• Corpus ID: 119582365

Shifted Schur Functions

  title={Shifted Schur Functions},
  author={Andrei Okounkov and Grigori Olshanski},
  journal={arXiv: Quantum Algebra},
The classical algebra $\Lambda$ of symmetric functions has a remarkable deformation $\Lambda^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur functions $s^*_\mu$, where $\mu$ 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… 

Quantum immanants, double Young-Capelli bitableaux and Schur shifted symmetric functions

In this paper are introduced two classes of elements in the enveloping algebra U ( gl ( n )) : the double Young-Capelli bitableaux [ S | T ] and the central Schur elements S λ ( n ) , that act in a

Central elements in U(gl(n)), shifted symmetric functions and the superalgebraic Capelli's method of virtual variables

In this work, we propose a new method for a unified study of some of the main features of the theory of the center of the enveloping algebra U(gl(n)) and of the algebra of shifted symmetric

The Pillowcase Distribution and Near-Involutions *

In the context of the Eskin-Okounkov approach to the calculation of the volumes of the different strata of the moduli space of quadratic differentials, the important ingredients are the pillowcase

Comultiplication Rules for the Double Schur Functions and Cauchy Identities

  • A. Molev
  • Mathematics
    Electron. J. Comb.
  • 2009
The dual Littlewood{Richardson coecien ts} provide a multiplication rule for the dual Schur functions and prove multiparameter analogues of the Cauchy identity.

On the Spectrum of the Derangement Graph

We derive several interesting formulae for the eigenvalues of the derangement graph and use them to settle affirmatively a conjecture of Ku regarding the least eigenvalue.

Littlewood-Richardson polynomials

Positivity results for Stanley's character polynomials

Interpolation Analogs of Schur Q-Functions

We introduce interpolation analogs of the Schur Q-functions — the multiparameter Schur Q-functions. We obtain for them several results: a combinatorial formula, generating functions for one-row and

Capelli elements in the classical universal enveloping algebras

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

Capelli identities for classical Lie algebras

Abstract. We extend the Capelli identity from the Lie algebra $\frak{gl}_N$ to the other classical Lie algebras $\frak{so}_N$ and $\frak{sp}_N$. We employ the theory of reductive dual pairs due to



Quantum Berezinian and the classical capelli identity

We study a superanalogue of the Yangian of the Lie algebra glmℂ. We apply our constructions to invariant theory.

The Capelli identity, the double commutant theorem, and multiplicity-free actions

0. The Capelli identity [Cal-3; W, p. 39] is one of the most celebrated and useful formulas of classical invariant theory [W; D; CL; Z]. The double commutant theorem [W, p. 91] is likewise a basic

The factorial Schur function

The application of the divided difference of a function to the inhomogeneous symmetric functions (factorial Schur functions) of Biedenharn and Louck is shown to lead to new relations and simplified

Symmetric functions and Hall polynomials

I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions

Schur functions: Theme and variations.

In this article we shall survey various generalizations, analogues and deformations of Schur functions — some old, some new — that have been proposed at various times. We shall present these as a


Shift Operators and Factorial Symmetric Functions

Remarks on classical invariant theory

A uniform formulation, applying to all classical groups simultaneously, of the First Fundamental Theory of Classical Invariant Theory is given in terms of the Weyl algebra. The formulation also

A New Tableau Representation for Supersymmetric Schur Functions

Abstract We give a new tableau definition for supersymmetric skew Schur functions, and obtain a number of properties of these functions as easy corollaries.

What Is Enumerative Combinatorics

The basic problem of enumerative combinatorics is that of counting the number of elements of a finite set. Usually are given an infinite class of finite sets S i where i ranges over some index set I