# Shifted Schur Functions

@article{Okounkov1996ShiftedSF, title={Shifted Schur Functions}, author={Andrei Okounkov and Grigori Olshanski}, journal={arXiv: Quantum Algebra}, year={1996} }

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…

## 248 Citations

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## References

SHOWING 1-10 OF 16 REFERENCES

### Quantum Berezinian and the classical capelli identity

- Mathematics
- 1991

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

- Mathematics
- 1991

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

- Mathematics
- 1993

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

- Mathematics
- 1979

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.

- Mathematics
- 1992

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

- MathematicsJ. Comb. Theory, Ser. A
- 1995

### Remarks on classical invariant theory

- Mathematics
- 1989

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

- Mathematics
- 1994

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

- Mathematics
- 1986

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…