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Journal
of
Algebra
www.elsevier.com/locate/jalgebra
Decomposing
modular
coinvariants
Müfit Sezer1
DepartmentofMathematics,BilkentUniversity,Ankara06800, Turkey
a r t i c l e i n f o a b s t r a c t
Articlehistory: Received18June2014
Availableonline24October2014 CommunicatedbyGunterMalle Keywords:
Coinvariants Modularactions
Weconsidertheringofcoinvariantsforamodular represen-tationofa cyclicgroupof primeorderp.Weshowthat the classesoftheterminalvariablesinthecoinvariantshave nilpo-tencydegreep andthatthecoinvariantsare afreemoduleover thesubalgebrageneratedby these classes.Anincidental re-sultwehaveisadescriptionofaGröbnerbasisfortheHilbert idealandadecompositionofthecorrespondingmonomial ba-sisforthecoinvariantswithrespecttothemonomialsinthe terminalvariables.
© 2014ElsevierInc.All rights reserved.
Introduction
LetV denoteafinitedimensionalrepresentationofafinitegroupG overafieldF.The inducedaction onthedual V∗ extendsas degreepreserving algebraautomorphisms to thesymmetricalgebraS(V∗) whichwedenotebyF[V ].Theringofinvariantpolynomials
F[V ]G :={f ∈ F[V ]| g(f)= f ∀g ∈ G} isagraded finitely generatedsubalgebra. The
HilbertidealI istheidealF[V ]G
+·F[V ] inF[V ] generatedbyinvariantsofpositivedegree.
Inthispaperwestudytheringofcoinvariantswhichisthequotientring
F[V ]G:= F[V ]/I.
E-mailaddress:sezer@fen.bilkent.edu.tr.
1
TheauthorissupportedbyagrantfromTübitak:112T113. http://dx.doi.org/10.1016/j.jalgebra.2014.08.059
Since G isfinite,F[V ]G isafinitedimensionalvectorspace. Notealsothatcoinvariants
arenaturallyaG-moduleasI isclosedundertheactionofG.Coinvariantsoftencontain information abouttheinvariants.Forinstance,if|G| isaunitinF,thenF[V ]G is
poly-nomial ifandonlyifF[V ]G satisfiesPoincaréduality,see[4,6,13].Eveninthemodular
case,thatiswhen|G| isdivisiblebythecharacteristicofF,someweakerversionsofthis equivalence stillholdinsmalldimensionssee [8,12].SinceF[V ]G isafinite dimensional
vector space,the largestdegreeof anon-zerocomponentisalsofinite.It iswell known thatinthenon-modularcasethetopdegreeisboundedbythegrouporderbutitcanbe arbitrarilylargeotherwise[7].Inmanymodularcasesthetopdegreeofthecoinvariants also coincides with the maximum degree of an indecomposable invariant and getting efficient bounds forthe topdegreehasbeen criticalwhen studying effectivegeneration of invariantrings,seeforexample[5,10].
We now fix our setup. Let p be a prime integer and F a field of characteristic p.
For the rest of the paper, G denotes the cyclic group of prime order p. Fix a gen-erator σ of G. There are exactly p indecomposable G-modules V1,. . . ,Vp over F and
each indecomposable module Vi is afforded by a Jordan blockof dimension i with 1’s
on the diagonal. Let V be an arbitrary G-module over F. Assume that V has l
sum-mands so we canwrite V =1≤j≤lVnj.Since adding atrivialsummand to a module
does not affect the coinvariants we assume as well that nj > 1 for 1 ≤ j ≤ l. We
set F[V ] = F[Xi,j | 1 ≤ i ≤ nj, 1 ≤ j ≤ l] and the action of σ is given by
σ(Xi,j) = Xi,j + Xi−1,j for 1 < i ≤ nj and σ(X1,j) = X1,j. Following the
estab-lished convention, we call thevariables Xnj,j for 1≤ j ≤ l terminal variables. We use
graded reverse lexicographic order on themonomials inF[V ] withX1,j <· · · < Xnj,j.
We denote the leadingmonomial of apolynomialf by LM(f ). Wealso use lower case lettersto denotetheimagesofthevariablesinthecoinvariants.
Certain examplesof coinvariantsfor modularrepresentationsof G are studiedin[9]
and for each example, among other things, a reduced Gröbner basis for the Hilbert ideal andthe corresponding monomialbasis forthecoinvariants aregiven. Thegoal of this paper is to demonstrate that some of the properties of the coinvariants and the Hilbert ideal thatare identified in the casesstudied inthatsource hold ingeneral for an arbitrary module V . We show that the Hilbert ideal I is generated by the orbit products of Xnj,j for 1 ≤ j ≤ l (which we denote by N (Xnj,j)) and polynomials in
A:= F[Xi,j| 1≤ i≤ nj− 1, 1≤ j ≤ l].NoticethatLM(N (Xnj,j))= X p
nj,j.Therefore
we getthat,apartfrom themonomialsXnpj,j for1≤ j ≤ l,nomonomial intheunique minimalgeneratingsetfortheleadtermidealofI isdivisiblebyanyXnj,jfor1≤ j ≤ l.
Wegoonto provethatthereisanisomorphismofgradedF-algebras F[xn1,1, . . . , xnl,l] ∼= F[t1, . . . , tl]/
tp1, . . . , tpl,
wheret1,. . . ,tlareindependentvariables.Moreover,weshowthatF[V ]Gisafreemodule
over F[xn1,1,. . . ,xnl,l].
1. Reductionsandresults
For apolynomial f ∈ F[V ] and 1≤ j ≤ l, let f(i,j) denote thei-th partial
deriva-tive of f with respect to Xnj,j. We first observe that the action of σ commutes with
differentiationwithrespecttoXnj,j.
Lemma1. Letf ∈ F[V ] and fix 1≤ j ≤ l. Thenσ(f(1,j))= σ(f )(1,j).ThereforeF[V ]G
isclosed underdifferentiation with respecttoXnj,j.
Proof. Sincebothdifferentiationandtheactionofσ are additiveitisenoughtoassume that f = gXk
nj,j, where g is a polynomial in the variables except Xnj,j and k is a
non-negativeinteger.Firstassumethatk ispositive.Noticethat
σf(1,j)= σkgXnkj−1,j= kσ(g)(Xnj,j+ Xnj−1,j) k−1.
Ontheotherhandwe alsohave
σ(f )(1,j)=σ(g)(Xnj,j+ Xnj−1,j) k(1,j)
= kσ(g)(Xnj,j+ Xnj−1,j) k−1.
Sincethe actionof σ does notincreasethe Xnj,j degree ofapolynomial,both sides of
theidentityintheassertionofthelemmaarezeroifk = 0.Hencetheresultfollows. 2 NoticethatsinceI consistsoffinite sumsfigi withfi ∈ F[V ]G+ andgi ∈ F[V ],by
the Leibniz rule and the previous lemma we get that I is closed under differentiation with respect to Xnj,j for 1 ≤ j ≤ l as well. We also note a combinatorialresult that
applies to any polynomial inF[V ] whose degree in a terminal variable is strictly less thanp.
Lemma2.Letf ∈ F[V ] beapolynomialofdegree k < p inoneoftheterminalvariables, say Xnj,j.Write f = fkX
k
nj,j+ fk−1X k−1
nj,j+· · · + f0, where fk,. . . ,f0 are polynomials
inthevariablesexceptXnj,j.Thenwehave
0≤i≤k (−1)i i! X i nj,jf (i,j)= f 0.
Proof. Fix d forsome0≤ d≤ k. Consider0≤i≤k (−1)i! iXnij,j(fdXndj,j)
(i,j).This
sum-mationisequalto 2 0≤i≤d (−1)i i! X i nj,j fdXndj,j (i,j) = 0≤i≤d (−1)i i! X i nj,jd· · · (d − i + 1)fdX d−i nj,j = 0≤i≤d (−1)i d i fdXndj,j.
Notice thatthis sumis f0 ifd= 0 andis zerootherwise. This givesthedesired
equal-ity. 2
Theorem3.TheHilbertidealI is generatedbytheorbitproductsN (Xnj,j) for1≤ j ≤ l
and polynomials in A. Moreover, the lead term ideal of I is generated by Xnpj,j for
1≤ j ≤ l andmonomialsin A.
Proof. Let f ∈ F[V ]G. By[11, Proposition 2.1] f hasadecomposition
f = g1N (Xn1,1) + g2N (Xn2,2) +· · · + glN (Xnl,l) + r,
where r ∈ F[V ]G is the normal form of f with respect to the Gröbner basis
{N(Xn1,1),. . . ,N (Xnl,l)} of the ideal (N (Xn1,1),. . . ,N (Xnl,l)) in F[V ]. Therefore we
mayassumethatf ∈ F[V ]G isofdegree< p asapolynomialinXnj,j for1≤ j ≤ l.Say,
asapolynomialinXn1,1,f hasdegreek < p andwritef = fkX
k
n1,1+fk−1X
k−1
n1,1+· · ·+f0,
wherefk,. . . ,f0arepolynomialsinallvariablesexceptXn1,1.ByLemma 1allderivatives
off withrespecttoXn1,1arealsoinvariant.Thentheidentity
0≤i≤k(−1) i i! X i n1,1f (i,1)=
f0 from the previouslemma gives thatf0 ∈ I. Furthermore, since thecoefficients of f
and f0 areunitsinthisidentity italsogivestheequalityoftheideals
f, f(1,1), . . . , f(k,1)=f0, f(1,1), . . . , f(k,1)
in F[V ]. Therefore f is in the ideal generated by f0,f(1,1),. . . ,f(k,1). Note that these
polynomials have degree < k in Xn1,1. Moreover since differentiation with respect to Xn1,1doesnotincreasethedegreewithrespecttoanyvariable,theirdegreeswithrespect
to other terminal variables are still strictly lessthan p. Therefore by inductionon the degreek wegetthatI isgeneratedbythenormsN (Xnj,j) for1≤ j ≤ l andpolynomials
that are ofdegree zeroin Xn1,1 and of degree < p inthe Xnj,j for2≤ j ≤ l. Since I
is closedunderdifferentiationwithrespect toany terminalvariable,wecanrepeatthis process byapplying theidentity ofthepreviouslemma (withrespectto other terminal variables)tothesuccessiveconstantterms.SowegetthatI isgeneratedbyN (Xnj,j) for
1≤ j ≤ l andpolynomialsinA.Finally,recallthatbyBuchberger’salgorithmaGröbner basisisobtainedbyreductionofS-polynomialsbypolynomialdivision,see[1, §1.7].But the S-polynomialof twopolynomials inA is alsoinA andviapolynomials inA italso reducestoapolynomialinA.Moreover,theS-polynomialofN (Xnj,j) andapolynomial
inA andtheS-polynomialofapairN (Xnj,j) andN (Xnj,j) reducetozerosincetheir
leadingmonomialsarepairwiserelativelyprimeforany1≤ j = j≤ l.Hencetheresult follows. 2
Remark 4.Let Λ denote theset of monomials inF[V ] that donot belong to the lead term ideal of I. Then the images of monomials in Λ in F[V ]G form a vector space
basis forF[V ]G.Theprevioustheorem givesus thatamonomialM ∈ A isinΛ ifand
only if M Xk1
n1,1· · · X
kl
(1+ t+ t2+· · · + tp−1)l dividesthe Hilbertpolynomial ofF[V ]G and thatthe vector
spacedimensionofF[V ]G isdivisiblebypl.Inthenexttheoremweshowthatthisisnot
acombinatorialaccident.
Theorem5.Wehave anisomorphismof gradedF-algebras
F[xn1,1, . . . , xnl,l] ∼= F[t1, . . . , tl]/
tp1, . . . , tpl,
where t1,. . . ,tl are independent variables. Moreover, F[V ]G is a free module over F[xn1,1,. . . ,xnl,l].Inparticular,
HF[V ]G(t)
(1+t+t2+···+tp−1)l ∈ Z[t],whereHF[V ]G(t) istheHilbert
polynomialof F[V ]G.
Proof. We first show that the nilpotency degree of xnj,j in F[V ]G is p for1 ≤ j ≤ l.
Pick 0 < i < p. Then xinj,j = 0 for some 1 ≤ j ≤ l implies that Xnij,j ∈ I. This is a contradiction to the description of the lead term ideal of I given by the previous theorem.Nextweshow thatxpnj,j = 0 for 1≤ j ≤ l.Equivalently,we proveXnpj,j ∈ I.
Without loss of generality takej = 1.Set N = N (X˜ n1,1)− X
p
n1,1. Notice that N is˜ a
polynomialofdegree < p inXn1,1. Callthis degreek. SoLemma 2 applies andwe get
0≤i≤k(−1) i i! X i n1,1N˜ (i,1)= f
0,wheref0 istheconstantterm(asapolynomialinXn1,1)
ofN .˜ But sincetheydifferbyXnp1,1, theconstanttermsofN and˜ N (Xn1,1) are equal.
Ontheother handsinceN (Xn1,1) is theorbitproduct of Xn1,1,alltermsinN (Xn1,1)
aredivisiblebyXn1,1.Itfollowsthatf0= 0.Moreover,sinceN and˜ N (Xn1,1) differbya
polynomialwhosederivativewithrespecttoXn1,1iszero,theirderivativesareallequal.
ButallderivativesofN (Xn1,1) areinvariantbyLemma 1.ItfollowsthatN˜
(i,1)∈ F[V ]G
for0< i≤ k.Thereforetheequality 0≤i≤k (−1)i! iXni1,1N˜(i,1)= 0 gives us thatN˜∈ I.
SoXnp1,1∈ I aswell.
Let t1,. . . ,tl be independent variables and consider the natural graded surjection
fromF[t1,. . . ,tl] toF[xn1,1,. . . ,xnl,l].Sincenilpotencydegreeofxnj,jisp for1≤ j ≤ l,
thekernel of this map contains theideal (tp1,. . . ,tpl). If this idealdoes notcontain the kernel, thenI containsapolynomialinXn1,1,. . . ,Xnl,l suchthatnomonomialinthat
polynomial is divisible by any Xnpj,j for 1 ≤ j ≤ l. This is a contradiction to the description of the lead term ideal of I in the previous theorem. So we establish the isomorphismintheassertionofthetheorem.
Secondly,letΛdenotethesetofmonomialsinA thatdonotbelongtotheleadterm idealofI,i.e.,Λ= Λ∩ A.Thenbythepreviousremarktheimagesofthesemonomials inF[V ]GgenerateF[V ]GasamoduleoverF[xn1,1,. . . ,xnl,l].Infacttheygeneratefreely
sinceM Xk1
n1,1· · · X
kl
nl,l is inΛ forallM ∈ Λ
and 0≤ k
j ≤ p− 1 for 1≤ j ≤ l and the
imagesofmonomialsinΛ formavectorspacebasisforF[V ]G.
FinalstatementfollowseasilyfromthesecondonebecausetheHilbertpolynomialof
F[t1,. . . ,tl]/(tp1,. . . ,t p
Acknowledgments
This studywasdoneduringmyvisittoUniversityofNebraskaonFulbrightVisiting ScholarProgram.IthankLuchezarAvramovformanyhelpfuldiscussions.Also,initially this paper was targeting indecomposable representations only. I thank Jim Shank for pointingoutthattheargumentgeneralizestodecomposablerepresentationsaswelland therefereeforhelpfulcomments.
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