ShiftedSymmetricFunctions

Okounkov–Olshanski shifted Schur and Jack polynomials, normalized characters, and Stanley–Feray–Sniady multirectangular character polynomials.

Load with Needs["ShiftedSymmetricFunctions`"].

Background on symmetricfunctions.com: Shifted Schur polynomials, Shifted Jack polynomials.

Functions and symbols

NormalizedCharacter

NormalizedCharacter[mu,lam] returns n(n-1)…(n-|mu|+1) chi^lam(mu 1^(n-|mu|))/dim(lam), for |mu|<=n=|lam|, and 0 otherwise.

ShiftedJackJEvaluate

ShiftedJackJEvaluate[mu,lam,a] evaluates J*_mu at the partition lam with Jack parameter a.

Background: Shifted Jack polynomials

ShiftedJackJPolynomial

ShiftedJackJPolynomial[mu,n,x,a] returns the hook-normalized shifted Jack J polynomial in x[1],…,x[n].

Background: Shifted Jack polynomials

ShiftedJackPEvaluate

ShiftedJackPEvaluate[mu,lam,a] evaluates P*_mu at the partition lam with Jack parameter a.

Background: Shifted Jack polynomials

ShiftedJackPPolynomial

ShiftedJackPPolynomial[mu,n,x,a] returns the shifted Jack P polynomial P*_mu in x[1],…,x[n] with Jack parameter a.

Background: Shifted Jack polynomials

ShiftedJackPSymmetric

ShiftedJackPSymmetric[mu,a,x] returns the legacy shifted Jack P symmetric function obtained from the Okounkov–Olshanski tableau formula. The alphabet x defaults to None.

Background: Shifted Jack polynomials

ShiftedSchurEvaluate

ShiftedSchurEvaluate[mu,lam] evaluates s*_mu at the partition lam using the determinant formula.

Background: Shifted Schur polynomials

ShiftedSchurPolynomial

ShiftedSchurPolynomial[mu,n,x] returns the Okounkov–Olshanski shifted Schur polynomial s*_mu in x[1],…,x[n].

Background: Shifted Schur polynomials

StanleyCharacterPolynomial

StanleyCharacterPolynomial[mu,p,q,d] returns the Stanley–Feray–Sniady normalized character polynomial in multirectangular coordinates p[1],…,p[d] and q[1],…,q[d].