Mixed tensor invariants of Lie color algebra
Santosha Pattanayak Preena Samuel







 MSC subject classification 2020: 13A50, 15A72, 16R30, 17B10, 17B70, 17B75.

Keywords: Picture invariants, Invariant theory, Lie color algebra, G-graded Lie algebra.

Mixed tensor invariants of Lie color algebra
    santosha Pattanayak, Preena Samuel
    September 9, 2024
============================================



     Department of Mathematics and Statistics, IIT Kanpur,
Kanpur, India

santosha@iitk.ac.in, preena@iitk.ac.in







§ ABSTRACT 
 In this paper, we consider the mixed tensor space of a G-graded vector space where G is a finite abelian group. We obtain a spanning set of invariants of the associated symmetric algebra under the action of a color analogue of the general linear group which we refer to as the general linear color group. As a consequence, we obtain a generating set for the polynomial invariants, under the simultaneous action of the general linear color group,  on color analogues of several copies of matrices. We show that in this special case, this is the set of trace monomials, which coincides with the set of generators given by Berele in <cit.>.




§ INTRODUCTION


Lie color algebras were introduced as ‘generalized Lie algebras’ in 1960 by Ree <cit.>, being also called color Lie superalgebras (see <cit.>). Since then, this kind of algebra has been an object of constant interest in mathematics, being also remarkable for the important role played in theoretical physics, especially in conformal field theory and supersymmetries. Lie color algebras have close relation with Lie superalgebras. Any Lie superalgebra is a Lie color algebra defined by the simplest nontrivial abelian group ℤ_2, while any Lie color  algebra defined by a finitely generated abelian group admits a natural Lie superalgebra structure. Unlike Lie algebras and Lie superalgebras, structures and representations of Lie color algebras are far from being well understood and also there is no general classification result on simple Lie color algebras. Some recent interest related to their representation theory and related graded ring theory can be found in <cit.>, <cit.>, <cit.>.

 In <cit.>, Procesi studied tuples (A_1, ⋯ ,A_k) of endomorphisms of a finite-dimensional vector space up to simultaneous conjugation by studying the corresponding ring of invariants. He showed that for an algebraically closed field F of characteristic zero the algebra of invariants F[(A_i)_jl]^GL_r(F) can be generated by traces of monomials in A_1, ⋯ A_k. 
The main tool used in the work of Procesi, in order to describe the invariants, is the Schur–Weyl duality. This tool was used in a similar way also to study more complicated algebraic structures than just a vector space equipped with endomorphisms. In <cit.>, Datt, Kodiyalam and Sunder applied this machinery to the study of finite-dimensional complex semisimple Hopf algebras. They were able to obtain a complete set of invariants for separating isomorphism classes of complex semi-simple Hopf algebras of a given dimension. This  they  accomplished by giving an explicit spanning set for the invariant ring of the mixed tensor space. These are called  “picture invariants". The picture invariants obtained in the case of complex semisimple Hopf algebras were also obtained by using techniques from Geometric Invariant Theory in <cit.>. In <cit.> these picture invariants were used to describe a finite collection of rational functions in the structure constants of a Lie algebra, which form a complete set of invariants for the isomorphism classes of complex semisimple Lie algebras of a given dimension. 

More recently, the invariant ring of the mixed tensor spaces was used to extend results of <cit.> to separate isomorphism classes of complex Hopf algebras of small dimensions <cit.>. The invariant ring of the mixed tensor space arises naturally in obtaining invariants for any such isomorphism classes. Here, we look at the G-graded analogue of this invariant ring. We define “graded picture invariants" which Λ_ϵ-linearly span this ring. 

 



In <cit.>, Fischman and Montgomery proved an analogue of the double centralizer theorem for coquasitriangular Hopf algebras and as a consequence they proved the double centralizer theorem for the general linear Lie color
algebra 𝔤𝔩_ϵ. In <cit.>, Moon reproves the double centralizer theorem for 𝔤𝔩_ϵ by relating its centralizer algebra to that of the general Lie superalgebra. For a G-graded vector space V, we set U:=V_Λ_ϵ.
The general linear color group is defined as the group of invertible elements in the set of all degree preserving Λ_ϵ-linear maps on U. We denote this group by _ϵ(U). Using the results of <cit.>, Berele obtained  a graded analogue of the Schur–Weyl
duality between the general linear color group and the symmetric group in <cit.>.  This result is then used to arrive at a generating set for the _ϵ(U)-polynomial invariants on color analogue of multiple copies of matrices, thereby extending some of the results of <cit.> to this setting.

In this paper, we extend Berele's results to the action of the general linear color group on the mixed tensor space ⊕_i=1^sU_b_i^t_i where  t_i, b_i are in ∪{0} for all i=1,… s. The case of invariants of the color analogues of d copies of matrices, as described in <cit.>, may be seen as a special case of the above by taking t_i=1=b_i for all i=1…,s. We show in Theorem <ref> that the ring of invariants of the G-graded symmetric algebra S(⊕_i=1^sU_b_i^t_i)^*, is generated by certain special invariants which are analogues of the “picture invariants" in <cit.>. We then show in Theorem <ref> that in the special case of t_i=1=b_i for all i=1…,s, this agrees with the results of <cit.>. Viewing a  superspace as a special case of a G-graded space, the invariants that we obtain here coincide with those obtained in <cit.>.


  To define polynomials on a G-graded vector space we use the notion of -valued polynomials over U, as introduced in <cit.>.  This is done in <ref>.  This description of polynomials over U is a suitable alternative to Berele's notion of a polynomial  as defined in <cit.> since we are interested in mixed tensor spaces which do not have a natural identification with matrices. In the special case of t_i=1=b_i for i=1,…,s, however, this notion of polynomials agrees with that of <cit.>. We have used this notion also in <cit.> to arrive at extensions of Berele's results in the supersetting. For defining this notion of a polynomial, we consider the -module of maps to  from the graded component of U corresponding to the identity element of G. This module is denoted as  ℱ(U_0,). Then using a G-graded analogue of the restitution map from the r-fold tensor space of U^* to  ℱ(U_0, ), we call the image under this map to be the space of homogeneous polynomials of degree r. The G-graded algebra of polynomials on U is then taken to be the direct sum of these spaces of homogeneous polynomials. We prove  that this algebra is isomorphic to the symmetric algebra of U^*, under the restitution map, analogous to loc. cit.. In this paper, we use the above notion of -valued polynomials on ⊕_i=1^sU_b_i^t_i and obtain a generating set for the polynomial invariants of a G-graded mixed tensor space.  We then show that in the special case when the mixed tensor space corresponds to several copies of the endomorphism space of U, the graded picture invariants are just trace monomials as given in <cit.>. This may be regarded as the G-graded analogue of Procesi's result in <cit.>. 
  
   
  
We now give an outline of the paper. In section 2 we review preliminaries of G-graded vector spaces and Lie color algebras and, we also recall the G-graded analogue of the Schur-Weyl duality. We also introduce in this section the notion of a general linear color group which is denoted as _ϵ(U) in <cit.> and recall the Schur-Weyl duality for it. In section 3 we introduce the notion of graded picture invariants and prove that these span the space of invariants of the symmetric algebra of the dual of the mixed tensor space. Using this result we give a spanning set for the polynomial invariants of the mixed tensor space and thereby show that the trace monomials span the  polynomial invariants for the action of the general linear color group on color analogues of several copies of matrices. 

Notation: Throughout this paper we work over the field of complex numbers ℂ. All modules and algebras are defined over ℂ and in addition all the modules are of finite dimension. We write ℤ_2={0̅,1̅} and use its standard field structure. For a finite abelian group G, we denote the identity element of G by ∘, while we reserve the symbols 0,1 to denote the usual complex numbers that they represent. 


§ PRELIMINARIES




 §.§ 
Definition: Let G be a finite abelian group.  A map ϵ: G × G →ℂ∖{0} is called a skew-symmetric bicharacter on G if the following identities hold, for all f, g, h ∈ G. 

(1) ϵ (f, g+h)=ϵ(f,g) ϵ(f,h),

(2) (g+h,f)=(g,f)(h,f),

(3) (g,h)(h,g)=1.

From the definition of ϵ it follows that ϵ(g,∘)=ϵ(∘,g)=1 and ϵ(g,g)=± 1, where ∘ denotes the identity element of G. 

For a bicharacter ϵ of G, we set G_0̅:={g ∈ G: ϵ(g,g)=1} and G_1̅=G ∖ G_0̅. Then there exists a group homomorphism θ:G →ℤ_2 such that θ(g)=0̅ for g ∈ G_0̅ and θ(g)=1̅ for g ∈ G_1̅. Moreover, G_0̅ is a subgroup of G of index 1 or 2. It easily follows that if g ∈ G_1̅ then -g ∈ G_1̅. 

Definition: For a finite abelian group G, a G-graded vector space is a vector space V together with a decomposition into a direct sum of the form V=⊕_g ∈ G V_g, where each V_g is a vector space. For a given g ∈ G the elements of V_g are then called homogeneous elements of degree g and we write |v| to denote the degree of v.

Any finite dimensional G-graded vector space 
V=⊕_g ∈ G V_g can be given a ℤ_2-grading via V=V_G_0̅⊕ V_G_1̅, where V_G_0̅=⊕_g ∈ G_0̅V_g and V_G_1̅=⊕_g ∈ G_1̅V_g.


Definition: Fix a pair (G,ϵ), where G is a finite abelian group and ϵ is a skew-symmetric bicharacter on G. A Lie color algebra L=⊕_g ∈ GL_g associated to (G, ϵ) is a G-graded ℂ-vector space  with a graded bilinear map [-,-]:L × L → L satisfying 

1. [L_g,L_h] ⊂ L_g+h for every g,h ∈ G,

2. [x, y]=-(x, y)[y, x] and 

3. (z, x)[x,[y, z]] + (x, y)[y,[z, x]] +(y, z)[z,[x, y]] = 0 for all homogeneous elements x,y,z ∈ L. 



 §.§ 
 Given a G-graded vector space V=⊕_g ∈ G V_g, the ℂ-endomorphisms of V, End_ℂ(V) is also G-graded, where 

    End_ℂ(V)_g={f: V → V: f(V_h) ⊂ V_g+h,    for    all    h ∈ G}

The general linear Lie color algebra, 𝔤𝔩_ϵ(V) is defined to be End_ℂ(V) with the Lie
bracket given by [x,y]_ϵ=xy-ϵ(x,y) yx. 

The k-fold tensor product V^⊗ k of V is also G-graded; V^⊗ k=⊕_g ∈ G (V^⊗ k)_g, where (V^⊗ k)_g=⊕_g=g_1+⋯ +g_kV_g_1⊗⋯⊗ V_g_k. 

We have an action of the symmetric group S_k on V^⊗ k as follows: 

The group S_k is generated by the transpositions s_1, s_2, ⋯ s_k-1, where s_i=(i,i+1). Then 

    s_i.(v_1 ⊗ v_2 ⊗⋯⊗ v_k)=ϵ(g,h) (v_1 ⊗ v_2 ⊗⋯⊗ v_i+1⊗ v_i ⊗⋯⊗ v_k),
 where v_i ∈ V_g and v_i+1∈ V_h. 
More generally, for σ∈ S_k and v=v_1 ⊗ v_2 ⊗⋯⊗ v_k where each v_i is a homogeneous element of V,
 
    σ.v=γ(v, σ^-1)(v_σ^-1(1)⊗ v_σ^-1(2)⊗⋯⊗ v_σ^-1(k)),
 where γ(v, σ)=∏_(i,j) ∈ Inv(σ)ϵ(|v_i|,|v_j|), with Inv(σ)={(i,j): i < j   and  σ(i) > σ(j)}. 
 
 We then extend the action to  V^⊗ k by linearity. We then have the following lemma.


    For two permutations σ, τ∈ S_k we have γ(v,στ)=γ(σ^-1v,τ)γ(v,σ).



Let w_i=v_σ(i) for all i. Then w_τ(i)=v_στ(i) for all i. 

The action of σ on v is given by σ.v=γ(v, σ^-1)(v_σ^-1(1)⊗ v_σ^-1(2)⊗⋯⊗ v_σ^-1(k))

We have τ^-1σ^-1.v=γ(v, στ)(v_στ(1)⊗ v_στ(2)⊗⋯⊗ v_στ(k)).

On the other hand τ^-1σ^-1.v=γ(v, σ)τ^-1.(v_σ(1)⊗ v_σ(2)⊗⋯⊗ v_σ(k))=γ(v, σ)τ^-1.w

=γ(v, σ) γ(w, τ)(w_τ(1)⊗ w_τ(2)⊗⋯⊗ w_τ(k))

=γ(v, σ) γ(σ^-1v, τ)(v_στ(1)⊗ v_στ(2)⊗⋯⊗ v_στ(k))

So we have the required identity. 


By fixing a set of homogeneous vectors v_1, ⋯ ,v_k of V such that |v_i|=a_i for all i, we set I to be the tuple (a_1,…, a_k) in G^k. The symmetric group S_k acts on G^k via  σ.(a_1,…, a_k):=(a_σ^-1(1),… , a_σ^-1(k)) for σ∈ S_k.
  Define γ(I,σ):=γ(v_1⋯ v_k,σ). Then the above relation may be rephrased as, γ(σ^-1 I,τ)γ(I,σ)=γ(I,στ).



We denote by Φ the resulting homomorphism: Φ: ℂ[S_k] → End_ℂ(V^⊗ k). 

On the other hand the general linear Lie color algebra, 𝔤𝔩_ϵ(V) acts on V^⊗ k by twisted derivation: 

    x.(v_1 ⊗ v_2 ⊗⋯⊗ v_k)=∑_i=1^k (∏_j <iϵ(α, g_j)) v_1 ⊗ v_2 ⊗⋯⊗ x.v_i ⊗⋯⊗ v_k,
 where x ∈𝔤𝔩_ϵ(V)_α and each v_j ∈ V_g_j. 

We denote by Ψ the resulting homomorphism: Ψ: 𝔤𝔩_ϵ(V) → End_ℂ(V^⊗ k).

The group G also act on V^⊗ k by 

    g(v_1 ⊗ v_2 ⊗⋯⊗ v_k)=∏_iϵ(g, g_i)(v_1 ⊗ v_2 ⊗⋯⊗ v_k),
 where g ∈ G and each v_i ∈ V_g_i.

We denote by η the resulting homomorphism: η: ℂ[G] → End_ℂ(V^⊗ k). We then have the double centralizer theorem for the general linear Lie color algebra. 

 Let A=Φ(ℂ[S_k]) and let B be the subalgebra of End_ℂ(V^⊗ k) generated by η(ℂ[G]) and Ψ(𝔤𝔩_ϵ(V)). Then A and B are centralizers of each other. 







 §.§ 
  Fix a pair (G,ϵ), where G is a finite abelian group and ϵ is a skew-symmetric bicharacter on G. We define a graded algebra Λ_ϵ which generalizes the infinite Grassmann algebra in the super setting. Let X=∪_g ∈ GX_g be a G-graded set, where each X_g is countably infinite. Let Λ be the free ℂ-algebra generated by X and we define Λ_ϵ:=Λ/I, where I is the ideal generated by the elements xy-ϵ(g,h)yx, for all g,h ∈ G and for all x ∈ X_g and y ∈ X_h. Then Λ_ϵ is also G-graded. 

If X={x_1, x_2, ⋯}, then the set {x_i_1x_i_2⋯ x_i_r: i_1 ≤ i_2 ≤⋯≤ i_r} is a basis of Λ_ϵ. For a fixed linear ordering of the elements of X, we set (N) to be the linear span of the basis vectors {x_i_1x_i_2⋯ x_i_r:1≤ i_1 ≤ i_2 ≤⋯≤ i_r≤ N} where N ∈ℤ_≥ 0. We then have the following filtration: 
    Λ_ϵ(N) ⊂Λ_ϵ(N+1),   for all   N ≥ 0.
 



 §.§ 
Given a G-graded vector space V=⊕_g ∈ G V_g, we set U:= V ⊗_ℂΛ_ϵ. Then U is also a G-graded Λ_ϵ-bimodule: U=⊕_g ∈ GU_g, where U_g=⊕_g_1+g_2=g V_g_1⊗ (Λ_ϵ)_g_2; x.u=ϵ(g,h)ux where x and u are of degrees g and h respectively. 

Let End_Λ_ϵ(U):={T ∈ End_ℂ(U): T(ux)=T(u)x    for    u ∈ U, x ∈Λ_ϵ}. Then End_Λ_ϵ(U) is also G-graded in a natural way via 
    End_Λ_ϵ(U)_g={T ∈ End_Λ_ϵ(U): T(U_h) ⊆ U_g+h,    for    all    h ∈ G}.


There is a natural embedding V↪ U given by v↦ v⊗ 1. Any ℂ-linear map T:V → V extends uniquely to an element of End_Λ_ϵ(U) and so we have End_Λ_ϵ(U) ≅ End_ℂ(V) ⊗_ℂΛ_ϵ. 

More generally,  for two G-graded vector spaces V and V', set U:=V_ℂ and U':= V'_ℂ, then we denote by Hom_Λ_ϵ(U,U')  to be the set of ℂ-linear maps from U to U' which commute with the right action of Λ_ϵ. Then Hom_ℂ(V, V')_ℂ≅ Hom_(U,U'). In particular, if we denote by U^* the G-graded space Hom_Λ_ϵ(U,Λ_ϵ), then U^*≅ V^*_ℂ.

With notation as above, it may be easily seen  that U_U'≅ (V_ℂ V')_ℂ via the map defined on the homogeneous elements by vλ v'λ'↦ v v'ϵ(|λ|,|v'|)λλ'. Here |λ|, |v'| denote the G-grading of λ and v' respectively. 

There is a pairing between U and U^*  given by uα↦ϵ(|α|,|u|)α(u) where u∈ U and α∈ U^*. This will be called the evaluation map and is denoted by ev. More generally, let V_1,…, V_k be G-graded vector spaces and let W_i:=V_i_ℂ. Let W=W_1⊕⋯⊕ W_k and τ be the permutation which takes (1,2,⋯, k,k+1, ⋯ ,2k) to (τ(1), τ(2), ⋯ ,τ(k),τ(k+1), ⋯ ,τ(2k))=(1,3,5, ⋯ ,2k-1, 2,4,6, ⋯ ,2k).  Then the group S_2k acts naturally on the 2k-fold tensor product (W⊕ W^*)^ 2k. Under this action the element τ∈ S_2k induces an isomorphism between the subspaces W_1^*⋯ W_k^*  W_1⋯ W_k and W_1^* W_1⋯ W_k^*  W_k of (W⊕ W^*)^ 2k. This isomorphism followed by the map W_1^* W_1⋯ W_k^*  W_k→ given by α_1 w_1⋯α_k w_k↦∏_iα_i(w_i) is called the evaluation map and is denoted by ev. The non-degeneracy of the pairing ev:W_1^*⋯ W_k^*  W_1⋯ W_k→ comes from noticing that this map is obtained by extending scalars to  of a non-degenerate pairing over ℂ. The isomorphism induced by this non-degenerate pairing will be denoted by ι:W_1^*⋯ W_k^* → (W_1⋯ W_k)^*




 §.§ Symmetric algebra on a graded vector space
   Let G be a finite abelian group and ϵ is a skew-symmetric bicharacter on G. Let V=⊕_g ∈ G V_g be a G-graded vector space over ℂ. The tensor algebra on V is T(V)=⊕_k ≥ 0 V^⊗ k. 

The symmetric algebra on V is defined to be S(V)=T(V)/I(V), where I(V) is the ideal of T(V) generated by elements of the form v ⊗ w-ϵ(g,h)w ⊗ v, where v and w are homogeneous elements of degrees g and h respectively. Note that the ideal I(V) is both ℤ-graded as well as G-graded.

We note that if we write V=⊕_g ∈ G V_g=V_G_0̅⊕ V_G_1̅, where V_G_0̅=⊕_g ∈ G_0̅ V_g and V_G_1̅=⊕_g ∈ G_1̅ V_g then since ϵ(v,v)=-1 for v ∈ V_G_1̅ we have v ⊗ v=0 in S(V). 

We define the d-th symmetric power of V, written S^d(V) to be the image of V^⊗ d in S(V). Since I(V) is both ℤ as well as G-graded, we have S(V)=⊕_d ≥ 0S^d(V) and each S^d(V) is G-graded. If V_G_1̅=V then S(V) is denoted as ⋀(V) and it is called the exterior algebra of V.

We then have the following lemma which will be used in the proof of the main theorem. 


    (1) Given any map f:V → W, between two G-graded vector spaces, there is a unique map of ℂ-algebras S(f):S(V) → S(W) carrying V to W.

    (2) For a G-graded vector space V, we have S(V ⊗_ℂΛ_ϵ)=S(V) ⊗_ℂΛ_ϵ.

    (3) For two G-graded vector spaces V and W we have S(V ⊕ W)=S(V) ⊗ S(W). 

    (4) If V is a G-graded vector space and {v_1, v_2, ⋯ ,v_n} is a basis of V consisting of homogeneous elements, then the set of all monomials of the form v_1^i_1..v_n^i_n such that ∑_j=1^ni_j=d and 0≤ i_j≤ 1 for v_i_j in V_G_1̅ form a basis of S^d(V). 


 (1) The map f induces a map from T(V) to T(W) carrying the ideal of relations in T(V) to the ideal of relations in T(W). So we have an induced map of ℂ algebras S(f):S(V) → S(W), which is unique by the construction. 

(2) It is clear that in the tensor algebra level the assertion holds, that is, T(V ⊗_ℂΛ_ϵ)=T(V) ⊗_ℂΛ_ϵ. The algebra Λ_ϵ⊗_ℂ S(V) is obtained by factoring out the ideal generated by elements of the form 1 ⊗ (x ⊗ y -ϵ(x,y)y ⊗ x) from T(V) ⊗_ℂΛ_ϵ. The element 1 ⊗ (x ⊗ y -ϵ(x,y)y ⊗ x) corresponds to (1 ⊗ x) (1 ⊗ y) -ϵ(x,y)(1 ⊗ y) ⊗ (1 ⊗ x) and these elements generate the ideal of relations in T(V ⊗_ℂΛ_ϵ). 

(3) The proof is straight forward. 

(4) We write V=⊕_g ∈ G V_g=V_G_0̅⊕ V_G_1̅, where V_G_0̅=⊕_g ∈ G_0̅ V_g and V_G_1̅=⊕_g ∈ G_1̅ V_g. Then S(V)=S(V_G_0̅) ⊗⋀(V_G_1̅), where S(V_G_0̅) and ⋀(V_G_1̅) are the symmetric and exterior algebras on V_G_0̅ and V_G_1̅ respectively. We then have the required result. 


   



 §.§ 
 For V=⊕_g ∈ G V_g and U=V ⊗_ℂΛ_ϵ, we set 
    M_ϵ(U):={T: U → U: T(U_g) ⊆ U_g,    for    all    g ∈ G}.
 With the identification,  End_Λ_ϵ(U) ≅ U ⊗_Λ_ϵ U^*, M_ϵ(U) is spanned by all u ⊗ f, where u and f are homogeneous of opposite degree, i.e., |u|=-|f|. We define the trace function from  M_ϵ(U) to (Λ_ϵ)_G_0 by tr(u ⊗ f)=ϵ(|u|, |f|)f(u) with the above identification. 

By choosing a basis {v_1, ⋯ ,v_r, v_r+1⋯ ,v_n} of V, where the first r of them are in V_G_0̅ and the last n-r are in V_G_1̅ and identifying M_ϵ(U) with the space of matrices we get that 
    tr(A)=∑_i=1^ra_ii-∑_i=r+1^na_ii.
 It satisfies tr(AB)=tr(BA) for A,B ∈ M_ϵ(U) (see Lemma 3.7 of <cit.>).

Definition: The general linear color group GL_ϵ(U) is defined to be the group of invertible elements in M_ϵ(U).

The group GL_ϵ(U) acts on the k-fold tensor product U^⊗ k diagonally: 


    T.(u_1 ⊗⋯⊗ u_k)=(Tu_1 ⊗⋯⊗ Tu_k).


We denote by Ψ the resulting homomorphism: Ψ:  GL_ϵ(U) → End_Λ_ϵ(U^⊗ k).

The action of the symmetric group on V^⊗ k defined in <ref> can be extended to an Λ_ϵ-linear action on U^⊗ k as follows: 

For σ∈ S_k and u=u_1 ⊗ u_2 ⊗⋯⊗ u_k ∈ U^⊗ k where each u_i is a homogeneous element of U,
 
    σ.u=γ(u, σ^-1)(u_σ^-1(1)⊗ u_σ^-1(2)⊗⋯⊗ u_σ^-1(k)),
 where γ(u, σ)=∏_(i,j) ∈ Inv(σ)ϵ(|u_i|,|u_j|), with Inv(σ)={(i,j): i < j   and  σ(i) > σ(j)}. 


As U=V ⊗Λ_ϵ is a Λ_ϵ-bimodule, U^⊗ k is also a Λ_ϵ-bimodule. The action of S_k and Λ_ϵ on U^⊗ k commute with each other. So the S_k action on U^⊗ k extends to an action of the group algebra Λ_ϵ[S_k]. 

We denote by Φ the resulting homomorphism: Φ:  Λ_ϵ[S_k] → End_Λ_ϵ(U^⊗ k).

Then in <cit.>, Berele proved a version of Schur-Weyl duality for the general linear color group.


    Let A be the subalgebra of End_Λ_ϵ(U^⊗ k) generated by Ψ(GL_ϵ(U)). Then the centralizer of A in End_Λ_ϵ(U^⊗ k) is Φ(Λ_ϵ[S_k]).





§ MIXED TENSOR SPACES
 For a G-graded vector space V over ℂ, one can define the mixed tensor space as the direct sum of the form ⊕_i=1^s(V^ b_iV^*^⊗ t_i) for an s∈ℕ and t_i,b_i∈ℕ∪{0}. For simplicity of notation we denote V^ b_iV^*^⊗ t_i by V_b_i^t_i. Since each summand in the mixed tensor space has a G-grading, there is a natural G-grading inherited by their direct sum ⊕_i=1^s(V_b_i^t_i). Taking U=V_ℂ and U^*=Hom_(U,)≅ V^*_ℂ, the mixed tensor space over  is the G-graded space ⊕_i=1^s(U^ b_i _U^*^⊗ t_i) for an s∈ℕ and t_i,b_i∈ℕ∪{0}. It may be seen that (⊕_i=1^s(V_b_i^t_i))_ℂ≅⊕_i=1^s(U^ b_i_U^*^⊗ t_i). We shall denote this mixed tensor space over  by W. 

Let m= V_G_0̅ and n= V_G_1̅. Fixing an ordering for the elements of G_0̅ and G_1̅, we list the elements of G as {∘,g_1,g_2,…} with the elements in G_0̅ appearing first. We then arrange the basis vectors of V, {e_i}_i=1^m+n, such that the first m vectors are from V_G_0̅ and the rest are from V_G_1̅; further, ordered linearly so that i<j implies |e_i| appears before |e_j| under the fixed ordering of elements of G. Here we use the notation |v|=h for v∈ V_h. Let {e_i^*}_i=1^m+n be the dual basis corresponding to {e_i}_i=1^m+n. We denote the image in U and U^* of the above basis elements under  the embedding V↪ U (and V^*↪ U^* respectively) also by the same notation. The mixed tensor space W then has a basis given in terms of the above bases of U and U^*.
Denote the element dual to the basis vector e_l_1⋯ e_l_b_i e^*_u_1⋯ e^*_u_t_i∈ U_b_i^t_i by T(i)_l_1… l_b_i^u_1… u_t_i. We denote the corresponding element in W^* also by the same notation. Notice that T(i)_l_1… l_b_i^u_1… u_t_i defines a linear map on W via the projection p_i:W→ U_b_i^t_i, i.e., T(i)_l_1… l_b_i^u_1… u_t_i(v_1,…,v_s)=T(i)_l_1… l_b_i^u_1… u_t_i(v_i) for (v_1,…,v_s)∈ W.  The G-grading of the element T(i)_l_1… l_b_i^u_1… u_t_i∈ W^* is given by ∑_i=1^b_ih_l_i-∑_i=1^t_ih_u_i where e_l_j∈ V_h_l_j and e^*_u_j∈ V_h_u_j^*. The set of all T(i)_l_1… l_b_i^u_1… u_t_i, where i=1,… , s and u_j, l_j are from {1,⋯, m+n}, forms a -basis of W^*. 



 §.§ Symmetric algebra of the mixed tensor space
   Let  S(W^*) be the symmetric algebra of W^*. We denote by ϖ:T(W^*)→ S(W^*)  the natural map from the tensor algebra of W^* to  S(W^*). We know that S(W^*) has a ℤ-grading given by  ⊕_r∈∪{0}S^r(W^*) where  S^r(W^*) is the image  under ϖ of T^r(W^*). The restriction of ϖ to T^r(W^*) is denoted as ϖ_r. By Lemma <ref>(2), S(W^*) can be identified with S(⊕_i=1^s(V_b_i^t_i)^*)_; further by Lemma <ref>(4)  and the remarks in <ref>, this  in turn is identified with [S((⊕_i=1^s(V_b_i^t_i)^*)_G_0̅)⋀((⊕_i=1^s(V_b_i^t_i)^*)_G_1̅)] _. Using the relations among the T(i)_l_1… l_b_i^u_1… u_t_i, i=1,… , s and u_j, l_j∈{1,⋯, m+n} which are given by the ideal I(W^*) of T(W^*), the latter identification yields a -basis for S(W^*) consisting of monomials in T(i)_l_1… l_b_i^u_1… u_t_i, i=1,… , s and u_j, l_j are from {1,⋯, m+n} where the variables T(i)_l_1… l_b_i^u_1… u_t_i  are  arranged in order such that basis vectors of (⊕_i=1^sV_b_i^t_i^*)_g_l come before those of (⊕_i=1^sV_b_i^t_i^*)_g_k whenever l<k and among them arranged from i=1… ,s;  the degree of each variable  T(i)_l_1… l_b_i^u_1… u_t_i∈ (⊕_i=1^sV_b_i^t_i^*)_G_1̅ in such a monomial being either 0 or 1. The monomials among the above whose total degree is r, form a basis  for S^r(W^*) over . 


For  each r∈ℕ and a s-tuple (m_1,…, m_s) such that 
m_1+⋯+ m_s=r, the tensor space T^m_1(U_b_1^t_1^*)⋯ T^m_s(U_b_s^t_s^*) is realised as a subspace of T^r(W^*). The image of the restriction of ϖ_r to this subspace is denoted as S^m_1,…,m_s(W^*). If the multidegree of the element T(i)_l_1… l_b_i^u_1… u_t_i∈ S(W^*) is set to be the s-tuple (0,…,0,1,0…,0) with 1 in exactly the i-th position then the monomials in the above listed basis of 
S^r(W^*) with multidegree (m_1,…,m_s) forms a basis of S^m_1,…, m_s(W^*). The space _i=1^sS^m_i((U_b_i^t_i)^*) can be  identified with S^m_1,…, m_s(W^*) under the map ϕ_1⋯ϕ_s↦ϕ_1⋯ϕ_s.  It is easy to see that the following diagram commutes, by checking it does on the basis elements of T^m_1(U_b_1^t_1^*)⋯ T^m_s(U_b_s^t_s^*):

    _i=1^s((U_b_i^t_i)^*)^ m_i@->>[r]^ϖ_m_1…ϖ_m_s@^(->[dd]   _i=1^sS^m_i((U_b_i^t_i)^*)[d]^≅
       S^(m_1,…, m_s)(⊕_i=1^s(U_b_i^t_i)^*)@^(->[d]
    T^r(W^*)@->>[r]^ϖ_m   S^r(W^*)


For (m_1,…, m_s), an s-tuple of non-negative integers such that  ∑_i m_i=m, the subspace T^m_1(U_b_1^t_1^*)⋯ T^m_s(U_b_s^t_s^*) of T^m(W^*) is a contains the direct sum of 
T^r_1((U_b_1^t_1^*)_0) T^n_1((U_b_1^t_1^*)_1)⋯ T^r_s((U_b_s^t_s^*)_0) T^n_s((U_b_s^t_s^*)_1)
for (r_1,…, r_s), (n_1,…, n_s), non-negative integers such that  r_j+n_j=m_j for each j=1… s.

Note that in the above diagram, all the maps are _ϵ(U)-equivariant; indeed, since the _ϵ(U)-action on (U_b_i^t_i)^*^ m_i (resp., W^*^ r) commutes with the S_m_i-action (resp., S_r-action) and when M=W or M=U^t_i_b_i, we note that S^m(M^*) can be identified under ϖ_r with the _ϵ(U)-invariant subspace e(r)(M^*^ r) where e(r):=1/r!∑_σ∈ S_rσ. Thus the _ϵ(U)-action descends to S^r(M^*) thereby making the  map ϖ_r a _ϵ(U)-equivariant map with respect to this action.


 §.§ Main Result

Keeping notations as above, let ∑_i m_ib_i=N and ∑_i m_it_i=N'. We have the following sequence of GL_ϵ(U)-invariant isomorphisms:

    (U^ (∑_i m_ib_i) (U^*)^(∑_i m_it_i))^*→(_i=1^s(U_b_i^t_i)^ m_i)^*→_i=1^s((U_b_i^t_i)^*)^ m_i

The first isomorphism is induced on the duals by the permutation action of μ∈ S_N+N' on the subspace(U⊕ U^*)^ (N+N') of (U⊕ U^*)^ (N+N') where μ takes (1,…, N+N') to (1,… b_1, N+1, … , N+t_1, b_1+1, … 2b_1, ⋯  ). The isomorphism above is _ϵ(U)-equivariant since the symmetric group action on (U⊕ U^*)^  (N+N') commutes with the _ϵ(U)-action induced on it. 
Let k=∑_i m_i and W=W_1⊕⋯⊕ W_k, where W_i=U_b_i^t_i. Then the group S_2k acts naturally on the 2k-fold tensor product (W⊕ W^*)^ 2k.  The second isomorphism in the above sequence is the inverse of ι as described in <ref> between W_1^*⋯ W_k^* and (W_1⋯ W_k)^*. More explicitly, the isomorphism is given on the dual basis by 
    T_w_1⋯ w_∑_i m_i↦p_ϵ(w,w)T(1)_w_1⋯ T(s)_ w_∑_i m_i
 where w_i∈ W_i is a basis vector and T(i)_w_i is the dual vector in W_i^*; here for w_i=e_l_1⋯ e_l_b_i e^*_u_1⋯ e^*_u_t_i∈ U_b_i^t_i the notation T(i)_w_i represents the linear map  T(i)_l_1… l_b_i^u_1… u_t_i by our earlier notation. The value of p_ϵ(w,w) for w=(w_1,…, w_k) where w_i∈ V_|w_i| is given by ∏_i=1^k(∏_i≤ j≤ kϵ(|w_i|,|w_j|)). This isomorphism is _ϵ(U)-equivariant since the symmetric group action on (W⊕ W^*)^  2k  commutes with the _ϵ(U)-action induced on it.

 By a standard argument, the space U^ NU^*^ N' has _ϵ(U)-invariants if and only if N=N', and so does its dual, (U^ NU^*^ N')^*. When N=N', the latter space can be identified with _(U^ N) via the non-degenerate pairing (_(U^ N)) (U^ NU^*^ N)→Λ_ϵ given by ⟨ A,vf^*⟩:= ev(A(v)f^*). The latter is ev(ν.τ. (A(v)f^*)) where τ∈ S_2N is as in <ref> (with k replaced by N) and ν∈ S_2N is the permutation (1  2)(3  4)⋯ (2N-1  2N). This gives a _ϵ(U)-invariant isomorphism 
    Θ:_(U^ N)→ (U^ NU^*^ N)^*.




  §.§.§ Graded picture invariants for colored spaces

Given an s-tuple (m_1,…,m_s) of non-negative integers such that ∑_k=1^s m_kt_k=∑_k=1^s m_k b_k=N and a σ∈ S_N we associate the polynomial φ_σ∈ S(W^*) given by 
    ∑_r_1,…, r_N∈{1,…, n}p_σ(r_1,…, r_N, m_1,…, m_s)∏_i=1^s∏_j=1^m_iT(i)_r'_(∑_p<im_pb_p+(j-1)b_i+1)⋯ r'_(∑_p<im_pb_p+jb_i)^r_(∑_p<im_pt_p+(j-1)t_i+1)⋯ r_(∑_p<im_pt_p+jt_i)
where r'_j:=r_σ j and p_σ(I,M) for an N-tuple I=(r_1,…,r_N) of N elements from {1,…  m+n} and M=(m_1,…, m_s)∈ (∪{0})^s such that ∑_k m_kt_k=∑_k m_k b_k=N takes the value in G given by 
    γ(μ^-1.(I,σ I^*),(ντσ̂μ)^-1)p(w,w)
where (r_1,…, r_N)^*:=(r_1^*,…, r_N^*), indicating that these are the indexes corresponding to the basis vectors e_r_1^*,…,e_r_N^*  in the dual space U^*, μ, τ,ν are as defined above and for a σ∈ S_N we define σ̂∈ S_2N as 

    σ̂(i)   =σ(i)   i≤ N 
    σ̂(i)   =i  i>N
 The vector w:=w_1 ⊗ w_2 ⊗⋯⊗ w_∑_i=1^sm_i where each w_∑_p<im_p+j:=e_r_∑_p<im_pb_p+(j-1)b_i+1⋯ e_r_∑_p<im_pb_p+jb_i e_r_σ^-1(∑_p<im_pt_p+(j-1)t_i+1)^*⋯ e_r_σ^-1(∑_p<im_pt_p+jt_i)^* for i=1,…,s;  j=1,… ,m_i.

The polynomials φ_σ as defined above[In (<ref>), the monomials should be considered modulo the anti-commutativity relations in S(W^*). See Remark <ref> above.] are called the graded picture invariants.

 


    


    With the above notation, the elements φ_σ linearly span S(W^*)^_ϵ(U).


     As was seen earlier in this section, the space (U^ NU^*^ N')^* has _ϵ(U)-invariants if and only if N=N'. By Theorem <ref>, we know that the _ϵ(U)-invariants of _(U^ N) are spanned over  by S_N. So, via the isomorphism Θ we get invariants on (U^ NU^*^ N)^*. Let (m_1,…,m_s) be an s-tuple of non-negative integers such that ∑_im_it_i=∑ m_ib_i=N, and σ∈ S_N. Going by the isomorphisms in (<ref>) and projecting Θ(σ) onto _i=1^sS^m_i(U_b_i^t_i^*) via ϖ_1⋯ϖ_s we arrive, under the natural identification of _i=1^sS^m_i(U_b_i^t_i^*) with S^m_1,…,m_s(W^*), at  the invariants  φ_σ defined above. Since S(W^*) is the direct sum  ⊕_r∈ℤ_≥ 0S^r(W^*), each summand of which in turn is a direct sum of S^m_1,…, m_s(W^*) as (m_1,…,m_s) varies over s-tuples of non-negative integers such that ∑_im_i=r, we get the required result.




 §.§ Restitution map and the polynomial ring on W_ø

For a G-graded vector space V let M:=V_ℂ  and M_o denote the graded component in M corresponding to the identity element o∈ G. In this section we define the polynomial ring over M_∘ and the `restitution map' from the space of multilinear maps on W to this polynomial ring. For this, let us consider the -module of all functions from M→, denoted by ℱ(M,). Let F^r:T^r(M^*)→ℱ(M,) be given by F^r(α)(v)=ev(α v⋯ v) where ev:T^r(M^*) T^r(M)→ is as defined in <ref>.



In other words, the following diagram commutes 
    
    _i=1^sS^m_i((U_b_i^t_i)^*)[d]^≅[r]^F^m_1… F^m_s   _i=1^sℱ((U_b_i^t_i)_0,)[dd]
    _i=1^s((U_b_i^t_i)^*)^ m_i@->>[ru]^ϖ_m_1…ϖ_m_s@->[rd]   S^(m_1,…, m_s)(⊕_i=1^s(U_b_i^t_i)^*)@^(->[d]
       S^r(W^*)[r]^F^r   ℱ(W_0,)


The symmetric group S_r acts on T^r(M) as described in <ref>. We then have the following analogue of  <cit.> 

    For σ=(i  i+1)∈ S_r, F^r(σ.α)(v)=ϵ(|α_i|,|α_i+1|)ϵ(|v|,|α_i|-|α_i+1|)F^r(α)(v). In particular, F^r(α)(v)=0 for α∈ I(M^*) and v∈ M_∘.
    


    The lemma follows from a simple calculations using the identity <ref>(1)  satisfied by ϵ.


Let ℱ(M_ø,) be the -module of all functions from M_ø→. 
 Let F^∙:=⊕_r=0^∞ F^r:T(M^*)→ℱ(M_ø,). As a consequence of the above lemma,  we note that this map factors through S(M^*). We continue to denote the induced map from S(M^*)→ℱ(M_ø,) also by F^∙, and its restriction to S^r(M^*) as F^r. 
 
 The next result allows us to define polynomials on M_o via the restitution map.  For the purpose of the proof, we fix a basis v_1,… ,v_k of V ordered such that |v_i|∈ G comes before |v_j|∈ G whenever i<j. (Here the elements of G are ordered as given in the beginning of <ref>). Let ϕ_1,…,ϕ_k be dual to the above basis. Denote the bases of M and M^* corresponding to the above bases also by the same notation.
 Let 𝒫^r(M_ø) be the image of F^r in ℱ(M_ø,). The space of polynomials on M_ø is the image of F^∙:=⊕_r=0^∞ F^r:S(M^*)→ℱ(M_ø,), denoted as 𝒫(M_ø). Then the following proposition is a G-graded analogue of (<cit.> ).
 

    The map F^∙ is an isomorphism from S(M^*)→𝒫(M_ø). 


     The surjectivity of the map F^∙: S(M^*)→𝒫(M_ø) is just a consequence of the definition of 𝒫(M_ø). To obtain the injectivity, we show the injectivity of each F^r. For this we consider an f∈ F^r. The symmetric algebra S(M^*) has a basis given by monomials in ϕ_i; the monomials whose total degree is r form a basis of S^r(M^*). Expressing f in terms of this basis, we have 
    f=∑_r_1+⋯+r_k=rλ_r_1,…,r_kϕ_1^r_1⋯ϕ_k^r_k.

     Let v=∑_ib_iλ_i∈ M_ø for some λ_i∈. Since v∈ M_ø, |b_i|=-|v_i| for all i. Evaluating F^r(f) at v, we get
    F^r(f)(v)=∑_r_1+⋯ +r_k=rλ_r_1,…,r_kλ_1^r_1⋯λ_k^r_k.
Choose N>0 such that λ_r_1,…,r_k∈(N) for all indices r_1,…,r_k such that ∑_i r_i=r. We now inductively choose λ_j for j=1,…,k such that λ_j∈ ((N_j)∖ (N_j-1))∩ X_-|b_j| for some suitable N_j>N_j-1; set N_0=N. For this choice of scalars λ_i, noting that the individual terms in the summation are distinct basis vectors of , we deduce that λ_r_1,…,r_k=0.    

The space 𝒫^r(M_ø) is called the space of polynomials of degree r on M_ø.  For f∈ S^r(M^*) and w∈ M_ø, we note that F^r(f)(w)=ev(f w^⊗ r) where w^⊗ r=w ⊗ w ⊗⋯⊗ w (r-times) and f∈ T^r(M^*) such that ϖ_r(f)=f; (recall, ϖ_r: T^r(M^*)→ S^r(M^*) is the natural quotient map).



 §.§ Polynomial invariants of W_ø

Let W be the mixed tensor space as defined in the begining of this section. Then the graded component of W corresponding to the identity element ø∈ G is W_ø=⊕_i=1^s(U_b_i^t_i)_ø. As described above, we obtain the restitution map  F^r:S^r(W^*)→ℱ(W_ø,).


For a tuple (m_1,…,m_s) such that ∑_i=1^sm_i=r, 
  let T_σ∈ (_i=1^s(U_b_i^t_i)^ m_i)^* be the linear map corresponding to σ∈ S_N obtained in <ref>. The graded picture invariants defined in <ref> are the images in S^m_1,…, m_s(W^*) of these linear maps T_σ, σ∈ S_N. Then we have,

    The graded picture invariants φ_σ∈ S^(m_1,…,m_s)(W^*) maps under restitution F^r to the element of ℱ(W_ø,) given by u↦ T_σ(u_1^⊗ m_1⋯ u_s^⊗ m_s) where u=(u_1…,u_s)∈ W_ø.


    As noted earlier in <ref>, the isomorphism  ι:T^r(W^*)→ (T^r(W))^* is obtained from the non-degenerate pairing T^r(W^*) T^r(W)→Λ_ϵ given by the evaluation map. Therefore, for any ϕ∈ T^r(W^*) we get a linear map ι(ϕ) on T^r(W) and the evaluation ι(ϕ)(w_1… w_r) is given by ev(ϕ w_1… w_r). In particular, F^r(ϕ)(u)=ι(ϕ)(u…u).
    
    The linear maps on U_b_i^t_i are regarded as linear maps on W via the projection p_i: W→ U_b_i^t_i. Denote the induced map on the dual spaces as p_i^*:U_b_i^t_i^*→ W^*.
    For a tuple (m_1,…,m_s) such that ∑_i=1^sm_i=r, T^m_1(U_b_1^t_1^*)… T^m_s(U_b_s^t_s^*) is a subspace of T^r(W^*), via _i p_i^*^ m_i. Similarly, _i=1^s(U_b_i^t_i)^ m_i is a direct summand of the tensor space T^r(W) so the projection   map : T^r(W)↠(_i=1^s(U_b_i^t_i)^ m_i) induces an injective map (_i=1^s(U_b_i^t_i)^ m_i)^*↪T^r(W)^*
given by  ψ↦ [(w_1… w_r)↦ψ∘(w_1… w_r)] for ψ∈(_i=1^s(U_b_i^t_i)^ m_i)^* and w_1… w_r∈ T^r(W). The non-degenerate pairing above restricted to the subspace T^m_1(U_b_1^t_1^*)… T^m_s(U_b_s^t_s^*) T^m_1(U_b_1^t_1)… T^m_s(U_b_s^t_s) via the above described maps, is a non-degenerate pairing and induces the isomorphism in (<ref>). For r=∑_i m_i, f_1⋯f_r∈_i=1^s(U_b_i^t_i^*)^ m_i and u∈ W_ø, we have 
    ι∘ (_ip_i^*^ m_i)(f_1⋯f_r)(u…u)=ev(p_1^*(f_1)⋯ p_s^*(f_r)u⋯u).
 
(Here the scaling factor involved is 1 since u∈ W_ø.)  
On the other hand, the isomorphism in (<ref>)  takes
 f_1⋯f_r to the linear map on T^r(W) given by w_1⋯ w_r ↦  ev(f_1⋯f_r pr(w_1⋯ w_r)). This linear map on  _i=1^s((U_b_i^t_i)^ m_i also is denoted by ι(f_1⋯f_r).  When w_i=u∈ W_ø for all i, the latter equals ev(f_1⋯f_r u_1^⊗ m_1⋯ u_s^⊗ m_s). 
 This in turn evaluates to the right hand side of the above equation.
 
 
 
 


In the construction of φ_σ in Theorem <ref>, let T_σ∈(_i=1^s((U_b_i^t_i)^ m_i)^* maps to φ_σ∈ S^r(W^*). Let φ_σ∈ T^m_1(U_b_1^t_1^*)… T^m_s(U_b_s^t_s^*) be such that ι(φ_σ)=T_σ and ϖ_r(_ip_i^*^ m_i(φ_σ))=φ_σ. By the above discussion, we have F^r(φ_σ)(u)=ev(_ip_i^*^ m_i(φ_σ)u…u)=ev (φ_σ u_1^⊗ m_1⋯ u_s^⊗ m_s) where u=(u_1…,u_s)∈ W_ø. As ι(φ_σ)=T_σ, the latter is T_σ( u_1^⊗ m_1⋯ u_s^⊗ m_s),  as required.

 




  §.§.§ Graded picture invariants in terms of traces

We now restrict to the case when W=(U_1^1)^s. Under the identification U_1^1≅ End_ (U), we define a product operation on U U^* as (vα). (wβ)=vα(w)β making the identification an isomorphism of G-graded algebras. Consider the trace function on U_1^1 given by tr(vα)=ϵ(|v|,|α|)α(v). With this notation, one may define the trace monomial tr_σ for a permutation σ∈ S_N as tr_σ(v_1ϕ_1⋯ v_Nϕ_N)=tr(v_i_1ϕ_i_1.v_i_2ϕ_i_2.⋯ .v_i_rϕ_i_r)tr(v_j_1ϕ_j_1.v_j_2ϕ_j_2.⋯ .v_j_tϕ_j_t)⋯

where σ^-1=(i_1  i_2 … i_r)(j_1 j_2⋯ j_t)⋯. This definition is dependent on the permutation σ and its cycle decomposition as well. However, if we restrict to v_1ϕ_1⋯ v_Nϕ_N coming from W_∘ then the definition is independent of the cycle decomposition of the permutation.  The proof of the following is analogous to that of Lemma 3.8 of <cit.>:
 (see <cit.>)
For a σ∈ S_N such that σ^-1=(i_1  i_2 … i_r)(j_1 j_2⋯ j_t)⋯∈ S_N, the _ϵ(U)-invariant map T_σ (as in Proposition <ref>) corresponds to the trace monomials tr_σ up to a scalar. Further, both the maps agree when restricted to the degree 0 part, ((U_1^1)_ø)^ N.



 The invariants in 𝒫(W_ø) for the induced action of _ϵ(U) such that the isomorphism is _ϵ(U)-equivariant are called the invariant polynomials on W_ø. We recover Theorem 5.6 of <cit.> as follows.
 


    The invariant polynomials for the simultaneous action of _ϵ(U) on ⊕_i=1^s(U_1^1)_o is spanned by the trace monomials _σ given by 
    _σ(A_1,…, A_s):= (A_f(i_1)⋯ A_f(i_r))(A_f(i_r+1)⋯ A_f(i_t))⋯
 where A_1,…, A_s∈⊕_i=1^sU_1^1, σ=(i_1 … i_r)(i_r+1… i_t)⋯∈ S_n and a map f:{1,…, n}→{1,…, s} as n varies over .


    The invariants in 𝒫(W_0) is the image of S(W^*)^_ϵ(U) which in turn is spanned by φ_σ, by Theorem <ref>. Proposition <ref> followed by Lemma <ref> then gives the required result.





Declaration of competing interest 

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Data availability

No data was used for the research described in the article.



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