Vector invariants of a class of pseudo-reflection groups and multisymmetric syzygies
aa r X i v : . [ m a t h . R T ] F e b Vector invariants of a class of pseudo-reflectiongroups and multisymmetric syzygies ∗ M. Domokos † R´enyi Institute of Mathematics, Hungarian Academy of Sciences,P.O. Box 127, 1364 Budapest, Hungary, E-mail: [email protected]
Abstract
First and second fundamental theorems are given for polynomialinvariants of a class of pseudo-reflection groups (including the Weylgroups of type B n ), under the assumption that the order of the groupis invertible in the base field. As a special case, a finite presentation ofthe algebra of multisymmetric polynomials is obtained. Reducednessof the invariant commuting scheme is proved as a by-product. Thealgebra of multisymmetric polynomials over an arbitrary base ring isrevisited. MSC: 13A50, 14L30, 20G05
Fix natural numbers n and q , and a field K . Apart from Theorem 2.7 andSection 5, we shall assume that n ! q is invertible in K , and assume that K contains a primitive q th root of 1. Denote by G = G ( n, q ) the subgroupof GL ( n, K ) consisting of the monomial matrices whose non-zero entriesare q th roots of 1. The order of G is n ! q n , and as an abstract group, G is isomorphic to the wreath product of the cyclic group C q of order q andthe symmetric group S n ; that is, G is isomorphic to a semi-direct product( C q × · · · × C q ) ⋊ S n . ∗ J. Lie Theory 19 (2009), 507-525. † Partially supported by the Bolyai Fellowship, OTKA K61116, and the LeverhulmeTrust. G on V = K n . Since G is generatedby pseudo-reflections, by the Shephard-Todd Theorem [37] (see [7] for auniform proof in characteristic zero, and [39] for the case when char( K ) ispositive and co-prime to the order of G ) the algebra K [ V ] G of polynomialinvariants is generated by algebraically independent elements. Now considerthe diagonal action of G on V m = V ⊕ · · · ⊕ V , the direct sum of m copiesof V . The algebra K [ V m ] G is no longer a polynomial ring if m ≥ G is not the trivial group). In the present paper we show a very short andsimple argument that yields simultaneously the generators of K [ V m ] G (firstfundamental theorem) and the relations among these generators (secondfundamental theorem). Our main result is Theorem 3.2, which providesan explicit finite presentation of K [ V m ] G ( n,q ) in terms of generators andrelations. In the proof we apply Derksen’s degree bound on syzygies [9] andideas of Wallach and Garsia [20].In the special case q = 1 we have G = S n , and K [ V m ] S n is the algebra ofmultisymmetric functions, which received much attention in the literature(see the references in Remark 2.6 (ii)). Our approach gives new insighteven in this special case, especially by its simplicity and transparency. Wemention also that no finite presentation of the algebra of multisymmetricfunctions appeared in prior work (apart from the case of char( K ) = 2 studiedin [15]).Another interesting special case is when q = 2, and the group G is theWeyl group of type B n . The generators of K [ V m ] G ( n, were determined in[23] and [22]. To the best of our knowledge, the second fundamental theoremhas never been considered in the literature when q > GL ( n, C )-invariant commuting scheme is reduced, see Theorem 4.1.The present paper joins the content of our preprints [12] and [13] (somedigressions from the preprints have been omitted). In addition, in Section 5we adjust our method for arbitrary base rings and clarify and strengthenthe known results in this case. In particular, in Theorem 5.5 we give anew characteristic free (infinite) presentation of the ring of multisymmetricpolynomials. Denote by x ( i ) j the function on V m mapping an m -tuple of vectors to the j th coordinate of the i th vector component. The coordinate ring K [ V m ] is2he mn -variable polynomial ring over K with generators x ( i ) j , i = 1 , . . . , m , j = 1 , . . . , n .There is a natural multigrading on K [ V m ] preserved by the action of G .Namely, f ∈ K [ V m ] has multidegree α = ( α , . . . , α m ), if it has degree α i inthe set of variables x ( i ) , . . . , x ( i ) n , for all i = 1 , . . . , m . Write K [ V m ] Gα forthe multihomogeneous component of K [ V m ] G with multidegree α . Considerthe symbols x (1) , . . . , x ( m ) as commuting variables, and denote by M ( q ) theset of non-empty monomials in the variables x ( i ) whose degree is divisible by q . In particular, M (1) is the set of all non-empty monomials in the variables x ( i ). For w = x (1) α · · · x ( m ) α m ∈ M ( q ) set w h j i = x (1) α j · · · x ( m ) α m j , anddefine [ w ] = n X j =1 w h j i . (These are polarizations of the usual power sum symmetric functions .) Notethat the multidegree of [ w ] is α . Since for all j = 1 , . . . , n and w ∈ M ( q ),the term w h j i is invariant with respect to the normal subgroup N consist-ing of the diagonal matrices in G , the S n -invariant polynomial [ w ] is G -invariant. Sometimes we shall think of the x ( i ) as the generic diagonalmatrices diag( x ( i ) , . . . , x ( i ) n ), i = 1 , . . . , m , and then [ w ] is nothing butthe trace of the diagonal matrix w . Proposition 2.1
The products [ w ] · · · [ w r ] with r ≤ n , w i ∈ M ( q ) consti-tute a K -vector space basis of K [ V m ] G ( n,q ) .Proof. An arbitrary monomial u ∈ K [ V m ] in the variables x ( i ) j can bewritten as u = u h i · · · u n h n i with a unique n -tuple ( u , . . . , u n ) of monomialsin M (1) ∪ { } . The algebra K [ V m ] N is spanned by the u with u , . . . , u n ∈M ( q ) ∪ { } . The action of S n permutes these monomials, and since G ( n, q )is generated by N and S n , the S n -orbit sums of the N -invariant monomialsof multidegree α form a basis in K [ V m ] Gα . For a multiset { w , . . . , w r } with r ≤ n , w i ∈ M ( q ), denote by O { w ,...,w r } the S n -orbit sum of the monomial w h i · · · w r h r i . Call r the height of this monomial multisymmetric function .Set T { w ,...,w r } = [ w ] · · · [ w r ].So the O { w ,...,w r } with multideg( w · · · w r ) = α form a basis in K [ V m ] Gα .Assume that the multiset { w , . . . , w r } contains d distinct elements withmultiplicities r , . . . , r d (so r + · · · + r d = r ), then expanding T { w ,...,w r } as a linear combination of monomial multisymmetric functions, the coef-ficient of O { w ,...,w r } is r ! · · · r d ! (which is non-zero by our assumption onthe characteristic of K ), and all other monomial multisymmetric functions3ontributing have height strictly less than r . This clearly shows the claim. (cid:3) Remark 2.2
After writing [12] we noticed that the special case q = 1 ofProposition 2.1 appears as Corollary 2.3 of [3] (see also Section 3 of [4] forthe interpretation needed here).Associate with w ∈ M ( q ) a variable t ( w ), and consider the polynomialring F ( q ) = K [ t ( w ) | w ∈ M ( q )]in infinitely many variables. Denote by ϕ ( q ) : F ( q ) → K [ V m ] G ( n,q ) the K -algebra homomorphism induced by the map t ( w ) [ w ]. This is asurjection by Proposition 2.1. Next we introduce a uniform set of elementsin its kernel. For a multiset { w , . . . , w n +1 } of n + 1 monomials from M ( q ),we associate an element Ψ( w . . . , w n +1 ) ∈ F ( q ) as follows. Write P n +1 forthe set of partitions λ = λ ∪· · ·∪ λ h of the set { , . . . , n +1 } into the disjointunion of non-empty subsets λ i , and denote h ( λ ) = h the number of parts ofthe partition λ . SetΨ( w , . . . , w n +1 ) = X λ ∈P n +1 ( − h ( λ ) h ( λ ) Y i =1 ( | λ i | − · t ( Y s ∈ λ i w s ) . (See Section 6 for examples.) Proposition 2.3
The kernel of ϕ ( q ) contains the element Ψ( w , . . . , w n +1 ) for arbitrary w , . . . , w n +1 ∈ M ( q ) .Proof. One way to see it is to specialize the fundamental trace identity of n × n matrices to diagonal matrices. Indeed, let Y (1) , . . . , Y ( n +1) be generic n × n matrices (so their entries generate an ( n + 1) n -variable commutativepolynomial ring). For a permutation π ∈ S n +1 with cycle decomposition π = ( i · · · i d ) · · · ( j . . . j e )set Tr π = Tr( Y ( i ) · · · Y ( i d )) · · · Tr( Y ( j ) · · · Y ( j e )) . X π ∈ S n +1 sign( π )Tr π = 0 (1)called the fundamental trace identity of n × n matrices. (This can be ob-tained by multilinearizing the Cayley-Hamilton identity to get an identitymultilinear in the n matrix variables Y (1) , . . . , Y ( n ), and then multiplyingby Y ( n + 1) and taking the trace; see for example [18] for details.) Thesubstitution Y ( i ) w i ( i = 1 , . . . , n + 1) in (1) yields the result. (cid:3) Remark 2.4
We shall benefit in Section 4 from the fact that in the aboveproof we descend from a statement about non-commuting matrices. A moreelementary proof is the following: express the ( n + 1)th elementary sym-metric function in n + 1 variables in terms of the first n + 1 power sumsusing the Newton formulae; then multilinearize this identity, and specializethe ( n + 1)th coordinates to zero. Theorem 2.5 (i) The kernel of the K -algebra homomorphism ϕ ( q ) is theideal generated by the Ψ( w , . . . , w n +1 ) , where w , . . . , w n +1 ∈ M ( q ) .(ii) The algebra K [ V m ] G ( n,q ) is minimally generated by the [ w ] , where w ∈M ( q ) with deg( w ) ≤ nq .Proof. (i) The coefficient in Ψ( w , . . . , w n +1 ) of the term t ( w ) · · · t ( w n +1 )is ( − n +1 , and all other terms are products of at most n variables t ( u ). Sothe relation ϕ ( q )(Ψ( w , . . . , w n +1 )) = 0 can be used to rewrite [ w ] · · · [ w n +1 ]as a linear combination of products of at most n invariants of the form [ u ](where u ∈ M ( q )). So these relations are sufficient to rewrite an arbitraryproduct of the generators [ w ] in terms of the basis given by Proposition 2.1.This obviously implies our statement about the kernel of ϕ ( q ).(ii) If w ∈ M ( q ) and deg( w ) > qn , then w can be factored as w = w · · · w n +1 where w i ∈ M ( q ). The term t ( w ) appears in Ψ( w , . . . , w n +1 )with coefficient − ( n !), which is invertible in K by our assumption on thecharacteristic. Therefore the relation ϕ ( q )(Ψ( w , . . . , w n +1 )) = 0 shows that[ w ] can be expressed as a polynomial of invariants of strictly smaller degree.It follows that K [ V m ] G ( n,q ) is generated by the [ w ], where w ∈ M ( q ) withdeg( w ) ≤ nq . This is a minimal generating system, because if deg( w ) ≤ nq for some w ∈ M ( q ), then [ w ] can not be expressed by invariants oflower degree, since there is no relation among the generators whose degreeis smaller than ( n + 1) q by (i). (cid:3) emark 2.6 (i) In the special case n = 1 the above theorem gives the(classically known) defining relations of the q -fold Veronese embedding ofthe projective space P m − in P d , where d = (cid:18) m + q − q (cid:19) − q = 1 our group is the symmetric group S n and K [ V m ] S n is called the algebra of multisymmetric polynomials . It is anold result proved by Schl¨afli [36], MacMahon [28], Weyl [43], that when K has characteristic zero, this algebra is generated by the polarizations of theelementary symmetric polynomials (we mention that Noether [29] used thisto prove her general degree bound on generating invariants of finite groups).The result on the generators was extended to the case char( K ) > n byRichman [34] (see also [40], [5], [17] for other proofs). Counterexamplesand further results on the generators in the case 0 < char( K ) ≤ n are dueto Fleischmann [16], Briand [5], Vaccarino [41]. The relations among thegenerators had been studied classically by Junker [24] [25], [26]. Workingover C as a base field, an infinite presentation was given by Dalbec [8](partly reformulating results of Junker [25]), and the special case q = 1(and K = C ) of Theorem 2.5 appears in this explicit form in a recent paperby Bukhshtaber and Rees [6] (using a different language and motivation).Working over an arbitrary base ring K , [41] gives K -module generators ofthe ideal of relations among a reasonable infinite generating system (seeSection 5).(iii) In the special case q = 2 our group G ( n,
2) is the Weyl group oftype B n , and the ring of invariants C [ V m ] G ( n, is isomorphic to the ringof invariants of the special orthogonal group SO (2 n + 1 , C ) acting via theadjoint representation on the commuting variety of its Lie algebra, and isalso isomorphic to the ring of invariants of the symplectic group Sp (2 n, C )acting on the commuting variety of its Lie algebra (see for example [22] forthis connection).(iv) In our point of view, Theorem 2.5 (i) is a close relative of the resultof Razmyslov [33] and Procesi [31] saying that all trace identities of n × n matrices are consequences of the fundamental trace identity. Their originalapproach is based on Schur-Weyl duality; Kemer in [27] gave an elementarycombinatorial proof valid on the multilinear level in all characteristic.(v) Identify V with the K -algebra of n × n diagonal matrices. Denoteby Mor S n ( V m , V ) the space of S n -equivariant polynomial maps from V m to V . Since S n acts on V by algebra automorphisms, Mor S n ( V m , V ) is a K -algebra with pointwise multiplication of maps V m → V . Identify thespace Mor( V m , V ) of all polynomial maps from V m to V with the algebraof n × n diagonal matrices with entries in K [ V m ] in the obvious way; then6e have the usual trace function on Mor( V m , V ), and it is an algebra withtrace in the sense of [32]. The special case q = 1 of Theorem 2.5 has thefollowing corollary (by some methods of [31]): if n ! is invertible in K , thenthe algebra Mor S n ( V m , V ) of S n -equivariant polynomial maps from V m to V is the trace-stable subalgebra generated by the generic diagonal matrices x (1) , . . . , x ( m ) in the algebra of diagonal matrices with entries from K [ V m ].Furthermore, it is the free object of rank m in the category of commutativealgebras with a trace satisfying the n th Cayley-Hamilton identity (in thesense of [32]). This latter result is due to Berele, see Theorem 2.1 in [3].An element of K [ V m ] is multilinear if it is multihomogeneous with multi-degree (1 , . . . , F ( q ) by setting the multidegreeof t ( w ) equal to the multidegree of w . The multilinear version of the abovetheorem is valid over an arbitrary base field. Abusing notation we keepon writing K [ V m ] G ( n,q ) even if the group G ( n, q ) is not defined over K (i.e.when K does not contain a primitive q th root of 1); in this case the notationrefers to the algebra of invariants of S n acting on the subalgebra of K [ V m ]spanned by monomials u = u h i · · · u n h n i , where u i ∈ M ( q ) ∪ { } for all i (with the notation of the proof of Proposition 2.1). Theorem 2.7
Let K be an arbitrary field (or even an arbitrary commuta-tive ring). Then the multilinear component of K [ V m ] G ( n,q ) coincides withthe multilinear component of the image of ϕ ( q ) . Moreover, any multilin-ear element in the kernel of ϕ ( q ) is contained in the ideal generated by Ψ( w , . . . , w n +1 ) , where w i ∈ M ( q ) and the multidegree of w · · · w n +1 be-longs to { , } m .Proof. The only point in the proof of Proposition 2.1 where we need an as-sumption on K is to guarantee that r ! · · · r d ! = 0. If we consider multilinearelements of K [ V m ] only, then all the r i here are automatically equal to 1.So the same argument yields that the multilinear elements in K [ V m ] of theform [ w ] · · · [ w r ] (with r ≤ n , w i ∈ M ( q )) constitute a K -basis in the mul-tilinear component of K [ V m ] G ( n,q ) . The relations from our statement areobviously sufficient to rewrite an arbitrary multilinear product of elements[ w ] (with w ∈ M ( q )) as a linear combination of such products with at most n factors. This implies the claim. (cid:3) Remark 2.8
As an immediate corollary one recovers the known fact thatthe invariant [ x (1) · · · x ( m )] ∈ K [ V m ] S n is indecomposable (can not be ex-pressed by lower degree multisymmetric polynomials) when 0 < char( K ) ≤ . Indeed, in this case each term of the fundamental relation is the productof at least two traces, hence there is no relation involving [ x (1) · · · x ( m )] asa term. This has interesting consequences for matrix invariants in positivecharacteristic, and for vector invariants of the orthogonal group; see [11]. Building on Derksen’s general degree bound for syzygies in [9], we derive afinite presentation from Theorem 2.5.Denote by J m ( q ) the ideal in K [ V m ] generated by the homogeneous G ( n, q )-invariants of positive degree (the so-called Hilbert ideal), and denoteby τ G ( n,q ) ( V m ) the minimal natural number d such that all homogeneouselements of degree ≥ d belong to J m ( q ). Lemma 3.1
We have τ G ( n,q ) ( V m ) = 1 + P nj =1 ( qj − .Proof. In the special case q = 1 (and K = C , m = 2) this lemma appears in[20] and is due to Wallach and Garsia. Define the exponent of a monomial Q i,j x ( i ) α ij j as ( β , . . . , β n ), where β j = P mi =1 α ij (in other words, β j is theexponent of x j if we specialize all the x ( i ) j to x j ). Introduce a partialordering on the set of monomials in K [ V m ]: a monomial u is smaller than v if they have the same multidegree, and the exponent of u is lexicographicallysmaller than the exponent of v . This partial ordering is compatible withmultiplication of monomials. We shall show that if β k ≥ qk for some k ∈{ , . . . , n } , where ( β , . . . , β n ) is the exponent of some monomial u in K [ V m ],then modulo J m ( q ), the monomial u can be rewritten as a linear combinationof smaller monomials. We need some details from the proof of the specialcase q = 1 of our lemma. To simplify notation when m = 1, write x j = x (1) j .Then it is shown in Lemma 3.2.1 in [20] that h k ( x k , . . . , x n ) ∈ J (1) for all k = 1 , . . . , n, (2)where h k denotes the k th complete symmetric polynomial of the arguments.Note that (2) shows that modulo J (1), the monomial x kk can be rewritten asa linear combination of smaller monomials. Applying to (2) the comorphismof the S n -equivariant morphism V m → V , ( v , . . . , v m ) v + · · · + v m weget h k ( m X i =1 x ( i ) k , . . . , m X i =1 x ( i ) n ) ∈ J m (1) for all k = 1 , . . . , n. (3)8ince the ideal J m (1) is multihomogeneous, all multihomogeneous compo-nents of the left hand side of (3) belong to J m (1). In particular, for all k = 1 , . . . , n , there exists a linear combination f k of k -linear monomialssmaller than x (1) k x (2) k · · · x ( k ) k such that x (1) k x (2) k · · · x ( k ) k − f k ∈ J k (1) . (4)(At this point it is essential that k ! is assumed to be invertible in K .) Nowtake an arbitrary k -tuple w (1) , . . . , w ( k ) ∈ M ( q ). The map x ( i ) j w ( i ) h j i induces a K -algebra homomorphism γ : K [ V k ] → K [ V m ] N . Clearly γ is an S n -equivariant map, whence γ ( J k (1)) ⊆ J m ( q ). Applying γ to (4) we getthat w (1) h k i · · · w ( k ) h k i − γ ( f k ) ∈ J m ( q ) , (5)where γ ( f k ) is a linear combination of monomials smaller than w (1) h k i · · · w ( k ) h k i .Now let w be a monomial in the variables x (1) k , x (2) k , . . . , x ( m ) k such thatdeg( w ) = qk . Then w can be factored as w = w (1) h k i · · · w ( k ) h k i with w ( i ) ∈M ( q ), whence modulo J m ( q ) the monomial w can be rewritten as a linearcombination of smaller monomials by (5). This clearly implies that thefactor space K [ V m ] /J m ( q ) is spanned by the images of the monomials whoseexponent ( β . . . , β n ) satisfies β k ≤ qk − k = 1 , . . . , n . Since J m ( q )is a homogeneous ideal, the factor space K [ V m ] /J m ( q ) inherits the gradingfrom K [ V m ]. This shows that the homogeneous components of degree > P nk =1 ( qk −
1) are all contained in J m ( q ), implying that τ G ( V m ) ≤ P nk =1 ( qk − m = 1, the coordinate ring K [ V ] is known to be a freemodule over K [ V ] G ( n,q ) , and the latter is a polynomial algebra in n genera-tors having degrees q, q, . . . , nq . This implies a formula for the dimensionsof the homogeneous components of the graded vector space K [ V ] /J , calledthe coinvariant algebra (see for example [7]). In particular, the highest de-gree non-zero homogeneous component has degree P nk =1 ( qk − τ G ( n,q ) ( V m ) ≥ P nk =1 ( qk − (cid:3) For a natural number d , consider the finitely generated subalgebra of F ( q ) given by F ( q, d ) = K [ t ( w ) | w ∈ M ( q ) , deg( w ) ≤ d ] . Theorem 3.2
The kernel of the K -algebra surjection F ( q, qn ( n + 1) − n + 2) → K [ V m ] G induced by t ( w ) [ w ]9 s generated as an ideal by the elements Ψ( w , . . . , w n +1 ) , where w , . . . , w n +1 ∈M ( q ) , and the degree of the product w · · · w n +1 is not greater than qn ( n +1) − n + 2 .Proof. A general result of Derksen [9] says that the ideal of relations in aminimal presentation of K [ V m ] G is generated in degree ≤ τ G ( V m ). There-fore our statement follows from Theorem 2.5, Lemma 3.1, and the generalLemma 3.3 below. (cid:3) For a finitely generated commutative graded K -algebra A denote by τ ( A ) the minimal non-negative integer τ such that A and its first syzygyideal are generated in degree τ . That is, let ρ : K [ x , . . . , x r ] → A bea surjective K -algebra homomorphism such that ρ ( x i ), i = 1 , . . . , r , is aminimal homogeneous generating system of A , and the grading on the poly-nomial algebra K [ x , . . . , x r ] is defined so that ρ preserves the grading. Let f , . . . , f s be a minimal homogeneous generating system of the ideal ker( ρ ).If ker( ρ ) = 0, then define τ ( A ) = max { deg( x i ) | i = 1 , . . . , r } , and otherwiseset τ ( A ) = max { deg( f j ) | j = 1 , . . . , s } . Lemma 3.3
Let ϕ : K [ t µ | µ ∈ M ] → A be a surjective homomorphism ofgraded algebras from a not necessarily finitely generated polynomial algebra,and let { f β | β ∈ B } be a homogeneous generating system of the ideal ker( ϕ ) .Then the kernel of the restriction of ϕ to the subalgebra K [ t µ | deg( t µ ) ≤ τ ( A )] is generated as an ideal by { f β | deg( f β ) ≤ τ ( A ) } .Proof. Denote by I the ideal of R = K [ t µ | deg( t µ ) ≤ τ ( A )] generated by { f β | deg( f β ) ≤ τ ( A ) } . Take a subset C ⊂ M such that the restriction ϕ : R = K [ t µ | µ ∈ C ] → A of ϕ is a minimal presentation of A . Then ker( ϕ )is generated in degree τ ( A ) and is contained in ker( ϕ ) = ( f β | β ∈ B ),hence ker( ϕ ) ⊂ I . Moreover, since im( ϕ ) = im( ϕ ), for each µ ∈ M thereis an element g µ ∈ R with deg( t µ ) = deg( g µ ) such that t µ − g µ ∈ ker( ϕ ).Obviously, if deg( t µ ) ≤ τ ( A ), then t µ − g µ ∈ I . It follows that R = R + I ,implying ker( ϕ ) = ker( ϕ ) + I = I . (cid:3) Remark 3.4
Since the Hilbert ideal of a subgroup H of G ( n, q ) contains theHilbert ideal of G ( n, q ), we have τ H ( V m ) ≤ τ G ( n,q ) ( V m ). Thus Lemma 3.1provides upper bounds on the degrees of the relations in a minimal presenta-tion of K [ V m ] H by the result of Derksen [9] cited above. Among subgroupsof G ( n, q ) are other series of pseudo-reflection groups, including the Weylgroups of type D n or the dihedral groups.10 Semisimple commutative representations
In this section we work over the field C of complex numbers (what is essen-tial for Theorem 4.1 is that the base field has characteristic zero). Write F m for the free associative algebra C h x , . . . , x m i , and A m for the commutativepolynomial algebra C [ x (1) , . . . , x ( m )]. Denote by rep( F m , n ) the space of m -tuples of complex n × n matrices, endowed with the simultaneous conjugationaction of GL ( n, C ). The points of this affine variety determine n -dimensionalrepresentations of F m in the obvious way, and the GL ( n, C )-orbits are ina one-to-one correspondence with the isomorphism classes of n -dimensionalrepresentations of F m . The algebraic quotient rep( F m , n ) //GL ( n, C ) param-eterizes the isomorphism classes of semisimple n -dimensional representationsof the free algebra, see [2]. The coordinate ring C [rep( F m , n )] is the n m -variable polynomial algebra generated by the entries of the generic n × n matrices Y (1) , . . . , Y ( m ). Denote by J the ideal in C [rep( F m , n )] generatedby the entries of the commutators Y ( i ) Y ( j ) − Y ( j ) Y ( i ), 1 ≤ i < j ≤ m .The quotient algebra C [rep( F m , n )] /J = C [rep( A m , n )] is the coordinatering of the scheme rep( A m , n ) of n -dimensional representations of A m (bydefinition of this affine scheme). It is a long standing open problem in com-mutative algebra whether this scheme is reduced or not, or equivalently,whether J is a radical ideal or not; see for example [21]. The commonzero locus of J (i.e. the set of C -points of the scheme rep( A m , n )) is theso-called commuting variety consisting of m -tuples of pairwise commuting n × n -matrices. So the question is whether J is the whole vanishing idealof the commuting variety or not. The embedding of the space V m of m -tuples of diagonal matrices into the commuting variety induces a surjectivehomomorphism β : C [rep( A m , n )] → C [ V m ]. Consider the homomorphism γ : F (1) → C [rep( A m , n )] GL ( n, C ) given by t ( w ) Tr( Y (1) α · · · Y ( m ) α m )for w = x (1) α · · · x ( m ) α m ∈ M (1), where we keep the notation Y ( j ) for theimages of the generic matrices under the natural surjection M ( n, C [rep( F m , n )]) → M ( n, C [rep( A m , n )] . It is well known that γ is surjective, see [38]; we use here that any GL ( n, C )-invariant in C [rep( A m , n )] lifts to an invariant on rep( F m , n ), since ourbase field has characteristic zero. The fundamental trace identity holdsin M ( n, C ) for an arbitrary commutative ring C (this is equivalent to theCayley-Hamilton theorem), therefore Ψ( w , . . . , w n +1 ) ∈ ker( γ ) for any w i ∈M (1). It follows by Theorem 2.5 that γ factors through a surjection γ : C [ V m ] S n → C [rep( A m , n )] GL ( n, C ) , [ w ] Tr( Y (1) α · · · Y ( m ) α m )11or w = x (1) α · · · x ( m ) α m ∈ M (1). So we have the surjections C [ V m ] S n γ −→ C [rep( A m , n )] GL ( n, C ) β −→ C [ V m ] S n . Since β ◦ γ is the identity map of C [ V m ] S n by definition of the maps γ and β , we conclude that they are isomorphisms. Thus we obtained the followingresult, supporting the conjecture that the commuting scheme is reduced: Theorem 4.1
The surjection C [rep( A m , n )] → C [ V m ] induced by the em-bedding of the space of m -tuples of diagonal matrices into the commut-ing variety restricts to an isomorphism C [rep( A m , n )] GL ( n, C ) ∼ = C [ V m ] S n .In particular, the radical of C [rep( A m , n )] contains no non-zero GL ( n, C ) -invariants. Remark 4.2
One may paraphrase the above theorem by saying that ”thescheme of n -dimensional semisimple representations of the commutative m -variable polynomial algebra is reduced”. (The weaker statement thatthe algebraic quotient of the commuting variety with respect to the ac-tion of GL ( n, C ) is isomorphic to V m /S n is well known, and is explainedfor example in Proposition 6.2.1 of [20].) In the special case m = 2, re-ducedness of C [rep( A m , n )] GL ( n, C ) was proved by different methods by Ganand Ginzburg, see Theorem 1.2.1 in [19]. For arbitrary m it appeared inour preprint [12], and is proved also by Vaccarino in his parallel preprint[arXiv:math.AG/0602660], that appeared in the meantime as [42]. Throughout this section K is an arbitrary base ring. We extend the methodof Section 2 to this case.For a monomial w ∈ M = M (1) and l ∈ { , . . . , n } we set σ l ( w ) = X ≤ i < ··· n , the equality (6) remains validalso if l > n . For a multidegree α = ( α , . . . , α s ) and w , . . . , w s ∈ M define S α ( w , . . . , w s ) ∈ K [ e r ( w ) | w ∈ M , r = 1 , . . . , n ]as the element obtained by making first the substitution x ( i ) w i , i =1 , . . . , s in the right hand side of (6), and replacing the symbols σ by e everywhere. Since σ l is identically zero for l > n , the equality (6) showsthat S α ( w , . . . , w s ) ∈ ker( φ ) if X α i > n. (7)There is another type of relations we shall need. Simplify the notationby writing x = x (1) in the special case m = 1. It is well known that K [ V ] S n is generated by the algebraically independent elements σ r ( x ), r = 1 , . . . , n ,hence for each k ≥ j ∈ { , . . . , n } there is a unique n -variable polyno-mial f j,k with integer coefficients such that σ j ( x k ) = f j,k ( σ ( x ) , . . . , σ n ( x )).Now given a monomial w ∈ M , consider the element Q j,k ( w ) = e j ( w k ) − f j,k ( e ( w ) , . . . , e n ( w )) ∈ ker( φ ) . For example, it is an easy exercise to verify that Q r, ( x ) = e r ( x ) − min { r,n } X i =max { , r − n } ( − r + i e i ( x ) e r − i ( x ) . As a first application of the relations (7) we derive a short proof of the degreebound of Fleischmann [16] for the generators of K [ V m ] S n . We call an ele-ment of K [ V m ] S n indecomposable if it is not contained in the K -subalgebragenerated by strictly lower degree elements, and call it decomposable other-wise. Obviously, a homogeneous element of K [ V m ] S n is indecomposable ifand only if it is not contained in the square A of the ideal A of K [ V m ] S n spanned by the homogeneous components of positive degree. Lemma 5.2
Let w = x (1) α · · · x ( m ) α m ∈ M be a monomial such that α j ≥ n/r for some j , and deg( w ) ≥ n/r . Then σ r ( w ) is decomposable.Proof. Apply induction on r . Clearly it is sufficient to show that if kr ≥ n ,then σ r ( x k y ) ∈ A (here we write x, y instead of x (1) , x (2)). Indeed, thegeneral case follows on substituting x, y by appropriate monomials. Fromthe relation φ ( S ( kr,r ) ( x, y )) = 0 we get( − r σ r ( x k y ) + X ( − r/i C { x ki y i } σ r/i ( x ki y i ) ∈ A , i > r (and C { x ki y i } is the numberof cyclic equivalence classes of reduced words that specialize to x ki y i ). Inparticular, this sum is empty if r = 1 and so σ ( x k y ) ∈ A for k ≥ n . If r >
1, then the assumptions in our lemma apply to all summands, hencethey belong to A by the induction hypothesis, forcing σ r ( x k y ) ∈ A . (cid:3) Corollary 5.3
The K -algebra K [ V m ] S n is generated by the elements σ n ( x ( i )) , i ∈ { , . . . , m } , and σ r ( x (1) α · · · x ( m ) α m ) , where r ∈ { , . . . , n } , α j < n/r for all j = 1 , . . . , m , and the greatest common divisor of α , . . . , α m is . Remark 5.4
In particular, K [ V m ] S n is generated in degree ≤ max { m ( n − , n } ; this is the main result of Fleischmann [16], where it is also shownthat this bound can not be improved in general. Using the method of [16]we showed in [11] that if K is a field of characteristic p and n = p k with k ∈ N , then σ n − ( x (1) · · · x ( m )) is indecomposable. Vaccarino [41] showsthat the σ r ( w ) with w reduced (not a power of a monomial with lowerdegree) generate the ring of multisymmetric functions, and refering to thebound from [16] deduces a finite system of generators. Our Corollary 5.3reduces further this generating system. A minimal generating system of K [ V m ] S n appears in [35]. Theorem 5.5
The ideal ker( φ ) is generated by S α ( w , . . . , w s ) and Q r,k ( w ) , where s ≥ , w, w , . . . , w s ∈ M ; P si =1 α i > n , P i = j α i ≤ n for all j =1 , . . . , s ; r ∈ { , . . . , n } ; k < n/r .Proof. Call the weight of a product σ r ( w ) · · · σ r s ( w s ) the integer r + · · · + r s .First we show that if the weight is > n , then modulo the first type re-lations in our statement, this product can be rewritten as a linear com-bination of products with strictly smaller weight. It is sufficient to dealwith the case when P i = j r i ≤ n for all j = 1 , . . . , s . Set α = ( r , . . . , r s ).Then S α ( x (1) , . . . , x ( s )) contains the term e r ( x (1)) · · · e r s ( x ( s )) with coef-ficient ±
1. In any other term e i ( u ) · · · e i d ( u d ) of S α , at least one of the u t has degree >
1, hence i + · · · + i d < r + · · · + r s . So the relation φ ( S α ( w , . . . , w s )) = 0 does what we need.Next we claim that using our relations, any product σ r ( w ) · · · σ r d ( w d )with weight ≤ n can be rewritten as a linear combination of σ i ( u ) · · · σ i t ( u t )with weight ≤ P r i and u , . . . , u t distinct. By Proposition 5.1 our theoremwill follow. Obviously, it is sufficient to verify the latter claim in the special15ase when all the w i are powers of the same monomial w . We may alsoassume that w = x , so we are working in the special case m = 1. Aninspection of the proof of Proposition 5.1 shows that σ r ( x j ) · · · σ r d ( x j d ) is alinear combination of monomial multisymmetric functions of height ≤ P r i ,and in the K -space spanned by such monomial multisymmetric functions,the products σ i ( x k ) · · · σ i t ( x k t ) with weight ≤ P r i and k , . . . , k t distinctform a basis. So we conclude that there are relations among the σ r ( x k ) ofthe desired form. Therefore it suffices to prove the special case m = 1 of ourtheorem.Since in the special case m = 1, K [ V ] S n is a polynomial ring generatedby σ ( x ) , . . . , σ n ( x ), all we need to show is that our relations are sufficient toexpress all σ r ( x k ) in terms of the n algebraically independent generators. Weapply a double induction: the first induction goes on deg( σ r ( x k )) = rk , andthe second induction goes on r . If ( k − r < n , then we may use our relation φ ( Q r,k ( x )) = 0 from our statement and we are done. Assume ( k − r ≥ n .Then we slightly adjust the argument in the proof of Lemma 5.2. Write l for the lower integral part of n/r . Then we have k > l and we have therelation φ ( S ( lr,r ) ( x, x k − l )) = 0 in our statement. This relation can be usedto express ( − r σ r ( x k ) + X ( − r/i C { x li y i } σ r/i ( x ki )(where the sum ranges over divisors i > r ) by lower degree elements.The statement then follows by the two induction hypotheses. (cid:3) Remark 5.6
Although the above theorem is far from yielding a finite pre-sentation of K [ V m ] S n , at least it gives a uniform presentation in the sensethat all relations are obtained from finitely many types by substituting dif-ferent monomials (note that the restrictions made on α force P si =1 α i ≤ n and s ≤ n + 1). For comparison we mention that in [41] the author consid-ers the generating system σ r ( w ) where w is reduced (i.e. is not a power ofa monomial with smaller degree), and describes the relations among thesegenerators in the following sense: he works in the ring of multisymmetricfunctions in infinitely many variables, where the analogues of σ r ( w ) ( r ∈ N , w ∈ M reduced) form an algebraically independent generating system, andobserves that the unique expressions of the monomial multisymmetric func-tions with height > n in terms of these generators form a K -basis in thespace of relations among these generators. Implementing this procedure inpractice one needs to produce formulae like our Q r,k ( w ) but with no boundon r and k , and one needs to produce expressions similar to S α ( w , . . . , w s )with no bound on α . 16 Calculations
Throughout this section we assume that n ! is invertible in K , so K [ V m ] S n isCohen-Macaulay (see for example [10] as a general reference for the invarianttheory of finite groups). To illustrate how to proceed to find a minimal pre-sentation we do calculations in concrete examples. An obvious homogeneoussystem of parameters (primary generators) in K [ V m ] S n is P = { [ x ( i )] , [ x ( i ) ] , . . . , [ x ( i ) n ] | i = 1 , . . . , m } . Write h P i for the ideal of K [ V m ] S n generated by P , and write K [ P ] forthe polynomial subalgebra of K [ V m ] S n generated by P . Note that to find secondary generators we need to get a vector space basis modulo h P i in K [ V m ] S n . Lemma 6.1
Assume that n ! is invertible in K . Let w be a monomial havingdegree ≥ n in one of the variables x (1) , . . . , x ( m ) , and having total degree ≥ n + 1 . Then [ w ] belongs to h P i .Proof. Assume for example that w has degree ≥ n in x (1). Then w canbe written as a product of n + 1 factors as x (1) x (1) · · · x (1) u for some non-empty monomial u . The relation Ψ n +1 ( x (1) , x (1) , . . . , x (1) , u ) = 0 showsthe claim, since each nontrivial partition of the multiset { x (1) , . . . , x (1) , u } contains a part consisting solely of x (1)s. (cid:3) Remark 6.2
From Lemma 6.1 and Proposition 2.1 we immediately getthe upper bound n ( n − m for the degrees of the secondary generators.However, the general method of Broer (see Theorem 3.9.8 in [10]) yields thebetter bound n ( n − m . n = 2 In this subsection we assume char( K ) = 2. The fundamental multilineartrace identity for diagonal 2 × xyz ] = 12 ([ xy ][ z ] + [ xz ][ y ] + [ yz ][ x ] − [ x ][ y ][ z ]) (8)Substitute z zw in (8), and eliminate traces of degree 3 using (8) to get[ xyzw ] = 14 ([ xz ][ y ][ w ] + [ xw ][ y ][ z ] + [ yz ][ x ][ w ] + [ yw ][ x ][ z ])+ 12 [ xy ][ zw ] −
12 [ x ][ y ][ z ][ w ] (9)17xchanging the variables y and z in (9) we obtain[ xzyw ] = 14 ([ xy ][ z ][ w ] + [ xw ][ y ][ z ] + [ yz ][ x ][ w ] + [ zw ][ x ][ y ])+ 12 [ xz ][ yw ] −
12 [ x ][ y ][ z ][ w ] (10)The difference of (9) and (10) yields[ xz ][ yw ] − [ xy ][ zw ] = 12 ([ xy ][ z ][ w ] + [ zw ][ x ][ y ] − [ xz ][ y ][ w ] − [ yw ][ x ][ z ]) (11)(compare with (4.17) in [30]). The specialization z = w in (11) leads to[ xz ][ yz ] = [ xy ][ zz ] + 12 ([ xy ][ z ][ z ] + [ zz ][ x ][ y ] − [ xz ][ y ][ z ] − [ yz ][ x ][ z ]) (12)The N m -graded Hilbert series of K [ V m ] S can be written as a rational func-tion by Molien’s formula: H ( K [ V m ] S ; t , . . . , t m ) = P [ m ] i =0 e i ( t , . . . , t m ) Q mj =1 (1 − t j )(1 − t j ) , where e i ( t , . . . , t m ) is the i th elementary symmetric function in t , . . . , t m . Itis easy to describe the Hironaka decomposition of K [ V m ] S . The secondarygenerators are S = { [ x ( i ) x ( i )] · · · [ x ( i k − ) x ( i k )] | ≤ i < · · · < i k ≤ m } . The substitution x x ( i ), y x ( j ), z x ( k ) in (12) gives a relation show-ing that [ x ( i ) x ( k )][ x ( j ) x ( k )] is contained in h P i for all i, j, k . Substitutions { x, y, z, w } → { x ( j ) , . . . , x ( j k ) } in the relation (11) show that[ x ( j ) x ( j )] · · · [ x ( j k − ) x ( j k )] − [ x ( j π (1) ) x ( j π (2) )] · · · [ x ( j π (2 k − ) x ( j π (2 k ) )]belongs to h P i for an arbitrary permutation π . These relations implythat the algebra of multisymmetric functions is generated by S as a mod-ule over the polynomial ring K [ P ]. The Hilbert series shows that it isa free module. These considerations show also that the specializations { x, y, z, w } → { x (1) , . . . , x ( m ) } in (11) generate the ideal of relations amongthe generators [ x ( i )] , [ x ( k ) x ( l )], and that a minimal system of relations con-sists of relations of degree 4. 18 .2 The case n = 3 , m = 2 In this subsection we assume char( K ) = 2 ,
3. The fundamental identityΨ ( x, y, z, w ) = 0 takes the form[ xyzw ] = 13 ([ xyz ][ w ] + [ xyw ][ z ] + [ xzw ][ y ] + [ yzw ][ x ])+ 16 ([ xy ][ zw ] + [ xz ][ yw ] + [ xw ][ yz ] + [ x ][ y ][ z ][ w ]) −
16 ([ xy ][ z ][ w ] + [ xz ][ y ][ w ] + [ xw ][ y ][ z ] ++[ yz ][ x ][ w ] + [ yw ][ x ][ z ] + [ zw ][ x ][ y ])Consider the following consequences:Ψ ( x, x, x, x ) = 0 Ψ ( y, y, y, y ) = 0 Ψ ( x, x, x, y ) = 0 (13)Ψ ( y, y, y, x ) = 0 Ψ ( x, x, x, xy ) = 0 Ψ ( y, y, y, xy ) = 0Ψ ( x, x, y, y ) = 0 Ψ ( x, x, x, y ) = 0 Ψ ( y, y, y, x ) = 0Ψ ( x, x, x, y ) = 0 Ψ ( x, x, x, xy ) = 0 Ψ ( y, y, y, yx ) = 0and Ψ ( x , x, y, y ) = 0 Ψ ( y , y, x, x ) = 0 (14)Ψ ( x , x , y, y ) = 0 Ψ ( y , y , x, x ) = 0Ψ ( x , x, y , y ) = 0Let us use the notation f ≡ g to indicate that f, g ∈ K [ V ] S are congru-ent modulo h P i . The above 17 relations in the order of listing imply thefollowing: [ x ] ≡ y ] ≡ x y ] ≡ xy ] ≡ x y ] ≡ xy ] ≡ x y ] ≡ [ xy ][ xy ] [ x y ] ≡ y x ] ≡ x y ] ≡ x y ] ≡ y x ] ≡ x y ] ≡ [ x y ][ xy ] [ y x ] ≡ [ y x ][ yx ][ x y ] ≡ [ x y ][ x y ] [ x y ] ≡ [ xy ][ xy ][ x y ] ≡ [ x y ][ xy ] + [ x y ][ xy ]19ecall that K [ V ] S is minimally generated as an algebra by P ∪{ [ xy ] , [ x y ] , [ xy ] } .The above congruences obviously show that as a module over K [ P ], the al-gebra K [ V ] S is spanned by S = { , [ xy ] , [ x y ] , [ xy ] , [ xy ] , [ x y ][ xy ] } . Using Molien’s formula one can easily compute H ( K [ V ] S ; t , t ) = 1 + t t + t t + t t + t t + t t (1 − t )(1 − t )(1 − t )(1 − t )(1 − t )(1 − t ) , and this implies that K [ V ] S is a free K [ P ]-module generated by S . Theabove considerations imply that the ideal of relations among the 21 genera-tors { [ x i y j ] | ≤ i, j ≤ , i + j ≤ , ( i, j ) = (0 , } is generated by the 17 relations (13) and (14). The relations (13) can beused to eliminate the 12 superfluous generators of degree ≥ { [ x i y j ] | i + j ≤ } . The bidegrees of these relations are (3 , , , , , C [ V m ] S is determined in [14] for arbitrary m .) References [1] S. A. Amitsur, On the characteristic polynomial of a sum of matri-ces, Lin. Multilin. Alg. 8 (1980), 177-182.[2] M. Artin, On Azumaya algebras and finite dimensional representa-tions of rings, J. Algebra 11 (1969), 532-563.[3] A. Berele, Trace identities for diagonal and upper triangular matri-ces, International J. Algebra and Computation 6 (1996), 645-654.[4] A. Berele and A. Regev, Some results on trace cocharacters, J. Al-gebra 176 (1995), 1013-1024.[5] E. Briand, When is the algebra of multisymmetric polynomials gen-erated by the elementary multisymmetric polynomials? Beitr¨ageAlgebra Geom. 45 (2004), no. 2, 353-368.206] V. M. Bukhshtaber and E. G. Rees, Rings of continuous functions,symmetric products, and Frobenius algebras (Russian), UspekhiMat. 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16 ([ xy ][ z ][ w ] + [ xz ][ y ][ w ] + [ xw ][ y ][ z ] ++[ yz ][ x ][ w ] + [ yw ][ x ][ z ] + [ zw ][ x ][ y ])Consider the following consequences:Ψ ( x, x, x, x ) = 0 Ψ ( y, y, y, y ) = 0 Ψ ( x, x, x, y ) = 0 (13)Ψ ( y, y, y, x ) = 0 Ψ ( x, x, x, xy ) = 0 Ψ ( y, y, y, xy ) = 0Ψ ( x, x, y, y ) = 0 Ψ ( x, x, x, y ) = 0 Ψ ( y, y, y, x ) = 0Ψ ( x, x, x, y ) = 0 Ψ ( x, x, x, xy ) = 0 Ψ ( y, y, y, yx ) = 0and Ψ ( x , x, y, y ) = 0 Ψ ( y , y, x, x ) = 0 (14)Ψ ( x , x , y, y ) = 0 Ψ ( y , y , x, x ) = 0Ψ ( x , x, y , y ) = 0Let us use the notation f ≡ g to indicate that f, g ∈ K [ V ] S are congru-ent modulo h P i . The above 17 relations in the order of listing imply thefollowing: [ x ] ≡ y ] ≡ x y ] ≡ xy ] ≡ x y ] ≡ xy ] ≡ x y ] ≡ [ xy ][ xy ] [ x y ] ≡ y x ] ≡ x y ] ≡ x y ] ≡ y x ] ≡ x y ] ≡ [ x y ][ xy ] [ y x ] ≡ [ y x ][ yx ][ x y ] ≡ [ x y ][ x y ] [ x y ] ≡ [ xy ][ xy ][ x y ] ≡ [ x y ][ xy ] + [ x y ][ xy ]19ecall that K [ V ] S is minimally generated as an algebra by P ∪{ [ xy ] , [ x y ] , [ xy ] } .The above congruences obviously show that as a module over K [ P ], the al-gebra K [ V ] S is spanned by S = { , [ xy ] , [ x y ] , [ xy ] , [ xy ] , [ x y ][ xy ] } . Using Molien’s formula one can easily compute H ( K [ V ] S ; t , t ) = 1 + t t + t t + t t + t t + t t (1 − t )(1 − t )(1 − t )(1 − t )(1 − t )(1 − t ) , and this implies that K [ V ] S is a free K [ P ]-module generated by S . Theabove considerations imply that the ideal of relations among the 21 genera-tors { [ x i y j ] | ≤ i, j ≤ , i + j ≤ , ( i, j ) = (0 , } is generated by the 17 relations (13) and (14). 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