Zero-dimensional Gorenstein algebras with the action of the symmetric group

Zero-dimensional Gorenstein Algebras with the Action of the Symmetric Group

HIDEAKIMORITA(*) - AKIHITOWACHI(**) - JUNZOWATANABE(***)

1. Introduction.

Suppose that A is a Gorenstein K-algebra with a strong Lefschetz element l2A. Put Lˆ l2End(A). Then it is possible to construct a degree 1 map D2End(A) such that fL;D;Hg is an sl(2)-triple, Hˆ[L;D], and the weight space decomposition coincides with the natural grading decomposition. This means that the eigenspaces ofHare precisely the homogeneous parts ofA. Suppose moreover that the symmetric group S_k acts onAas permutation of the variables and that the invariant linear formlˆx₁‡x₂‡ ‡x_n is a strong Lefschetz element. Then, as is ob- vious, the vector space KerLis fixed, hence the spaces KerL\A_iare fixed by the action ofSk. Thus in such a case an irreducible decomposition ofA can be constructed by first decomposing the kernel of the multiplication map

(x₁‡ ‡x_k):A!A …1†

into irreducible spaces and then applying the map D repeatedly to the constituents of the decomposition of KerL. (Or equivalently, first decom- pose KerDand then applyL.)

An obvious example of an Artinian Gorenstein algebra in which lˆx₁‡ ‡x_k is a strong Lefschetz element is the equi-degree mono-

(*) Indirizzo dell'A.: Muroran Institute of Technology, Muroran 050-8585, Japan.

E-mail: morita@muroran-it.ac.jp

(**) Indirizzo dell'A.: Division of Comprehensive Education, Hokkaido Institute of Technology, Sapporo 006-8585, Japan.

E-mail: wachi@hit.ac.jp

(***) Indirizzo dell'A.: Department of Mathematics, Tokai University, Hiratsu- ka 259-1292, Japan.

E-mail: watanabe@sm.u-tokai.ac.jp

mial complete intersection

A(n;k)ˆK[x1; ;x_k]=(xⁿ₁; ;xⁿ_k):

It is possible to identifyA(n;k) with the tensor space (Kⁿ)^k

as (GL(n)S_k)-modules. The purpose of this paper is to understand the strong Lefschetz property ofA(n;k) withla strong Lefschetz element in the context of the Schur-Weyl duality.

LetY^T be the Young symmetrizer inK[S_k] corresponding to a Young tableauT. One might ask first what is the Hilbert series of

S_kY^T(A(n;k));

the maximal isotypic submodule ofA(n;k) corresponding to the shape ofT.

It turns out that it equals the q-analog of the Weyl dimension formula.

Since it does not seem to be well known and since we have been unable to find a good reference, we give a proof for it in section 6. This is stated in Proposition 27.

Proposition 27 is not a consequence of the strong Lefschetz property of A(n;k), but Corollary 36, which states that theq-analog of the Weyl di- mension formula is unimodal and symmetric, may be thought of as a consequence of the factlis a strong Lefschetz element.

Terasoma-Yamada [8] considered the coinvariant algebra ofS_k, AˆK[x₁; ;x_k]=(e₁; ;e_k);

…2†

whereei is the elementary symmetric polynomial of degreei. Note that this is also a Gorenstein algebra with the action ofSk. They remark that the Hilbert series ofS_kY^T(A) is obtained as theq-analog of the hook-length formula. Thus our situation is similar to theirs, but it is not true that the Hilbert series of S_kY^T(A) is unimodal and symmetric. The algebra A above has the strong Lefschetz property, but no symmetric linear form is a strong Lefschetz element, which causes the difference.

In section 7, we consider the irreducible representation r^l:GL(n)!GL(W^l)

of the general linear groupGL(n) on a finite dimensional vector spaceW^l and show how the Jordan matrix with a single eigenvaluea,

J(a;n)2GL(n)

is transformed byr^l. This is done again by identifying the Artinian algebra

A(n;k) with the tensor space (Kⁿ)^k as (GL(n)Sk)-modules so that the Schur-Weyl duality applies to A(n;k). Here the consideration of the be- havior oflas a multiplication operator is indispensable. The two extremal cases ofA(n;k) treated in Sections 3 and 4 should be regarded as examples which show that our method is a new way to understand some classical results.

One other result of this paper is Theorem 22 in Section 5. We apply Theorem 19 to determine the minimal number of generators of the ideal

(x²₁; ;x²_k):(x₁‡ ‡x_k)

in the polynomial ring. By analyzing Specht polynomials involving only square-free monomials, it is possible to determine a minimal generating set of the ideal.

The authors thank the referee very much for helpful comments and suggestions.

2. Preliminaries.

2.1 ±A list of notational conventions.

Here is a list of notation which we are going to fix throughout the paper.

Details follow the list.

K denotes an algebraically closed field of characteristic 0.

lˆ(k₁;k₂; ;k_r)`kindicates thatlis a partition of a positive in- tegerk. The same notation is used to indicate a Young diagram of sizek. It is assumed thatk₁k₂ k_r>0. The lengthroflis denoted byl(l).

IfJ2End(V) is nilpotent, we call the partitionl(J)ˆ(k1; ;kr) the conjugacy class of J, indicating the sizes of Jordan blocks in the Jordan decomposition ofJ.

A Young tableauTis a Young diagramlwith a numbering of boxes with integers 1;2; ;k. In this caselis the shape ofT.

Y^Tdenotes the Young symmetrizer defined by the Young tableauT.

It is an element of the group algebraK[Sk].

Suppose thatTis a Young tableau andlis the shape ofT. IfV is an S_k-module, then Y^l(V) denotes S_kY^T(V). Thus Y^l(V) is the maximal submodule ofVconsisting of modules isomorphic to the Specht moduleV^l. We may regard Y^l( ) as a functor to extract the isotypic component be- longing tol.

DTdenotes the Specht polynomial defined by a Young tableauT.

2.2 ±The equi-degree monomial complete intersection as a tensor space.

LetRˆK[x₁; ;x_k] be the polynomial ring. The partial degree of a homogeneous polynomialf 2Ris the maximum degree offwith respect to a single variable. ByA(n;k) we denote the vector subspace ofRconsisting of polynomials of partial degree at most n 1. Let I be the ideal Iˆ(xⁿ₁; ;xⁿ_k) ofR. Then as vector spaces we have the decomposition

RˆA(n;k)I:

The vector spaceA(n;k) may be regarded as the tensor space (Kⁿ)^k. Since A(n;k)R=Ithe vector spaceA(n;k) has the structure of a commutative ring. PutAˆA(n;k). A basis ofAcan be the set of monomials of partial degree at most (n 1):

fxⁱ₁¹ xⁱ_k^kj0i1; ;ikn 1g An element ofAis expressed uniquely as

XF(i1;i2; ;ik)xⁱ₁¹xⁱ₂² xⁱ_k^k:

With the identificationA(Kⁿ)^kthe general linear groupGL(n) acts on the vector spaceAas the tensor representation. Let

F:GL(n)!GL(A) …3†

be the representation. Explicitly, ifgˆ(gab)2GL(n) then

F(g)(x^j₁¹ x^j_k^k)ˆ X^{n 1}

aˆ0

g_aj1x^a₁

!

X^{n 1}

aˆ0

g_ajkx^a_k

! : …4†

Here the indicesa;bof the matrix entries forgˆ(g_ab)2GL(n) range over 0;1; ;n 1. At the same time the symmetric groupS_kacts onAas the permutation of the variables.

We are interested in the decomposition ofAinto irreducibleS_k-mod- ules. According to the Schur-Weyl duality it will give us an irreducible decomposition ofAas (GL(n)S_k)-modules.

2.3 ±Young tableaux andSpecht polynomials.

A partition of a positive integer kis a way to express kas a sum of positive integers. If we say thatlˆ(k₁; ;k_r) is a partition ofk, it means

that kˆk1‡ ‡krandk1 kr>0. A partition ofkis identified with a Young diagram of size kin a well known manner. Thus the same notationlˆ(k₁; ;kr) denotes a Young diagram of sizeP

k_iwith rows of k_iboxes,iˆ1; ;r, aligned left. A Young tableauTis a Young diagraml whose boxes are numbered with integers 1; ;kin any order. In this case we say thatlis the shape ofT. A Young tableau is standard if every row and column is numbered increasingly.

A Young tableauTdefines a Specht polynomial, denotedD_T, as follows:

PutIˆ f1;2; ;kgand

I_jˆ fa2Ija is in thejth column of Tg:

Define

DjˆY

a;b2a5bIj

(xa xb);

and finally,

D_T ˆY

j

D_j; …5†

wherejruns over all columns.D_Tis the Specht polynomial defined by the Young tableauT.

2.4 ±Nilpotent matrices andJordan bases.

Let M(k) denote the set of kk matrices with entries in K. Let J2M(k) be a nilpotent matrix. Let

niˆrankJⁱ rankJ^i‡1 foriˆ0;1; ;p;

wherepis the least integer such thatJ^p‡1ˆO. Thenbl:ˆ(n0;n1; ;np) is a partition ofk. We denote the dual partition ofblbyl(J). (cf. Definition 1 below.)

Now suppose thatTis a Young tableau of sizek. Using the numbering ofTdefine the matrixJˆ(aij)2M(k) by

aijˆ 1 if jis next to and on the right of iinT;

0 otherwise:

( …6†

It is easy to see that the matrixJis nilpotent andl(J) is the shape ofT.

We call any matrix defined as above for a Young tableau TaJordan ca- nonical form(of a nilpotent matrix).

Let V be a vector space of dimension k. If a basis of V is fixed, we may identify M(k) and End(V). Suppose that J2End(V) is nil- potent. A Jordan basis for J is a basis of V on which J is in a Jordan canonical form. (According to the definition of a Jordan basis just defined above, any permutation of basis elements of a Jordan basis is a Jordan basis.) Since K is algebraically closed, there exists a Jordan basis. Note that if two nilpotent elements J and J⁰ are conjugate, then l(J)ˆl(J⁰). We make a definition of the ``conjugacy class'' of a nilpotent endomorphism, with a slight abuse of language, as follows.

DEFINITION1. LetVbe ak-dimensional vector space overK. Suppose thatJ2End(V) is nilpotent. We say that a partition (k1; ;kr) ofkis the conjugacy classofJif the Jordan canonical form ofJconsists of the Jordan blocks of sizesk₁; ;k_r. We denote byl(J) the conjugacy class ofJ.

REMARK2. Note that the notationl(J) coincides with the previously definedl(J) for a nilpotent matrix. In fact, if we putn_iˆdim ImJⁱ=ImJ^i‡1, then the sequenceblˆ(n0;n1; ;np) is a partition ofk. One sees easily that the dual partitionltoblis the conjugacy class ofJ.

LetJ2End(V) be nilpotent with the conjugacy classlˆl(J). Suppose thatBVis a Jordan basis forJ. Then it is possible to place the elements ofBinto the boxes of the Young diagraml(J) bijectively in such a way that it satisfies the following conditions:

e;e⁰2Bande⁰ˆJe,e⁰is next to and on the right of e:

e2KerJ,eis at the end of a row.

( …7†

(cf. Equation (6).)

REMARK 3. As explained above, by choosing a bijection between the elements of a Jordan basis for J and the boxes of the Young diagram l(J), it is possible to identify a Jordan basis Bfor J and the Young diagram l(J). With this identification the rightmost boxes of l(J) form a basis for KerJ. Also the boxes of the first column of l(J) coincide with fb2Bjb62ImJg. Once l(J) is known, KerJ\B de- termines B. Similarly the diagram l(J) and the subset BnImJB determine B.

2.5 ±The strong Lefschetz property.

LetV ˆL^c

iˆ0V_ibe a finite dimensional graded vector space. The Hilbert series ofVis the mapi7!dimV_i, which we usually write as the polynomial P( dimV_i)qⁱ. (For convention we let dimV_iˆ0 for i<0 ori>c.) Let J2End(V) be a map of degree one so J consists of the graded pieces Jj_Vi:V_i!V_i‡1. ThenJis nilpotent. We say that J2End(V) is a strong Lefschetz elementif the restricted map

J^{c 2i}j_Vi:V_i!V_{c i}

is bijective for alliˆ0;1; ;[c=2]. (Such an endomorphism exists only if the Hilbert series ofV is symmetric and unimodal.)

DEFINITION4. Let AˆL^c

iˆ0A_i be an Artinian graded K-algebra. De- note by:A!End(A) the regular representation ofA. (I.e.,a(b)ˆab for a;b2A.) We say thatA has the strong Lefschetz property, if there exists a linear form l2A such that l2End(A) is a strong Lefschetz element. We call such a linear forml2Aa strong Lefschetz element ofA as well asl2End(A).

PROPOSITION5. Suppose that AˆL^c

iˆ0A_iis a graded Artinian K-algebra with a symmetric Hilbert seriesP

h_iqⁱ. Let l be a linear form of A. Thenl is a strong Lefschetz element if andonly ifl(l)is the dual partition to (h⁰₀;h⁰₁; ;h⁰_c), which is a permutation of (h0;h1; ;hc) put in the de- creasing order.

PROOF. See [3] Proposition 18. p

PROPOSITION 6. With the same notation as above, suppose that J2End(A)is a strong Lefschetz element. Then any homogeneous basis of KerJ can be extended uniquely to a homogeneous Jordan basis for J.

PROOF. Let P

h_iqⁱ be the Hilbert series of A. Since J is a strong Lefschetz element, we have

dim (KerJ\A_i)ˆ 0 if h_ih_i‡1; h_i h_i‡1 if h_i>h_i‡1: (

…8†

Letsbe the greatest integer such thaths 1<hsand let m_iˆh_i h_{i 1}; foriˆ0;1;2; ;s:

Sinceh_iˆh_{c i}for 0ic, we may rewrite the equation (8) as dim (KerJ\A_{c i})ˆ m_i foriˆ0;1;2; ;s;

0 otherwise.

(

Now letBbe a homogeneous basis of KerJgiven arbitrarily. We have to find a basis ofAcontainingBon whichJis in the Jordan canonical form. Put B_{c i}ˆB\A_{c i}foriˆ0;1; ;s. ThenB_{c i}is a basis of KerJ\A_{c i}. Since the restricted mapJ^{c 2i}:A_i!A_{c i}is bijective, there is a finite setB_iA_i such that#B_iˆm_iand such thatJ^{c 2i}(B_i)ˆB_{c i}. It is easy to show that the set

B:e ˆG^s

iˆ0

fJ^j(B_i)jjˆ0;1;2; ;c 2ig

is linearly independent and hence is a basis ofA. It is easy to see thatBeis a desired basis. The uniqueness is obvious since the choice of the finite setB_i

is unique foriˆ0;1; ;s: p

The following result is proved in [9] and plays an important role in this paper.

THEOREM7. Let AˆA(n;k)be the same as defined inSection 1. Then A has the strong Lefschetz property andx1‡x2‡ ‡xk is a strong Lef- schetz element of A.

3. A(n;2), the two fold tensor ofKⁿ.

If kˆ2, then the Gorenstein algebra AˆK[x1; ;x_k]=(xⁿ₁; ;xⁿ_k) takes the form

AˆK[x₁;x₂]=(xⁿ₁;xⁿ₂):

Write x;y for x₁;x₂. Recall that we identify AˆA(n;2) as a vector subspace ofRˆK[x;y] andRˆA(xⁿ;yⁿ). Denote by:A!End(A) the regular representation of the Artinian algebraA. Put Jˆ (x‡y).

Since J is a strong Lefschetz element, we may use Proposition 6 to construct a Jordan basis for J. First we would like to construct a homogeneous basis for KerJ.

The Hilbert series ofAis the following sequence.

degree 0 1 2 n 2 n 1 n 2n 3 2n 2

dim 1 2 3 n 1 n n 1 2 1

SinceJis a strong Lefschetz element,J:A_i!A_i‡1is either injective or surjective. Hence we have

dim(KerJ\Ai)ˆ 0; if i5n 1;

1; if n 1i2n 2:

(

Since dim(KerJ\A_i) is at most one, a homogeneous basis of KerJ is uniquely determined up to constant multiple. For dˆn 1;

n;n‡1; ;2n 2, put b_dˆX^d

jˆ0

( 1)^jx^{d j}y^j; andb_dˆb_dmod (xⁿ;yⁿ):

Then sinceb_d(x‡y)ˆx^d‡1y^d‡1, we haveb_d2KerJfordn 1. Thus we have

KerJ\A_dˆ hb_di for dˆn 1;n; ;2n 2:

…9†

BecauseJ is a strong Lefschetz element, there is an elementai2Aifor each in 1 such that J^{2n 2 2i}(a_i)ˆb_{2n 2 i}. Note that J^j(a_i) are all symmetric ifiis even and are alternating ifiis odd. Now we have proved the following.

THEOREM8. The set G

n 1

iˆ0

fJ^j(ai)jjˆ0;1;2; ;2n 2 2ig

is a homogeneous Jordan basis for J2End(A). The basis element J^j(ai)is symmetric if i is even and alternating if i is odd. The conjugacy class of J is given by

l(J)ˆ(2n 1;2n 3; ;3;1):

PROOF. The first part was treated more generally in Proposition 6. The second part follows immediately from the definition ofb_{2n 2 i}anda_i. The

third statement follows from Proposition 5. p

Now we consider the representation F:GL(n)!GL(A)

as introduced in Section 2.2. (For the definition ofFsee (3) and (4).) Recall that the special linear group SL(2) has a unique irreducible module of dimensionifor eachi>0. We denote it byV(i 1). Fixn>0 and let

C:SL(2)!GL(n) …10†

be the irreducible representation corresponding to the moduleV(n 1).

We may considerAˆA(n;2) as anSL(2)-module via the composition SL(2)!^C GL(n)!^F GL(A):

…11†

PROPOSITION9. With the same notation above A decomposes into SL(2)- modules as

AV(2n 2)V(2n 4) V(0):

PROOF. Abbreviate rˆFC, so r:SL(2)!GL(A). The group homomorphismrinduces a Lie algebra homomorphism

dr:sl(2)!gl(A):

It is well known that the irreducible decomposition ofris determined by that ofdr. Now recall that the Lie algebrasl(2) is the vector space spanned by three elementse;f;hwith the bracket relations

[e;f]ˆh;[h;e]ˆ2e;[h;f]ˆ 2f:

It is easy to see that to decomposeAinto irreduciblesl(2)-modules is to decomposedr(e) into Jordan blocks (cf. [9]). Notice thatdC(e) is nilpotent and may be considered as a single Jordan block by conjugation sinceCis irreducible. Consequentlydr(e)2gl(A) may be considered as the multi- plication map(x‡y) by definition ofranddr. Hence the assertion follows

from Theorem 8. p

REMARK10. The isomorphism in Proposition 9 is known as the Clebsch- Gordan decomposition of the tensor productV(n 1)V(n 1) assl(2)- modules. We may also regard it as a consequence of Pieri's formula as follows. Let r_l denote the irreducible representation of GL(n) corre- sponding to a Young diagram l with l(l)n. Pieri's formula (e.g. [6, I

(5.16)]) applied to the character proves that r_mr_(r)M

l

r_l;

where the sum is taken over all partitionslsuch thatl=mis a horizontalr- strip. Whennˆ2 andmˆ(r) and allr_lare restricted toSL(2), it gives us the decompostion of Proposition 9 withrreplacingn 1.

With the same notation as above let WˆV(n 1)V(n 1). Let W ˆWsWa be the decomposition of W into the symmetric and alter- nating tensors respectively. (It is well known that the spacesW_s andW_a are irreducible GL(n)-modules, which we take for granted.) Via the re- presentation (10) the spacesW_s andW_aare also SL(2)-modules. The fol- lowing proposition shows how Ws and Wa decompose into irreducible SL(2)-modules.

PROPOSITION11. If n is even, then

W_s V|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}(2n 2)V(2n 6) V(2)

n=2

;

and

W_aV(2n|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}4)V(2n 8) V(0)

n=2

:

If n is odd, then

Ws V|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}(2n 2)V(2n 6) V(0)

(n‡1)=2

;

and

W_aV(2n|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}4)V(2n 8) V(2)

(n 1)=2

:

PROOF. Identify WˆA(n;2). Then Ws is the space spanned by the symmetric polynomials inAandWathe alternating polynomials. Hence the assertion follows immediately from Theorem 8 and Proposition 9. p

REMARK12. LetrˆFCbe the composite SL(2)!^C GL(n)!^F GL(W)

as defined in (11). Letrˆr_sr_abe the decomposition corresponding to WˆWsWa. Lethˆdiag q ¹;q

2SL(2). The plethysm formula (e.g.

[6, (I.8) Example 9)]) can be used to obtain the diagonal form for the ma- tricesr_s(h) andr_a(h) and the Jordan decomposition for

dr 0 0

1 0

ˆL:

As is conceived, it gives us the same decomposition as Proposition 11. In particular, it provides us with another proof for the fact that x‡y is a strong Lefschetz element in the Artinian algebraR=(xⁿ;yⁿ).

4. A(2;k), thek-fold tensor ofK².

Throughout this section we fixAˆA(2;k). SoAis the subspace of the polynomial ring K[x₁; ;x_k] spanned by square-free monomials. At the same timeAis endowed with the algebra structure with the identification:

AˆK[x₁; ;x_k]=(x²₁; ;x²_k):

We putLˆ (x₁‡x₂‡ ‡x_k) and Dˆ @

@x₁‡ ‡ @

@x_k:

We think of L and D as operating on the polynomial ring Rˆ

ˆK[x1; ;xk]. Recall thatRˆAI, whereIˆ(x²₁; ;x²_k). We denote by Dj_A the restricted mapDonA. Similarly by Lj_A we denote the map R=I!R=I induced by L. Thus Lj_A;Dj_A2End(A). Let H be the com- mutatorHˆ[Lj_A;Dj_A]. SoH2End(A).

PROPOSITION13. (a)hLj_A;Dj_A;Hiis ansl(2)-triple.

(b)There exists a Jordan basis B for Lj_A such that BnIm (Lj_A) is a basis ofKer (Dj_A).

PROOF. (a) LetJ_‡ˆ 0 0

1 0

;J ˆ 0

1 0 0

. Then the three elements hJ_‡;J ;[J_‡;J ]i, where [J_‡;J ]ˆ 1

0 0

1

, is ansl(2)-triple. This proves the case kˆ1. Let E2 be the 22 identity matrix. Then one sees that, using square free monomials as a basis ofA, the mapLj_Ais represented by

the matrix

X^k

iˆ1

E2 E2

|‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚}

i 1

J‡E|‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚}2 E2 k i

and similarlyDj_Aby X^k

iˆ1

E2 E2

|‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚}

i 1

J E|‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚}2 E2 k i

:

We induct on k to show that the three matrices Lj_A;Dj_A, and H:ˆ[Lj_A;Dj_A] satisfy the required relations [H;Lj_A]ˆ2Lj_A, and [H;Dj_A]ˆ 2Dj_A.

(b) See Humphreys ([4] pp. 31-34). p

The following theorem enables us to construct a Jordan basis forLj_A. THEOREM14. For iˆ0;1;2; ;[k=2], the vector space(KerD)\Aiis spannedby the Specht polynomial of degree i. The Specht polynomials arising from the standard Young tableaux form a basis of(KerD)\A.

Proof of Theorem 14 is postponed to the end of Proposition 18.

LEMMA15. KerDˆK[fxi xjj1i;jkg].

PROOF. Recall that RˆK[x₁; ;x_k] and KerDˆ ff 2RjDf ˆ0g.

Since KerDis a subalgebra ofR, we have

KerDK[fxi xjj1i;jkg]:

The right hand side is isomorphic to the polynomial ring in (k 1) variables.

Noticing that Dis surjective of degree 1, it is easily verified that they

coincide by comparing the Hilbert series. p

LEMMA16. Put V ˆ(KerD)\A. Then dimV_iˆMax k

i

k i 1

;0

;

where V_iˆV\A_i.

PROOF. First note that the Hilbert series ofAis given byP k i qⁱ.

The graded vector spaceAhas the strong Lefschetz property withLj_A a strong Lefschetz element. By Proposition 13, the mapsDj_Aand Lj_A are alike except that Dj_A is a degree 1 map. ThusDj_A:A_i!A_{i 1} is either injective or surjective. Thus the assertion follows. p LEMMA17. Let T be a Young tableau with the shapel. Suppose thatl has size k. LetDTbe the Specht polynomial defined by T.

(1) DTˆ1()lhas one row.

(2) D_T2A andD_T6ˆ1()lhas two rows.

(3) Suppose thatD_T2A. Then the degree ofD_Tis equal to the sec- ondterm ofl.

(4) DT2Ker [Dj_A:A!A]\Ai if lˆ(k i;i).

PROOF. (1), (2) and (3) are immediate from the definition of D_T. (4)

follows from Lemma 15. p

We need one more proposition to prove Theorem 14.

PROPOSITION18. Suppose thatlˆ(k i;i)is a Young diagram.

(a) The number of standard Young tableaux of shape l is k

i

k i 1

.

(b) The set of Specht polynomials defined by the standard Young tableaux of a fixedshapelis linearly independent.

PROOF. (a) Suppose thatTis a standard Young tableau. If the boxkis removed fromTit is a Standard Young tableau of sizek 1. Thus the in- duction works. (Details are left to the reader.) (b) Suppose thatTis stan- dard. Then one notices easily that the head term ofD_T in the reverse lex- icographic monomial order is the product of monomials in the second row.

Thus the proof is complete. p

PROOF OFTHEOREM14. Immediate by Lemmas 15, 16, 17 and Propo-

sition 18. p

Now our main theorem of this section is stated as follows.

THEOREM19. Let JˆLj_AandV_dˆ(KerDj_A)\A_d. Put hˆ[k=2], and m_iˆ k

i

k i 1

for iˆ0;1;. . .;h.

(1) The conjugacy classl(J)of J is given by

l(J)ˆ

(k|‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}‡1; ;k‡1

m0

;k|‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}1; ;k 1

m1

;k|‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}3; ;k 3

m2

; ;1;|‚‚‚‚{z‚‚‚‚} ;1

mh

); if k is even;

(k|‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}‡1; ;k‡1

m0

;k|‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}1; ;k 1

m1

;k|‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚}3; ;k 3

m2

; ;2;|‚‚‚‚{z‚‚‚‚} ;2

mh

); if k is odd:

8>

>>

<

>>

>:

(2) For0dh, and0ik 2d, the vector space Jⁱ(Vd)

is an irreducible S_k-module of isomorphism typelˆ(k d;d).

(3) For any Specht polynomialD_T2V_d, the vector space hDT;J(DT);J²(DT) ;J^{k 2d}(DT)i

is an irreducible GL(2)-module of isomorphism typelˆ(k d;d).

(4) An irreducible decomposition of A as S_k-modules is given by AˆM^h

dˆ0

M

k 2d

iˆ0

Jⁱ(V_d)

! :

In particular the irreducible S_k-module of type lˆ(k d;d)occurs (k‡1 2d)times, andthe irreducible GL(2)-module of type(k d;d) occurs mdtimes.

PROOF. (1) See Proposition 5. (2) In Lemma 17 and Proposition 18 we showed that the spaceV_dis spanned by the Specht polynomials defined by the standard Young tableau of shape (k d;d). It is well known that this is irreducible. (Cf. [8].) Alsolⁱ(Vd) is isomorphic toVdunless it is trivial be- causex1‡ ‡x_kisS_k-invariant. (3) Letrbe the composition

SL(2)!^C GL(2)!^F GL(A);

where C is the natural injection and Fis the tensor representation. To decomposeAinto irreducibleGL(2)-modules is the same as to decompose it as SL(2)-modules. This is obtained by decomposing the Lie algebra re- presentation:

dr:sl(2)!gl(A):

Now the assertion follows from Lemma 13. (4) Clear from (1), (2) and (3).

p

EXAMPLE 20. Let kˆ4. Let lˆ (x1‡x2‡x3‡x4)2End(A(2;4)).

We exhibit the Jordan basis ofl. The Hilbert series ofAis (1; 4; 6; 4; 1).

The derived sequence is (1; 3; 2).

1. The Specht polynomial of degree 0 is 1.

2. The Specht polynomials of degree 1 (corresponding to the standard Young tableau with shape lˆ(3;1)) are a:ˆx₁ x₂;b:ˆx₁ x₃ andc:ˆx₁ x₄.

3. The Specht polynomials of degree 2 (corresponding to the standard Young tableau with shape lˆ(2;2)) are f :ˆ(x1 x2)(x3 x4) andg:ˆ(x1 x3)(x2 x4).

The bases for the irreducible decomposition of AˆA(2;4) as GL(2)- modules are:

1. h1;l;l²;l³;l⁴i

3. ha;la;l²aiandhb;lb;l²biandhc;lc;l²ci 3. hfiand hgi

We have 2⁴ˆ51‡33‡12. When lⁱ is expanded, all terms which contain a square of a variable should be regarded zero. With this conventionlⁱis equal to theith elementary symmetric polynomial multi- plied byi!.

5. Application to the theory of Gorenstein rings.

Put AˆR=(x²₁; ;x²_k) and lˆx₁‡x₂‡ ‡x_k 2A. Using the notation of Proposition 13, we have lˆLj_A. Since we have obtained a Jordan basis for Dj_A:A!A, it gives us a basis for 0:l as well as for Ker [D:A!A]. (cf. Proposition 13.) However, it does not necessarily a minimal ideal basis for 0:l. In this section we would like to exhibit a minimal basis of the ideal 0:l. Denote by ( )^?:A!Athe ``Hodge dual'' ofA.

Namely, define

M^? ˆ(x1 xk)=M

for a monomialM2A. By linearity this is extended to define the dual map A!A. The following lemma is easy to see, and proof is omitted.

LEMMA21. Letlˆ(r;s)be a Young diagram, with r‡sˆk, andletD be a Specht polynomial of shapel. Then for any integer i,1ik,

(1) @

@x_iDis either0or a Specht polynomial of shape(r‡1;s 1).

(2) @

@x_iF

_?

ˆxi(F^?)for any F2A:

(3)

@

@x1‡ ‡ @

@xk

F

_?

ˆ(x₁‡ ‡x_k)(F^?)for any F2A:

In the next theoremD_T denotes the Specht polynomial defined by the Young tableau of some shapel.

THEOREM22. In the polynomial ring R, put Iˆ(x²₁; ;x²_k):(x₁‡ ‡x_k):

Then the minimal number of generators of the ideal I is k‡ k

h

k h‡1

:

Here h is such that hˆk=2or hˆ(k‡1)=2according as k is even or odd.

If k is even, then

Iˆ(x²₁; ;x²_k)‡ fD^?_Tjshape(T)ˆ(h;h)gR;

and if k is odd,

Iˆ(x²₁; ;x²_k)‡ fD^?_Tjshape(T)ˆ(h;h 1)gR:

(If k is even, it is the same if?is dropped.)

PROOF. It is enough to prove the second part. Put V ˆ

ˆKer [D:A!A], and letV_ibe the homogeneous part of degreeiso that we have the decompositionVˆL

V_i. First notice that

@

@x1V_i‡ ‡ @

@xkV_iˆV_{i 1} …12†

by Lemma 21 (1). This is equivalent to

x₁V_i^?‡ ‡x_kV_i^?ˆV_i^?₁ …13†

by Lemma 21 (2). We have already proved that the vector space V is spanned by the Specht polynomials. Hence by Lemma 21 (3), we have that 0:lis spanned byD^?_Tfor variousT. The above equality (13) shows that 0:l

is, as an ideal, generated byV_h^?. p

COROLLARY23. Let AˆK[x1; ;xk]=(x²₁; ;x²_k), andlet l be the linear element lˆx1‡ ‡xkof A. Then the minimal number of generators of the ideal(0:l)is

k h

k h‡1

:

Here h is as inTheorem 22.

PROOF. Immediate by Theorem 22. p

THEOREM24. As before let AˆR=(x²₁; ;x²_k), andlet y2A be a gen- eral linear form of A. Then the Macaulay type of A=(y)is the hth Catalan number 1

h‡1 2h

h . Here h is as inTheorem 22. Equivalently if we put Iˆ(x²₁; ;x²_k;Y), where Y is a general linear form of the polynomial ring andif we write a minimal free resolution of R=I as

0!F_k!F_{k 1}! !F₁!R!R=I!0;

then we haverank F_kˆ 1 h‡1

2h h .

PROOF. The Macaulay type ofA=(y) is equal to the minimal number of generators of 0:y. It is well known that this is also equal to the last rank of the minimal free resolution of 0:y. Now it suffices to notice that

k h

k h‡1

ˆ 1 h‡1

2h h ;

wherehˆk=2 or (k‡1)=2 as in Theorem 22. p

6. The Hilbert series of the ring of invariants ofA(n;k).

In this section we letAˆK[x1; ;x_k]=(xⁿ₁; ;xⁿ_k) wherenandkare arbitrary positive integers. Let G:ˆS_k act onA by permutation of the variables. In the next theorem we would like to exhibit the ring of in- variantsA^Gand the Hilbert series ofA^G.

THEOREM25.

A^GˆK[e1; ;ek]=(pn;pn‡1; ;pn‡k 1):

Here edis the elementary symmetric polynomial of degree d and pdis the power sum pdˆx^d₁‡ ‡x^d_k. Hence the Hilbert series of A^Gis:

h_A^G(q)ˆ(1 qⁿ)(1 q^n‡1) (1 q^{n‡k 1}) (1 q¹)(1 q²) (1 q^k) PROOF. Consider the exact sequence

0!(xⁿ₁; ;xⁿ_k)!R!A!0:

Since chKˆ0, we have the exact sequence

0!(xⁿ₁; ;xⁿ_k)^G!R^G!A^G!0:

Note that R^GˆK[e1; ;e_k]. The socle degree of A is nk k, and A_{nk k}ˆ h(e_k)^{n 1}i. Sinceeⁿ_k ¹is fixed under the action ofG, this shows that the maximum degree of elements ofA^Gisnk k.

Put A⁰ˆR^G=(p_n;p_n‡1; ;p_{n‡k 1}). Obviously we have a natural sur- jection:

A⁰!A^G!0 …14†

which we would like to prove to be an isomorphism. First note that the rational series in the statement of this theorem is the Hilbert series ofA⁰. (This can be obtained using the factR^GˆK[e₁;e₂; ;e_k].) This shows that A⁰andA^Ghave the same socle degree, which is equal tonk k. SinceA⁰is an Artinian Gorenstein ring, the one dimensional vector space of the maximum degree is the unique minimal ideal of the ring A⁰. This shows that the map (14) cannot have a non-trivial kernel. This completes the

proof. p

REMARK26. 1. Since limq!1

(1 qⁿ)(1 q^n‡1) (1 q^{n‡k 1})

(1 q¹)(1 q²) (1 q^k) ˆ n‡k 1 k

; we have dimA^Gˆh_AG(1)ˆ n‡k 1

k

. This is expected sinceA^Gis the irreducible GL(n)-module corresponding to the trivial l, which is the symmetric tensor space.

2. One may conceive that the Hilbert series of Y^l(A) where Y^l is a Young symmetrizer corresponding to lˆ(k₁; ;k_r)`k should be ob- tained as a q-analog of the dimension formula of the irreducibleGL(n)- module in the decomposition of (Kⁿ)^k. This can be proved using the q-

dimension formula ([5] Proposition 10.10). We indicate an outline of proof in the next proposition.

Recall that we have the isomorphismA(n;k)(Kⁿ)^kas vector spaces.

The symmetric groupS_kacts onA(n;k) as permutations of variables. Also the general linear group GL(n) acts on A(n;k) as the tensor repre- sentation. Recall that Y^l(A(n;k))ˆSkY^T(A(n;k)), where the map Y^T:A(n;k)!A(n;k) is a Young symmetrizer with shapel`k with at most n parts. It is well known that Y^l(A(n;k)) is an irreducible

GL(n)S_k

… †-module and every irreducible GL(n)-module is obtained as an irreducible component ofY^l(A(n;k)) for somel. (For details see, e.g., [2] pp. 336-339.)

PROPOSITION27. Forlˆ(k₁;. . .;k_r)`k with rn, the Hilbert series of the module Y^l(A(n;k))as a graded subspace of A(n;k)is

h_Y^l_(A(n;k))(q)ˆq^n(l) Y

1i5jn

[k_i k_j‡j i]

[j i] : Here n(l):ˆP

j1(j 1)kj,[a]denotes1 q^a

1 q for any positive integer a, and k_jˆ0for j>r.

Before we give a proof we review theq-dimension formula [5, § 10.9,

§ 10.10] forGL(n). LetWbe a finite dimensional irreducibleGL(n)-module with highest weightl, andW ˆL

m Wmits weight space decomposition. The q-dimension ofW is defined by

dim_qW ˆ X

m:weight ofW

( dimW_m)q^{hl m;di}; …15†

whereh;idenotes the standard inner product onRⁿ, anddis ``the half sum of the positive roots'' defined by (n 1;n 3;. . .; n‡1)=22Rⁿ. Note that the exponent hl m;diis the number of the simple roots oc- curring inl m, sinceha;di ˆ1 for any simple roota.

Theq-dimension formula [5, Proposition 10.10] says dimqWˆ Y

1i5jn

[k_i k_j‡j i]

[j i] ; …16†

where W is a finite dimensional irreducible GL(n)-module with highest weightlˆ(k1;. . .;kn).

PROOF OF PROPOSITION 27. - Let GL(n) act on the n-dimensional K-vector space K[x]=(xⁿ) through the vector representation with the basis 1;x;x²;. . .;x^{n 1}. Then x^{j 1} is a weight vector with weight

e_jˆ(0;. . .;1;. . .;0) (only jth entry is 1). Therefore the weight of a

weight vector decreases by a single simple root (like e_j e_j‡1), as its degree increases by one. This principle holds also for A(n;k), since A(n;k)ˆK[x1;. . .;xk]=(xⁿ₁;. . .;xⁿ_k) is isomorphic to the tensor product K[x1]=(xⁿ₁)_{K K}K[x_k]=(xⁿ_k) as GL(n)-modules. In particular, the homogeneous polynomial with the least degree in Y^l(A(n;k))A(n;k) is the highest weight vector.

It follows from the principle above and the note after (15) that the Hilbert series and the q-dimension of Y^l(A(n;k)) differ only by a power of q. Notice that the q-dimension dimqY^l(A(n;k)) starts with q⁰ followed by higher terms, while the Hilbert series h_Y^l_(A(n;k))(q) starts with q^d, where d is the degree of the monomial of the highest weight vector. Therefore it suffices to show that the degree of a monomial with weight lˆ(k1;. . .;kr)`k (rn) is equal to n(l).

Since the weight of 1 ise1ˆ(1;0;. . .;0) in theGL(n)-moduleK[x]=(xⁿ), the weight of 1 inA(n;k)'(K[x]=(xⁿ))^kiske₁ˆ(k;0;. . .;0). The degree of a monomial with weightlis

hke1 l;di ˆ h(k k1; k2; k3;. . .; kr);di

ˆ k₂‡2k₃‡ ‡(r 1)k_r:

We thus have proved the assertion. p

7. The multiplicative group of the Artinian algebras.

We need the following proposition from commutative algebra. Very important to us is the corollary that follows, which says that the generic form of degreedhas the largest rank among all elements of orderd. The proof for the case where M is principal is given in [10, Proposition A in Appendix]. Since the proof goes verbatim, we omit the proof.

PROPOSITION28. Let(R;m)be an Artinian local ring with residue field K andM a finite R-module. Let m₁; ;m_sbe arbitrary elements ofm, and let lˆx1m1‡x2m2‡ ‡xsms with xi2R. Let X1; ;Xs be in- determinates over R, let QˆR[X1; ;Xs]be the polynomial ring andlet

Me ˆMRQ=Y(MRQ):

where Y ˆX1m1‡ ‡Xsms. Let U be the set of non-zero-divisors of Q andletabe the ideal(X1 x1; ;Xs xs)Q. Then we have

length_QU(MeU)length_Q=a(Me _QQ=a):

…17†

Furthermore, there exists a radical idealbK[X1;. . .;Xs]such that the inequality(17)is the equality if andonly if the ideal(X1 x1;. . .;Xs xs) does not containb.(x_idenotes the image of x_iin K.)

COROLLARY29. Suppose that AˆL^c

iˆ0A_iis a standard graded K-alge- bra, where K is an infinite field. Let MˆL^b

iˆaM_i be a finite graded A- module. Letmbe the maximal ideal of A. Let d be any positive integer.

Then the minimal value

MinfdimM=yMg;

where y ranges overm^d, is attainedby a homogeneous element y of degree d. If M has the strong Lefschetz property, and if z2A₁is a strong Lefschetz element for M, then we have

dimM=z^dMˆMinfdimM=yMjy2m^dg;

or equivalently, the rank of the linear mapz^d:M!M is greater than or equal to the rank ofy:M!M for any y2m^d.

PROOF. Since m^dˆL

jdA_j and since m^d=m^d‡1A_d, any represen- tative of aK-basis ofm^d=m^d‡1is a minimal generating set of the idealm^d and vice versa. Thus the first part is a direct consequence of Proposition 28.

(We apply the proposition by lettingm1; ;ms be a set of minimal gen- erating set of m^d.) To see the second part suppose that y2m^d is any homogeneous element. The linear mapy:M!Mdecomposes into the sum of piece-wise linear maps y:M_i!M_i‡d. Hence the rank of the linear mapy:M!Mdoes not exceed

X^{b d}

iˆa

MinfdimMi;dimMi‡dg:

…18†

(Here we have setMb‡iˆ0 fori>0.) Now ifzis a strong Lefschetz ele- ment and ifyˆz^d, then the rank ofy:M!Mis equal to the integer given in (18) above. In fact the linear map z^d:M_i!M_i‡d is either in- jective or surjective as is easily deduced from the definition. Thus we have shown that the rank of the linear mapz^d:M!Mis greater than or equal

to the rank ofy⁰:M!Mfor anyy⁰2m^d. This is equivalent to the second

assertion of the corollary. p

Let AˆL^c

iˆ0A_ibe an Artinian standard gradedK-algebra. We denote by A the multiplicative group ofA. Obviously it is an Abelian algebraic group with the underlying algebraic set

(a0;a1; ;ac)2AˆM

Aijai2Ai;a06ˆ0 : We may regard it as an algebraic subgroup ofGL(A).

EXAMPLE30. LetAˆK[x]=(xⁿ). Thenf 2Ainduces an automorph- ismf :A!A. If we write

f ˆa₀‡a₁x‡a₂x²‡ ‡a_{n 1}x^{n 1}; thenf is represented by the matrix

a0 a1 a2 an 2 an 1

a0 a1 an 2

a0 an 3

... .. . ...

.. . ...

a_o 0

BB BB BB B@

1 CC CC CC CA :

This way we regard the groupAas the algebraic subgroup ofGL(n). p LetV be ann-dimensional vector space and let

F:GL(V)!GL(V^k)

be the tensor representation. Now letV ˆK[x]=(xⁿ) be as in Example 30 above and letAˆA(n;k)ˆK[x1; ;xk]=(xⁿ₁; ;xⁿ_k). Define the map

F⁰:V!A by

f(x)7!f(x₁)f(x₂) f(x_k):

Then it is easy to see thatF⁰is a group homomorphism. As described above we may identifyVas a subgroup ofGL(V) andAa subgroup ofGL(V^k).

Note thatF⁰is nothing but the restriction ofF. In other words we have the commutative diagram:

…19†

where the vertical maps are natural inclusions.

ByJ(a;n) we denote thennmatrix

It is an elementary fact that if a matrixMhas a single eigenvaluea, thenM decomposes into a direct sum of such blocks as follows:

This is known as the Jordan canonical form of M. We denote this matrix by

J(a;n1) J(a;nr):

DEFINITION 31. Let VˆP^b

iˆaV_i be a graded K-vector space with the Hilbert series

h(q):ˆhV(q)ˆX^b

iˆa

hiqⁱ;

where we assume that Va6ˆ0andVb6ˆ0. Suppose that h(q)is symmetric andunimodal. Then thedual Hilbert seriesof V, or of h(q), is defined to be the descending sequence of positive integers

u₁;u₂; ;u_s which satisfy

h(q)ˆ 1 1 q

X^r

iˆ1

q^{(a‡b‡1 ui)=2} q^{(a‡b‡1‡ui)=2} : …20†

REMARK32. With the same notation as above, the Hilbert seriesh(q) of V gives a partition of the positive integer dimV. In fact dimV ˆ

ˆha‡h_a‡1‡ ‡h_b. (If one prefers, the summands may be arranged in the increasing or decreasing order.) The dual Hilbert series of V is nothing but the dual partition of the Hilbert series. For example if h(q)ˆq^a‡3q^a‡1‡6q^a‡2‡7q^a‡3‡6q^a‡4‡3q^a‡5‡q^a‡6, then the dual Hilbert series of V is 7;5;5;3;3;3;1, independent of the shift bya.

Now we may state the main theorems of this section.

THEOREM33. LetF:GL(n)!GL((Kⁿ)^k)be the tensor representation of the general linear group GL(n). Let MˆJ(a;n). Then a^kis the unique eigenvalue ofF(M)andthe Jordan canonical form ofF(M)is given by

J(a^k;u1) J(a^k;ur);

where u₁; ;u_r is the dual Hilbert series of the polynomial h(q)ˆ

ˆ(1‡q‡ ‡q^{n 1})^k.

PROOF. Put AˆA(n;k)ˆK[x1; ;xk]=(xⁿ₁; ;xⁿ_k). Then, since (1‡ ‡q^{n 1})^kis the Hilbert series ofA, it suffices to prove that (1)a^kis the unique eigenvalue of F(M) and (2) F(M) a^k has the same Jordan canonical form as that of a strong Lefschetz element ofA. (See Proposition 6.) PutV ˆK[x]=(xⁿ). We have the commutative diagram (19) as shown above. With the natural embedding V!GL(n) the element a‡x corresponds to the matrix J(a;n). Thus F⁰(a‡x)ˆF(M). Now put l⁰ˆF⁰(a‡x) a^k. Then we have

l⁰ˆ(a‡x1) (a‡xk) a^kˆa^{k 1}(x1‡ ‡xk)‡polynomial inxi of degree2:

Putlˆx1‡ ‡xk. Then we have 1

a^{k 1}l⁰

_i

lⁱ modm^i‡1; iˆ1;2; :

Thus by Proposition 28 we have that the rank of (l⁰ⁱ) is equal to the rank of (lⁱ) for alli1. This shows thatF⁰(a x) a^kandlhave the same Jordan canonical

form. Now the proof is complete. p

THEOREM34. Letl`k with at most n parts andlet W^lbe an irreducible GL(n)-module corresponding to l. Denote by f^l:GL(n)!GL(W^l) the corresponding irreducible representation of GL(n). Put

A(n;k)ˆK[x₁; ;x_k]=(xⁿ₁; ;xⁿ_k):

Suppose that the sequence (f|‚‚‚‚‚{z‚‚‚‚‚}1; ;f1

m1

;f|‚‚‚‚‚{z‚‚‚‚‚}2; ;f2 m2

; ;f|‚‚‚‚‚{z‚‚‚‚‚}s; ;fs ms

)

is the dual Hilbert series of Y^l(A(k;n)). Then the Jordan canonical form of f^l(J(a;n))is given by

J(a^k;f1) J(a^k;f1)

|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}

m1=m

J(a|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}^k;f2) J(a^k;f2)

m2=m

J(a|‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚‚}^k;fs) J(a^k;fs)

ms=m

;

where m is the number of the standard Young tableaux of shapel.

Before proving this theorem, we collect some basic facts in a lemma.

LEMMA35. Put AˆA(n;k)ˆK[x1; ;x_k]=(xⁿ₁; ;xⁿ_k). Letl`k and let T be a standard Young tableau with shapel. We identify A with(Kⁿ)^k. Then

(1) Y^l(A)A is an irreducible GL(n)… Sk†-module.

(2) The number of times that the irreducible GL(n)-module W^l occurs in Y^l(A)is equal to the dimension of the irreducible S_k-module that occurs in Y^l(A).

(3) Suppose that W Y^l(A)is an irreducible GL(n)-module. Let m be the dimension of any irreducible Sk-module that occurs in Y^l(A). Then h_Y^l_(A)(q)ˆmhW(q).

(4) Let lˆx1‡ ‡x_k. Then l is a strong Lefschetz element for Y^l(A). Hence the endomorphisml2End(Y^l(A))decomposes as

J(0;u₁) J(0;u_r);

where u1; ;uris the dual Hilbert series of Y^l(A).

PROOF. (1) and (2) are immediate from the Schur-Weyl duality. Let WY^l(A) be an irreducible GL(n)-module and letT₁; ;T_m be all the standard Young tableaux with shapel. Then we have

Y^l(A)ˆS_kWˆY^T1W Y^TmW:

Thus (3) follows. We prove (4). WriteLfor the linear mapl2End(A). Let D2End(A) be the degree 1 map such thatD;L;[D;L] is ansl(2)-triple.

We note thatDis compatible with the action ofS_kas well asL. LetZbe any homogeneous basis for 0:l. Then the set S

i0Dⁱ(Z) is the Jordan basis forL, and the subsetY^l(A)\ S

i0Dⁱ(Z)

is a Jordan basis ofLj_Yl(A). This shows

(4). (cf. Proposition 7.) p

PROOF OFTHEOREM34. - We use the notation of the lemma above. It is well known thatW^lappears as a submodule of (Kⁿ)^k. First recall that the dimension of an irreducibleS_k-module inY^l(A) is equal to the number of the standard Young tableaux of shape l. To prove the assertion of the theorem it suffices to show that (1)F⁰(a‡x) restricted on the submodule Y^l(A) has the unique eigenvaluea^kand (2)l⁰:ˆF⁰(a‡x) a^kdecomposes into Jordan blocks in the same way as the multiplicationl2End(Y^l(A)) decomposes into Jordan blocks. (It should be noticed beforehand that lY^l(A)Y^l(A) andl⁰W^lW^l, which is easy to see.) By (4) of the lemma above,lˆx₁‡ ‡x_kis a strong Lefschetz element forY^l(A). Hence the

proof is complete. p

COROLLARY36. The graded vector space Y^l(A)has a unimodal sym- metric Hilbert series for anyl`k.

PROOF. The assertion is an immediate consequence of the fact that l2Ais a strong Lefschetz element forY^l(A). (cf. [5, Exercise 10.12].) p

REFERENCES

[1] W. FULTON,Young Tableaux, London Mathematical Society Student Text 35, Cambridge University Press, 1997.

[2] R. GOODMAN - N. R. WALLACH, Representations andInvariants of the Classical Groups, Cambridge University Press, 1998.

[3] T. HARIMA- J. WATANABE,The finite free extension of an Artinian K-algebra with the strong Lefschetz property, Rend. Sem. Mat. Univ. Padova,110(2003), pp. 119-146.

[4] J. E. HUMPHREYS,Introduction to Lie Algebras and Representation Theory, Springer-Verlag, 1972.

[5] V. G. KAC, Infinite-dimensional Lie algebras, third ed., Cambridge Uni- versity Press, Cambridge, 1990.

[6] I. G. MACDONALD, Symmetric functions andHall Polynomials, 2nd ed., Oxford Science Publications, London, 1995.

[7] T. MAENO, Lefschetz property, Schur-Weyl duality and a q-deformation of Specht polynomial, Comm. Algebra,35(2007), pp. 1307-1321.

[8] T. TERASOMA- H. -F. YAMADA,Higher Specht polynomials for the symmetric group, Proc. Japan Acad.,69(1993), pp. 41-44.

[9] J. WATANABE,The Dilworth number of Artinian rings andfinite posets with rank function, Adv. Stud. Pure Math.,11(1987), pp. 303-312.

[10] J. WATANABE,m-full ideals, Nagoya Math. J.,106(1987), pp. 101-111.

[11] H. WEYL, Classical groups, their invariants andrepresentations, 2nd Edition, Princeton, 1946.

Manoscritto pervenuto in redazione il 20 giugno 2007.