# Invariant theory for the commuting scheme of symplectic Lie algebras

aa r X i v : . [ m a t h . R T ] F e b Invariant theory for the commuting schemeof symplectic Lie algebras

Tsao-Hsien Chen and

Ngô Bảo Châu

Abstract

We prove the Chevalley restriction theorem for the commuting scheme of symplec-tic Lie algebras. The key step is the construction of the inverse map of the Chevalleyrestriction map called the spectral data map. Along the way, we establish a certain mul-tiplicative property of the Pfafﬁan which is of independent interest.

Let k be a ﬁeld of characteristic zero, or of large prime characteristic. Let G be a reduc-tive group of k , g its Lie algebra. For every d ∈ N , we consider the commuting scheme C d g consisting of elements ( x , . . . , x d ) ∈ g d such that [ x i , x j ] = i , j ∈ {

1, . . . , d } . Thecommuting scheme has always been of some interest in invariant theory but it was only re-cent that it appears as a primordial object in the study of moduli space of Higgs bundles forhigher dimensional varieties. It is also poorly understood. For d =

2, Richardson proved thatthe open subscheme of C g deﬁned by the condition that x , x are regular and semi-simpleis dense smooth and irreducible. Although the reducedness of C g has the status of a folkloreconjecture, there is a little evidence to expect C d g to be reduced in general. For d ≥ C d g isin general reducible, and components other than the closure of the regular semisimple opensubscheme of C d g are unlikely to be reduced.In this paper, we are more concerned with the categorical quotient C d g (cid:12) G = Spec ( k [ C d g ] G ) where k [ C d g ] G is the ring of G -invariant functions on C d g . We note that G acts on g by theadjoint action, and hence on g d by the diagonal adjoint action. This action leaves the com-muting subscheme C d g stable and we are interested in generalizing the Chevalley restrictiontheorem asserting a description of the categorical quotient C d g (cid:12) G in terms of a Cartan sub-algebra t of g equipped with the Weyl group action W . Let T denote a maximal torus of G ,1eligne’s construction t its Lie algebra. The Weyl group W = N G ( T ) / T acts on T and t . The embedding t d → g d factors through the commuting variety C d g and it induces a homomorphism of algebras c : k [ C d g ] G → k [ t d ] W because the restriction of a G -invariant function to t d is obviously W -invariant. We conjecturethat c is always an isomorphism. The classical Chevalley restriction theorem corresponds tothe case d =

1. Note that, since t d (cid:12) W is known to be normal and reduced, the conjecturewill imply C d g (cid:12) G is normal and reduced.For G = GL n , this conjecture was proved by Vaccarino in [ ] (not being aware of Vac-carino’s result, we also reprove it in [ ] ). Vaccarino’s proof in the case G = GL n relies on theconstruction of a map in the opposite direction, to be called the spectral data map s : k [ t d ] W → k [ C d g ] G which is due to Deligne in the case G = GL n . Once the spectral data map is constructed, toprove the Chevalley restriction theorem for commuting schemes it is enough to prove that s ◦ c and c ◦ s are identities. For this, one can use a a result of Procesi in [ ] which providesa system of generators of the ring k [ g d ] G .In this paper we will prove the Chevalley restriction theorem for the commuting schemeof the symplectic Lie algebra in following the same line on thought. The main novelty isthe construction of the spectral data map for G = Sp n . One should note that for generalreductive groups, even in the case d =

1, we don’t know to construct this map withoutassuming ﬁrst the Chevalley restriction theorem. For symplectic groups, a key ingredient inour construction is a certain multiplicative property of the Pfafﬁan. This is an elementaryfact of linear algebra which seems to be new and of independent interest.

We will ﬁrst recall Deligne’s beautiful construction of the spectral data map for GL n in [ ] . As a preparation, we will ﬁrst recall Roby’s concept of polynomial laws whichwill provide a convenient language for Deligne’s construction (see [ ] and also [ ] ).Let A be a commutative ring. For every A -module V , we will denote V A the functor R → V ⊗ A R from the category of A -algebras to the category of sets. If V and N are A -modules, wewill denote P ( V , N ) the set of morphism of functors f : V A → N A . In case N is not explicitlymentioned, we will understand that N = A , i.e., P ( V ) = P ( V , A ) and call P ( V ) the set ofpolynomial laws on V .The connection between polynomial law and usual polynomial can be explained as fol-lows. Let S A = A [ X , . . . , X d ] be the polynomial algebra with free variables X , . . . , X d . Forevery ﬁnite set of elements v , . . . , v d ∈ V , we have an element X v + · · · + X d v d ∈ V ⊗ A S A . If2eligne’s construction f is a polynomial law on V , then f v = f ( X v + . . . X d v d ) is an element of S A i.e. a polynomialof variables X , . . . , X n with coefﬁcients in A . If V is a free A -module and v , . . . , v d form abasis of V , then the polynomial f v ∈ R determines the polynomial law f .A polynomial law f on V is said to be homogenous of degree n if for every A -algebra R and element v ∈ V ⊗ A R we have f ( um ) = u n f ( m ) for every u ∈ R × . If V is a free A -moduleand v , . . . , v d form a base of V , then f is homogenous of degree n if and only if f v is ahomogenous polynomial of degree n .If V is an A -module, we denote T nA ( V ) the n th fold tensor power V with itself over A which is equipped with an action of the symmetric group S n . We denote TS nA ( V ) , the n thmodule of symmetric tensors of V that is the submodule of T nA ( V ) consisting of elements ﬁxedunder S n , to be differentiated from S nA ( V ) , the n th symmetric power of V that is the largestquotient of T nA ( V ) on which S n acts trivially. We have a map V → TS n V given by v v ⊗ n .Roby proved that if V is a free A -module, then there is a canonical bijection between the setof homogeneous polynomial laws f of degree n and the set of homogenous polynomial h ofdegree 1 on TS eA ( V ) characterized by the equality f ( v ) = h ( v ⊗ n ) . We note that Roby statesthis theorem [

5, Theorem IV.1 ] with the divided power module instead of the symmetrictensors modules. These modules coincide however in the case where V is a free A -module [

5, Propositions III.1, IV.5 ] .If moreover, V is an A -algebra, which is free as an A -module, and if f is a multiplicativehomogenous polynomial law of degree n on V i.e. if f ( x y ) = f ( x ) f ( y ) , then the corre-sponding degree 1 homogenous polynomial law on TS nA ( V ) is a homomorphism of algebrasTS eA ( V ) → A [

3, Proposition 2.5.1 ] .We now consider the group G = GL n whose Lie algebra is the space of matrices M n whichis also equipped with a structure of algebras. Let T n be the diagonal torus of GL n and t n its Liealgebra. The Weyl group W is the symmetric group S n acting on T n and t n by permutation ofcoordinates. The commuting scheme of GL n will be denoted by C dn . We will recall Deligne’sconstruction of a GL n -invariant map s : C dn → t dn (cid:12) S n which roughly records the joint eigenvalues of set of commuting matrices.This map can be easily described at level of points in an algebraically closed ﬁeld. If x , . . . , x d ∈ gl n ( k ) are commuting matrices, we can equipped the n -dimensional vector space V = k d with a structure of k [ X , . . . , X d ] -module. Since V is ﬁnite dimensional, it is supportedby a ﬁnite subscheme of A d = Spec ( k [ X , . . . , X d ]) V = M α ∈ k n V α where V α is annihilated by a power of the maximal ideal deﬁning α . We then set s ( α ) to bethe 0-cycle s ( V ) = X α dim ( V α ) α k -point of t dn (cid:12) S n . It is not clear how to generalize this cycle construction forcommuting matrices with values in an arbitrary test ring R .Now let A be the coordinate ring of C dn and x = ( x , . . . , x d ) ∈ C dn ( A ) the tautological A -point of C dn . Let S = k [ X , . . . , X d ] be the polynomial algebra of variables X , . . . , X d . Thepoint x gives rise to a homomorphism of algebras p x : S ⊗ k R → gl n ( A ⊗ k R ) with p x ( X i ) = x i for every k -algebra R . By composing with the determinant map we get apolynomial law on S f x : S ⊗ k R → A ⊗ R given by f x = det ◦ p x which is homogenous of degree n and multiplicative. By Roby’s theo-rem, this is equivalent with a homomorphism of algebras h x : TS n ( S ) → A which is a R -point of Spec ( TS n ( S ( V ))) = t dn (cid:12) S n . Since det is G -invariant, h x is also G -invariant. As a result, we obtain the spectral data map s : TS n ( S ) → A G .We note that this construction works without any restriction on the characteristic of the baseﬁeld k . Instead of the determinant, our construction of the spectral data map for symplectic groupsrelies on the Pfafﬁan function and its multiplicative property. The Pfafﬁan is a homogenousform of degree n on the space of antisymmetric forms on k n . Since the Pfafﬁan is a square-root of the determinant one may ask the question whether it enjoys the same multiplicativeproperty as the determinant. A priori the question is ill-posed for the product of antisymmet-ric matrices is not antisymmetric. We will show it is indeed possible to prove a multiplicativeproperty of a function closely related the Pfafﬁan. This elementary result seems to be newand of independent interest.In this section, we assume that the base ﬁeld k is of characteristic zero or of odd primecharacteristic. Let V be a 2 n -dimensional k -vector space. A bilinear form on V is an elementof the vector space Hom k ( V , V ∗ ) which is equipped with an involution given by x x ∗ . Weobserve that Hom k ( V , V ∗ ) is canonically isomorphic to V ∗ ⊗ k V ∗ equipped with the involution v ∗ ⊗ v ∗ v ∗ ⊗ v ∗ . We have a decomposition into eigenspaces of this involution V ∗ ⊗ k V ∗ = S V ∗ ⊕ Λ V ∗ V ∗ corresponds tp the space of symmetric bilinear forms on V and Λ ( V ∗ ) the spaceof alternating bilinear forms on V .There is a canonical map S n ( Λ V ∗ ) → Λ n V ∗ given by the super-commutative multiplication law in the exterior algebra Λ • V ∗ = L ni = Λ i V ∗ .It follows that we have a degree n homogenous polynomial law µ : Λ V ∗ → Λ n V ∗ which associates to an alternating form ω ∈ Λ V ∗ the image of ω n ∈ S n ( Λ V ∗ ) in the line Λ n V ∗ . We note that ω ∈ Λ V ∗ is a non-degenerate alternating bilinear form if and only if µ ( ω ) is a non-zero vector of Λ n V ∗ . If it is the case, we will say that ω is a symplectic form.We also note that µ is a square-root of the determinant in the following sense. A bilinearform b on V induces a bilinear form on the determinant line Λ n V det ( b ) ∈ Hom k ( Λ n V , Λ n V ∗ ) = Λ n V ∗ ⊗ Λ n V ∗ .Then for every alternating form ω ∈ Λ V ∗ we have the identitydet ( ω ) = µ ( ω ) ⊗ µ ( ω ) . (3.1)We will now ﬁx a symplectic form ω ∈ Λ V ∗ and consider the symplectic group G = Sp n of all linear transformations of V that preserve ω . The Lie algebra g of G is the subspace of gl ( V ) of matrices x ∈ gl ( V ) such that ω ( x u , v ) = − ω ( u , x v ) for all vectors u , v ∈ V . This is equivalent to say that the identity ω x = − x ∗ ω holds inHom k ( V , V ∗ ) . In this case we have ( ω x ) ∗ = − x ∗ ω ∗ = x ∗ ω = ω x and therefore ω x ∈ S V ∗ . In other words, the map x ω x induces an isomorphism of k -vector spaces g → S V ∗ .We will also consider the decomposition gl ( V ) = g ⊕ g + where g + is the subspace ofmatrices x ∈ End ( V ) such that ω ( x u , v ) = ω ( u , x v ) for all vectors u , v ∈ V . The map x ω x induces an isomorphism of k -vector spaces g + → Λ V ∗ . We will deﬁne a Pfafﬁan norm N + : g + → A by the equalityN + ( x ) µ ( ω ) = µ ( ω x ) for every k -algebra R and x ∈ g + ( R ) . Note that since µ ( ω ) is a generator of the free R -module Λ n V ∗ ⊗ k R the above equality deﬁnes N + ( x ) ∈ R uniquely.5ultiplicative property of the PfafﬁanApplying the identity (3.1) to x ω ∈ Λ V ∗ , we get the equalitydet ( ω x ) = µ ( ω x ) ⊗ µ ( ω x ) in the line ( Λ n V ∗ ) ⊗ . It follows that det ( x ) = N + ( x ) (3.2)for all x ∈ g + . Note that since this equality is valid for all x ∈ g + ⊗ k R for all k -algebra R ,det = N + can be seen as an equality in the coordinate ring of g + .As a square-root of the determinant, we may expect that the function N + satisﬁes a mul-tiplicative property as the determinant does. However, the multiplicativity does not makesense a priori as the subspace g + of gl ( V ) is not stable under matrix multiplication. We notethat for x , y ∈ g + , x y ∈ g + if and only if x y = y x . The multiplicicativity of the Pfafﬁan normN + then makes sense as an identity in the coordinate ring of the commuting subscheme C g + of g + × g + . Proposition 3.1.

The equality N + ( x y ) = N + ( x ) N + ( y ) holds in the coordinate ring of C g + .Proof. This is equivalent to prove that for every point ( x , y ) ∈ C g + ( R ) with values in anarbitrary k -algebra R , the identity N + ( x y ) = N + ( x ) N + ( y ) holds in R . For this we introducenew formal variables α , β and consider commuting elements1 + α x , 1 + β y ∈ g + ( R [ α , β ]) with values in the polynomial ring R [ α , β ] . We also have ( + α x )( + β y ) ∈ g + ( R [ α , β ]) because x y = y x . We now have elements P x = N + ( + α x ) , P y = N + ( + β y ) and P x y = N + (( + α x )( + β y )) in R [ α , β ] which are polynomial with free coefﬁcients equal to 1. Wealso note that P x ∈ R [ α ] is a polynomial of degree at most n in α whose coefﬁcient of α n isN + ( x ) , P y ∈ R [ β ] is a polynomial of degree at most n in β whose coefﬁcient of β n is N + ( y ) ,and P x y = N + (( + α x )( + β y )) ∈ R [ α , β ] is a polynomial of degree at most n in bothvariables α and β whose coefﬁcient of α n β n is N + ( x y ) . To prove N + ( x y ) = N + ( x ) N + ( y ) , itis enough to prove the equality of polynomials P x y = P x P y .Since the ring of polynomials R [ α , β ] embeds in the ring of formal series R [[ α , β ]] , itis enough to prove the equality P x y = P x P y in R [[ α , β ]] . We note that P x , P y , P x y are nowinvertible elements of R [[ α , β ]] which is a limit of thickening of R . Using the fact that thesquare map G m → G m is étale (here we use the assumption char ( k ) > g ∈ R [[ α , β ]] × with free coefﬁcient g ∈ R × , and for every square-root f ∈ R × of g , there existsa unique f ∈ R [[ α , β ]] × with free coefﬁcient f such that f = g .Using the equality (3.2) we have P x y = P x P y . The fact that P x , P y , P x y have free coefﬁ-cients 1 implies now the equality P x y = P x P y as formal series, and thus as polynomials.6pectral data map for symplectic groupsWe observe that it is possible to prove the equality N + ( x y ) = N + ( x ) N + ( y ) holds for everypoint C g + with value in a ﬁeld using their simultaneous triangulation. This implies that theequality N + ( x y ) = N + ( x ) N + ( y ) holds in the reduced quotient of k [ C g + ] . However we donot know whether k [ C g + ] is reduced. Let V be a 2 n -dimensional k -vector space equipped with a symplectic form ω ∈ Λ V ∗ . Thegroup G = Sp n is the subgroup of GL n preserving ω . The Lie algebra g = sp n consistsof elements x ∈ gl ( V ) such x ∗ ω = − ω x . We have an orthogonal complement g + of g in gl n consisting of x ∈ gl ( V ) such that x ∗ ω = ω x . We observe that if matrices x , y ∈ g are commuting matrices then we have x y ∈ g + . We have also noted that for commutingelements x , y ∈ g + we have x y ∈ g + . If x ∈ g and y ∈ g + are commuting matrices then wehave x y ∈ g .Let A denote the coordinate ring k [ C d g ] of the commuting scheme and ( x , . . . , x d ) ∈ g ( A ) d the universal sequence of commuting matrices. Let S = k [ X , . . . , X d ] be the polynomial ringwith d variables. The commutation property implies that for every k -algebra R there existsa morphism of rings p : S ⊗ k R → gl n ( A ⊗ k R ) sending X i x i . For every a = ( a , . . . , a d ) ∈ Z d ≥ the image p ( X a . . . X a d d ) lies in g ( R ) or g + ( R ) depending on whether a + · · · + a d is odd or even. If S + denotes the subalgebra of S generated by the monomials X a . . . X a d d with a + · · · + a d even then we have p ( S + ) ∈ g + ( R ) .Let us denote p + : S + ⊗ k R → g + ( A ⊗ k R ) the restriction of p to S + . Composing with thePfafﬁan norm N + : g + → A , we have a polynomial lawN + : S + ⊗ k R → A ⊗ k R which is homogenous of degree n and multiplicative according to Proposition 3.1. By Roby’stheorem, this gives rise to a morphism of k -algebras s : (( S + ) ⊗ n ) S n → A satisfying s ( q ⊗ n ) = N + ( p + ( q )) for all q ∈ S + . Since N + is G -invariant, the induced map s isalso G -invariant, and as a result, the image of s is contained in A G .Let τ denote the involution of S given by τ ( X i ) = − X i . We note that S + is the subalgebraof S of ﬁxed points of τ . As a result, (( S + ) ⊗ n ) S n can be identiﬁed with the subalgebra of S ⊗ n of ﬁxed points under ( Z / Z ) n ⋊ S n . This will permit us to identify (( S + ) ⊗ n ) S n with k [ t d ] W via some explicit choice of the Cartan algebra. This choice will be ultimately irrelevant as wewill prove that s is an inverse to the map c that doesn’t depend on the choice of the Cartanalgebra. 7pectral data map for symplectic groupsLet V be now the standard 2 n -dimensional vector space k n with basis e , . . . , e n and ω the standard symplectic form given by ω ( e i , e j ) = i + j = n +

11 if i + j = n + i ≤ n − i + j = n + i ≥ n + t of diagonal matrices of the form diag ( b , . . . , b n , − b n , . . . , − b ) will then be aCartan algebra of g equipped with the obvious action of W = {± } n ⋊ S n . The coordinatering of t is then the polynomial ring k [ b , . . . , b n ] where the b i are the coordiantes given byentries of the diagonal matrix as above. Let B = k [ t d ] denote the coordinate ring of t d and ( y , . . . , y d ) ∈ t d ( B ) the tautological B -point of t d . We consider the elements b j ( y i ) ∈ B with1 ≤ i ≤ d and 1 ≤ j ≤ n and the isomorphism of algebras β : S ⊗ n → B given by β ( X j , i ) = b j ( y i ) (4.1)where X j ,1 , . . . , X j , d are the coordinates of the j th copy of S . By restriction we have an iso-morphism of algebras β : (( S + ) ⊗ n ) S n = ( S ⊗ n ) ( Z / Z ) n ⋊ S n → k [ t d ] W .It follows that we have a morphism of algebras s : k [ t d ] W → k [ C d g ] G (4.2)such that s ( β ( q ⊗ n )) = N + ( p + ( q )) (4.3)for all q ∈ S + . It remains to prove that the spectral data map s is inverse to the Chevalleyrestriction. Theorem 4.1.

The map c : k [ C d g ] G → k [ t d ] W is an isomorphism with the inverse given by thespectral data map s : k [ t d ] W → k [ C d g ] G of (4.2) . We need to show that the compositions c ◦ s and s ◦ c are equal to the identities. To this end,we introduce a set of generators for the rings k [ t d ] W and k [ C d g ] G respectively, and then wecheck the desired property on those generators. Following Procesi as in [ ] these functionsare constructed as certain traces. We only use the assumption that k is of characteristic zeroor of prime characteristic large enough to prove that these trace functions form a systemof generators. We record this fact to facilitate the work of those who may want make theassumption on the characteristic explicit. 8pectral data map for symplectic groupsFor every a = ( a , . . . , a d ) ∈ Z d ≥ , we deﬁne the element φ a ∈ A given by φ a = tr ( x a · · · x a d d ) (4.4)where ( x , . . . , x d ) ∈ C d g ( A ) is the universal point of the commuting scheme. Since the traceis G -invariant, we have φ a ∈ A G . We note that if a + · · · + a d is odd then x a · · · x a d d ∈ g ( A ) and if a + · · · + a d is even then x a · · · x a d d ∈ g + ( A ) . It follows that φ a = a + · · · + a d isodd. For this reason we will only consider the functions φ a with a + · · · + a d even. Proposition 4.2.

The functions φ a with a ∈ Z d ≥ form a set of generators of k [ C d g ] G .Proof. The ﬁrst statement is consequence of a result of Procesi [

4, Theorem 10.1 ] . Procesidescribe a set of functions on gl ( V ) d invariant under the diagonal action of the symplecticgroup G which generate the ring k [ gl ( V ) d ] G . Since C d g is clearly a closed subscheme of gl ( V ) d „ the restriction map k [ gl ( V ) d ] G → k [ C d g ] G is surjective (this is true in characteristiczero because G is linearly reductive, and therefore it is also true for prime characteristiclarge enough). It follows that the restriction of Procesi’s functions form a set of generators of k [ C d g ] G . Using the fact that x , . . . , x d are commuting elements of the symplectic Lie algebra,it is easy to see that Procesi’s functions restrict to our functions φ a . Proposition 4.3. If ψ a = c ( φ a ) then the functions ψ a generate k [ t d ] W and we haves ( ψ a ) = φ a . Proof.

Let B = k [ t d ] denote the coordinate ring of t d and ( y , . . . , y d ) ∈ t d ( B ) the tautological B -point of t d . We have observed that B is a polynomial algebras of the variables b j ( y i ) for1 ≤ i ≤ d and 1 ≤ j ≤ n . By computing the trace of the matrix y a · · · y a d d we get ψ a = ¨ a + · · · + a d is odd,2 P nj = Q di = b j ( y i ) a i if a + · · · + a d is even, (4.5)We derive from these explicit formulas that the functions ψ a for a ∈ Z d ≥ with a + · · · + a d even generate B W . Here we need to assume the characteristic of k is either zero or no lessthan 2 n in order to perform usual manipulation with symmetric polynomials.In order to prove the equality s ( ψ a ) = φ a for a ∈ Z d ≥ with a + · · · + a d even, we willcalculate the determinant of the element p + ( θ a ) = t id − x a · · · x a d d ∈ g + ( A ⊗ k R ) where θ a = t − X a · · · X a d d ∈ R ⊗ k S + with the test ring R = k [ t ] being the algebra of polyno-mials in one variable t . 9eferencesOn the one hand, with the usual formula for the characteristic polynomial we havedet ( p + ( θ a )) = t n − φ a t n − + terms of lower degrees in t On the other hand after (3.2) we have det ( p + ( θ a )) = N + ( θ a ) where N + ( θ a ) = s ( β ( θ ⊗ na )) by (4.3). Using the identiﬁcation (4.1), the image of the element θ ⊗ na ∈ R ⊗ k S ⊗ n in R ⊗ B isgiven by the formula β ( θ ⊗ na ) = n Y j = ( t − d Y i = b j ( y i ) a i )= t n − n X j = d Y i = b j ( y i ) a i t n − + terms of lower degrees in t It follows that det ( p + ( θ a )) = t n − s ( ψ a ) t n − + terms of lower degrees in t Comparing the coefﬁcient of t n − in the two polynomial expression of det ( p + ( θ a )) ∈ A [ t ] we derive the desired equality s ( ψ a ) = φ a .To ﬁnish the proof of Theorem 4.1 we observe that the propositions above imply that thecompositions c ◦ s and c ◦ s are equal to the identities on the generators ψ a and φ a respectively. Acknowledgement

The research of Tsao-Hsien Chen is supported by NSF Grant DMS-2001257 and the S. S.Chern Foundation. The research of

Ngô Bảo Châu is supported by NSF grant DMS-1702380and the Simons foundation.

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