χ )⊗q.
λ(1),...,λ(k)
Denote H(q) = S × ··· × S . Given 1 ≤ i ≤ k, group S acts on
qn1 qnk qni
{x(i),...,x(i) }. We can induce the Hq-action on M to the H(q)-action. Con-
1,1 q,ni
sider the decomposition of M =F[H(q)]g into irreducible components,
q
χ (M)=f t χ .
H(q) ρ(1),...,ρ(k) ρ(1),...,ρ(k)
f ρ(1)⊢qn1X,...,ρ(k)⊢qnk
It follows by the Richardson–Littlewood rule that for any 1 ≤ i ≤ k, either ρ(i) =
qλ(i) or ρ(i) is obtained from λ(i) by putting down one or more boxes of D .
qλ(i)
Then by Lemma 2 we have Φ(ρ(i)) ≥ Φ(qλ(i)) = Φ(λ(i)). Now, Lemma 1 implies
the inequality
k(d+1)2
1
(19) d ···d > d ···d .
ρ(1) ρ(k) (cid:18)qn(1)(cid:19) qλ(1) qλ(k)
10 D.D.REPOVSˇANDM.V.ZAICEV
Recall that g is not an identity of A. Hence there exist integers p ,...,p ≥0
q q,1 q,k
such that p +···+p =d(k) and
q,1 q,k
c (A)≥d ···d .
qn1+pq,1,...,qnk+pq,k ρ(1) ρ(k)
In particular,
qn(1)
(20) cgr (A)≥ d ···d .
n(q) (cid:18)qn ,...,qn (cid:19) ρ(1) ρ(k)
1 k
Note thatforanypartitionµ⊢m,ν ⊢m withΦ(µ)≥Φ(ν)itfollowsbyLemma
1 that
1 1 1
d ≥ Φ(µ)m ≥ Φ(ν)m ≥ d .
µ md2+d md2+d md2+d+1 ν
Then by Lemma 6 and (19), the right hand side of (20) is not less than
1 2k(d+1)2 1 2kq n(1) q
d ···d .
(cid:18)qn(1)(cid:19) (cid:18)n(1)(cid:19) (cid:20)(cid:18)n ,...,n (cid:19) λ(1) λ(k)(cid:21)
1 k
Now, (17) implies the following inequality
2k(d+1)2 2kq 2k(d+1)2q
1 1 1
cgr (A)≥ (a−ε)n(q).
n(q) (cid:18)qn(1)(cid:19) (cid:18)n(1)(cid:19) (cid:18)d(n(1)+1)(cid:19)
Since a>1, by the assumptions of the lemma we then have
(a−ε)n(q)
(a−ε)qn(1) ≥
aqd
for all small enough ε. Hence
n cgr(A)>D(a−ε),
n
q
where D =D D ,
1 2
2k(d+1)2 2k 2k(d+1)2 d
1 n(q) 1 n(1) 1 n(1) 1 n(1)+d
D = , D = .
1 (cid:18)n(q)(cid:19) 2 (cid:18)n(1)(cid:19) (cid:18)d(n(1)+1)(cid:19) (cid:18)a(cid:19)
For small δ ,δ >0 one canchoose n(1) such that D >(1−δ ) and D >(1−δ )
1 2 1 1 2 2
for all n(q),q ≥1. Finally, we can take δ ,δ small enough and get the inequality
1 2
n cgr(A)>(1−δ)(a−ε),
n
q
for all n=n(q), q =1,2,... . (cid:3)
Remark 1. In the proof of the previous lemma we used associativity and com-
mutativity of Γ only for getting relation (18). In case of an arbitrary groupoid Γ
the element Q in (18) can be non-homogeneous in Γ-grading and hence an ideal I
generated by Q in A can be strictly less than A. But if A is simple in a non-graded
sense then I =A and relation (18) and Lemma 7 hold.
Forcompletingtheproofofthemainresultweneedthefollowingremark. Denote
by Ann A the annihilator of A.