The study of the center ζ(n) of the enveloping algebra U(gl(n)) of the general linear Lie algebra gl(n, C), and the study of the algebra Λ∗ (n) of shifted symmetric polynomials have noble and rather independent origins and motivations. The theme of central elements in U(gl(n)) is a standard one in the general theory of Lie algebras, see e.g. [18]. It is an old and actual one, since it is an offspring of the celebrated Capelli identity (see e.g. [11], [14], [21], [22], [36], [41], [42]), relates to its modern generalizations and applications (see e.g. [1], [24], [25], [29], [30], [31], [32], [40]) as well as to the theory of Yangians (see, e.g. [27], [28]).
Capelli bitableaux [S|T] and their variants (such as Young-Capelli bitableaux and double Young-Capelli bitableaux ) have been proved to be relevant in the study of the enveloping algebra U(gl(n)) = U(gl(n), C) of the general linear Lie algebra and of its center ζ(n). To be more specific, the superalgebraic method of virtual variables (see, e.g. [4], [5], [6], [7], [8], [9], [10]) allowed us to express remarkable classes of elements in U(gl(n)), namely, – the class of Capelli bitableaux [S|T] ∈ U(gl(n)) – the class of Young-Capelli bitableaux [S| |T| ] ∈ U(gl(n)) – the class of double Young-Capelli bitableaux [ |S| | |T| ] ∈ U(gl(n)) as the images - with respect to the Adgl(n)-adjoint equivariant Capelli devirtualization epimorphism - of simple expressions in an enveloping superalgebra U(gl(m_0|m_1 + n)).
Capelli (determinantal) bitableaux are generalizations of the famous column determinant element in U(gl(n)) introduced by Capelli in 1887 [11] (see, e.g. [9]). Young-Capelli bitableaux were introduced by the present authors several years ago [5], [6], [7] and might be regarded as generalizations of the Capelli column determinant elements in U(gl(n)) as well as of the Young symmetrizers of the classical representation theory of symmetric groups (see, e.g. [42]). Double Young-Capelli bitableaux play a crucial role in the study of the center ζ(n) of the enveloping algebra ([8], [10]).
In plain words, the Young-Capelli bitableau [S| |T| ] is obtained by adding a column symmetrization to the Capelli bitableau [S|T] and turn out to be a linear combination of Capelli bitableaux (see, e.g [10], Proposition 2.13). The double Young-Capelli bitableau [ |S| |T| ] is obtained by adding a further row skew-symmetrization to the Young-Capelli bitableau [S| |T| ] [10] turn out to be a linear combination of Young-Capelli bitableaux (see, e.g [10]) and, therefore, it is in turn a linear combination of Capelli bitableaux.
Capelli bitableaux are the preimages - with respect to the Koszul linear U(gl(n))- equivariant isomorphism K from the enveloping algebra U(gl(n)) to the polynomial algebra C[M_{n,n}] = Sym(gl(n)) ([26], [7], [9]) - of the classical determinantal bitableaux (see, e.g. [19], [17], [16], [20], [4]). Hence, they are ruled by the straightening laws and the set of standard Capelli bitableaux is a basis of U(gl(n)). The set of standard Young-Capelli bitableaux is another relavant basis of U(gl(n)) whose elements act in a nondegenerate orhogonal way on the set of standard right symmetrized bitableaux (the Gordan-Capelli basis of C[M_{n,n}]) and this fact leads to explicit complete decompositions of the semisimple U(gl(n))-module C[M_{n,n}] (see, e.g. [4], [5]). The linear combinations of double Young-Capelli bitableaux
(1) S_λ(n) = 1 H(λ) \times S [ |S| | |S| ] ∈ U(gl(n)), H(λ) the hook coefficient of the shape λ,
where the sum is extended to all row (strictly) increasing tableaux S on conjugate shape/partition of λ are central elements of U(gl(n)).
We called the elements S_λ(n) the Schur elements.
The Schur elements S_λ(n) are the preimages - with respect to the Harish-Chandra isomorphism - of the elements of the basis of shifted Schur polynomials S^∗_λ|n of the algebra Λ∗(n) of shifted symmetric polynomials [38], [33].
Hence, the Schur elements are the same [10] as the quantum immanants ([38], [31], [32], [33]) , first presented by Okounkov as traces of fusion matrices ([31], [32]) and, recently, described by the present authors as linear combinations (with explicit coefficients) of “diagonal” Capelli immanants [8]. Presentation (1) of Schur elements/quantum immanants doesn’t involve the irreducible characters of symmetric groups. Furthermore, it is better suited to the study of the eigenvalues on irreducible gl(n)−modules and of the duality in the algebra ζ(n), as well as to the study of the limit n → ∞, via the Olshanski decomposition (see, Olshanski [34], [35] and Molev [27], pp. 928 ff.)
We will consider a special class of Capelli bitableaux, namely the class of Capelli-Deruyts bitableaux.
These elements are Capelli bitableaux of the form K_λ = [Der∗ _λ |Der_λ] ∈ U(gl(n)),
where λ = (λ_1 ≥ λ_2 ≥ · · · ≥ λ_p) is a partition with λ1 ≤ n, and
– Derλ is the Deruyts tableaux of shape λ, that is the Young tableau of shape λ:
and
– Der∗ λ is the reverse Deruyts tableaux of shape λ, that is the Young tableau of shape λ.
Capelli-Deruyts bitableaux arise, in a natural way, as generalizations to arbitrary shapes
λ = (λ_1 ≥ λ_2 ≥ · · · ≥ λ_p) of the well-known Capelli column determinant elements
H(n)_ n ∈ U(gl(n)),
introduced by Alfredo Capelli [11] in the celebrated identities that bear his name (see, e.g. [11], [14], [21], [22], [36], [41], [42], [1], [24], [25], [29], [30], [31], [32], [40]).
Capelli-Deruyts bitableaux K_p n of rectangular shape λ = n p = ( p times n, n, n, · · · , n) are of particular interest since they are central elements in the enveloping algebra U(gl(n)).
The main problems we will study are the following:
- Given a gl(n, C)-highest weight vector v_µ of weight µ = (µ_1 ≥ µ_2 ≥ . . . ≥ µ_n), with µi ∈ N for every i = 1, 2, . . . , n, v_µ is an eigenvector of the action of the Capelli-Deruyts bitableau K_λ. Find an explicit form for the eigenvalues.
- Is it possible to expand the Capelli-Deruyts bitableau K_λ ∈ U(gl(n)) as a polynomial, with explicit coefficients, in the Capelli generators H_k^ (j) of the centers of the enveloping algebras U(gl(k)), k = 1, 2, . . . , n, j = 1, 2, . . . , k?
- Study further properties/applications of Capelli-Deruyts bitableaux.
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