Related papers: General Capelli-type identities
We prove Capelli type identities which involve the whole universal enveloping algebra $U(gl(n))$ and matrix elements of irreducible representations of the symmetric group. These identities generalize higher Capelli identities for the center…
We propose a universal matrix Capelli identity and explain how to derive Capelli identities for all quantum immanants in the Reflection Equation algebra and in the universal enveloping algebra U(gl_(M|N)).
We consider remarkable central elements of the universal enveloping algebra of the general linear algebra which we call quantum immanants. We express them in terms of generators $E_{ij}$ and as differential operators on the space of…
We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant…
We study the image of the universal $R$-matrix for the Yangian $Y(gl_N)$ with respect to the evaluation homomorphism of $Y(gl_N)$ to the enveloping algebra $U(gl_N)$. We use the fusion procedure as defined by I. Cherednik. As a corollary we…
Capelli identities are shown to facilitate the construction of representations of various Heisenberg algebras that arise in many-particle quantum mechanics and the construction of holomorphic representations of many Lie algebras by Vector…
Inspired by the Capelli identities for group determinants obtained by T\^oru Umeda, we give a basis of the center of the group algebra of any finite group by using Capelli identities for irreducible representations. The Capelli identities…
We get several identities of differential operators in determinantal form. These identities are non-commutative versions of the formula of Cauchy-Binet or Laplace expansions of determinants, and if we take principal symbols, they are…
We propose a new approach to a unified study of determinants, permanents, immanants, (determinantal) bitableaux and symmetrized bitableaux in the polynomial algebra $C[M_{n, n}]$ as well as of their Lie analogues in the enveloping algebra…
We extend the Capelli identities (1890) from the Lie algebra $gl_N$ to the other two classical Lie algebras $so_N$ and $sp_N$. We employ the theory of reductive dual pairs due to Howe. Our technique comes from the representation theory of…
By using the notion of a quantum double we introduce analogs of partial derivatives on a Reflection Equation algebra, associated with a Hecke symmetry of GL(N) type. We construct the matrix L=MD, where M is the generating matrix of the…
We show how the use of superalgebraic methods sheds new light on some classical themes of representation theory and it leads to significant simplifications of traditional proofs.
The Capelli identities claim $det(A)det(B) = det(AB+correction)$ for certain matrices with noncommutative entries. They have applications in representation theory and integrable systems. We propose new examples of these identities,…
We apply the technique of affine Hecke algebras to the invariant theory of the "queer" Lie superalgebra $q_N$. We give explicit formulas for the elements of a distinguished basis in the centre of $U(q_N)$, determined by "vanishing"…
We present and study two families of polynomials with coefficients in the center of the universal enveloping algebra. These polynomials are analogues of a determinant and a characteristic polynomial of a certain non-commutative matrix,…
We apply the recently introduced idempotents for the Sergeev superalgebra to construct quantum immanants for the queer Lie superalgebra ${\mathfrak q}_N$ as central elements of its universal enveloping algebra. We prove universal odd and…
This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent, and it is easy to calculate their eigenvalues on…
A quantum Capelli identity is given on the multiparameter quantum general linear group based on the $(p_{ij}, u)$-condition. The multiparameter quantum Pfaffian of the $(p_{ij}, u)$-quantum group is also introduced and the transformation…
We introduce and study quantum Capelli operators inside newly constructed quantum Weyl algebras associated to three families of symmetric pairs. Both the center of a particular quantized enveloping algebra and the Capelli operators act…
For any complex classical group $G=O_N,Sp_N$ consider the ring $Z(g)$ of $G$-invariants in the corresponding enveloping algebra $U(g)$. Let $u$ be a complex parameter. For each $n=0,1,2,...$ and every partition $\nu$ of $n$ into at most $N$…