arXiv subjects

Andrei Okounkov

Publications and source records attributed to Andrei Okounkov.

76 records · Page 5Linked to original sources

arXiv1996

Shifted Schur functions II. Binomial formula for characters of classical groups and applications

Let G be any of the complex classical groups GL(n), SO(2n+1), Sp(2n), O(2n), let g denote the Lie algebra of G, and let Z(g) denote the subalgebra of G-invariants in the universal enveloping algebra U(g). We derive a Taylor-type expansion for finite-dimensional characters of G (binomial formula) and use it to specify a distinguished linear basis in Z(g). The eigenvalues of the basis elements in highest weight g-modules are certain shifted (or factorial) analogs of Schur functions. We also study an associated homogeneous basis in I(g), the subalgebra of G-invariants in the symmetric algebra S(g). Finally, we show that the both bases are related by a G-equivariant linear isomorphism σ: I(g)\to Z(g), called the special symmetrization.

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arXiv1996

Young Basis, Wick Formula, and Higher Capelli 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 of $U(gl(n))$ introduced in the author's paper \cite{Ok}. The main role in the proof play the Jucis-Murphy elements.

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arXiv1996

Shifted Schur Functions

The classical algebra $Λ$ of symmetric functions has a remarkable deformation $Λ^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s^*_μ$, where $μ$ ranges over the set of all partitions. The main significance of the shifted Schur functions is that they determine a natural basis in $Z(\frak{gl}(n))$, the center of the universal enveloping algebra $U(\frak{gl}(n))$, $n=1,2,\ldots$. The functions $s^*_μ$ are closely related to the factorial Schur functions introduced by Biedenharn and Louck and further studied by Macdonald and other authors. A part of our results about the functions $s^*_μ$ has natural classical analogues (combinatorial presentation, generating series, Jacobi--Trudi identity, Pieri formula). Other results are of different nature (connection with the binomial formula for characters of $GL(n)$, an explicit expression for the dimension of skew shapes $λ/μ$, Capelli--type identities, a characterization of the functions $s^*_μ$ by their vanishing properties, `coherence property', special symmetrization map $S(\frak{gl}(n))\to U(\frak{gl}(n))$. The main application that we have in mind is the asymptotic character theory for the unitary groups $U(n)$ and symmetric groups $S(n)$ as $n\to\infty$. The results of this paper were used in \cite{Ok1--3}.

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arXiv1996

Quantum Immanants and Higher Capelli Identities

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 matrices. These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants.

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