SymmetricFunctions

Monomial, elementary, complete homogeneous, power-sum, Schur and forgotten bases with fast transition matrices; several alphabets; Hall and Jack inner products; plethysm; Kostka, inverse Kostka, Littlewood–Richardson and Kronecker coefficients; skew Schur, Schur P and Q, Jack, Hall–Littlewood, Macdonald P/J and modified Macdonald H~ (Haglund’s convention), LLT, k-Schur, Lah and Petrie functions; the Delta and nabla operators.

Load with Needs["SymmetricFunctions`"].

Background on symmetricfunctions.com: Standard symmetric functions, Schur polynomials, Littlewood–Richardson coefficients, Plethysm, Jack polynomials, Hall–Littlewood polynomials, Macdonald P polynomials, Modified Macdonald polynomials, LLT polynomials.

Functions and symbols

AugmentedMonomialSymmetric

AugmentedMonomialSymmetric[lam, x] returns the augmented monomial symmetric function indexed by the integer vector lam in alphabet x. The alphabet x defaults to None.

BigSchurSymmetric

BigSchurSymmetric[lam, x] returns the big Schur symmetric function indexed by partition lam in alphabet x.

ChangeFunctionAlphabet

ChangeFunctionAlphabet[expr, to, from] changes the symmetric-function expressions in expr from alphabet from to alphabet to. The source alphabet defaults to None.

ClearSymmetricFunctionsCache

ClearSymmetricFunctionsCache[] removes all stored results of memoized symmetric functions, such as SkewSchurSymmetric, MacdonaldHSymmetric and KroneckerCoefficient. Transition matrices between the classical bases are kept.

CoefficientsSum

CoefficientsSum[expr] returns the sum of coefficients.

CompleteHSymbol

CompleteHSymbol[lam, x] represents the complete homogeneous symmetric-function basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

CompleteHSymmetric

CompleteHSymmetric[lam, x] returns the complete homogeneous symmetric function indexed by the partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

CylindricSchurSymmetric

CylindricSchurSymmetric[{lam, mu}, d, x] returns the cylindric Schur symmetric function of shape lam/mu and shift d in alphabet x. The shift d and alphabet x default to 0 and None.

DeltaOperator

DeltaOperator[f, g, q, t] applies the Garsia Delta operator Delta_f to g: it multiplies each modified Macdonald H~_mu by f[B_mu], where B_mu = sum over cells (r, c) of mu of q^(c-1) t^(r-1) (Haglund’s convention).

DeltaPrimOperator

DeltaPrimOperator[f, g, q, t] applies the primed Delta operator to symmetric functions f and g with parameters q and t.

ElementaryESymbol

ElementaryESymbol[lam, x] represents the elementary symmetric-function basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

ElementaryESymmetric

ElementaryESymmetric[lam, x] returns the elementary symmetric function indexed by the partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

ForgottenSymbol

ForgottenSymbol[lam, x] represents the forgotten symmetric-function basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

ForgottenSymmetric

ForgottenSymmetric[lam, x] returns the forgotten symmetric function indexed by the partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

FunctionAlphabets

FunctionAlphabets[expr] returns the list of alphabets occurring in a symmetric-function expression.

HallInnerProduct

HallInnerProduct[f, g] returns the Hall inner product of symmetric functions f and g.
HallInnerProduct[f, g, {q, t}, x] computes the q,t inner product in alphabet x, which defaults to None.

HallLittlewoodMSymmetric

HallLittlewoodMSymmetric[lam, q, x] returns the Hall-Littlewood M symmetric function indexed by partition lam with parameter q. The alphabet x defaults to None.

Background: Hall–Littlewood polynomials

HallLittlewoodPSymbol

HallLittlewoodPSymbol[lam, x] represents a Hall-Littlewood P basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Hall–Littlewood polynomials

HallLittlewoodPSymmetric

HallLittlewoodPSymmetric[lam, t, x] returns the Hall-Littlewood P symmetric function indexed by partition lam with parameter t. The alphabet x defaults to None.

Background: Hall–Littlewood polynomials

HallLittlewoodTSymmetric

HallLittlewoodTSymmetric[lam, q, x] returns the transformed Hall-Littlewood symmetric function indexed by partition lam with parameter q. The alphabet x defaults to None.

Background: Hall–Littlewood polynomials

HookFilterMap

HookFilterMap[f, t, [x]] applies the hook filter map to a symmetric function f, returning a polynomial in t. The map sends SchurSymbol[lam] to t*(t-1)^(lam[[1]]-1) if lam is a hook shape (first part arbitrary, all remaining parts equal to 1), and to 0 otherwise.

InternalProduct

InternalProduct[f, g, x] computes the internal (Kronecker) product of symmetric functions f and g in alphabet x. The alphabet x defaults to None.

JackInnerProduct

JackInnerProduct[f, g, a] returns the Jack inner product of symmetric functions f and g with parameter a, which defaults to 1.
JackInnerProduct[f, g, a, x] computes the inner product in alphabet x, which defaults to None.

Background: Jack polynomials

JackJSymmetric

JackJSymmetric[lam, a, x] returns the Jack J symmetric function indexed by partition lam with parameter a. The alphabet x defaults to None.

Background: Jack polynomials

JackLowerHook

JackLowerHook[mu, a, {r, c}] returns the Jack lower hook at box {r, c} of partition mu. JackLowerHook[mu, a] returns the product of lower hooks over mu. The parameter a defaults to 1.

Background: Jack polynomials

JackPSymmetric

JackPSymmetric[lam, a, x] returns the Jack P symmetric function indexed by partition lam with parameter a. The alphabet x defaults to None.

Background: Jack polynomials

JackUpperHook

JackUpperHook[mu, a, {r, c}] returns the Jack upper hook at box {r, c} of partition mu. JackUpperHook[mu, a] returns the product of upper hooks over mu. The parameter a defaults to 1.

Background: Jack polynomials

KroneckerCoefficient

KroneckerCoefficient[lam,mu,nu] returns the Kronecker coefficient.

kSchurSymmetric

kSchurSymmetric[mu, k, t, x] returns the k-Schur function indexed by partition mu, with mu1 <= k. The parameter t defaults to 1 and the alphabet x defaults to None.

LahSymmetricFunction

LahSymmetricFunction[n, k, x] returns the Lah symmetric function with parameters n and k in alphabet x. The alphabet x defaults to None.

Background: Lah symmetric functions

LahSymmetricFunctionNegative

LahSymmetricFunctionNegative[n, k, x] returns the negative Lah symmetric function with parameters n and k in alphabet x. The alphabet x defaults to None.

Background: Lah symmetric functions

LLTSymmetric

LLTSymmetric[nu, q, x] returns the LLT symmetric function associated with the tuple nu of shapes with parameter q. The alphabet x defaults to None.

Background: LLT polynomials

LoadBasisMatrices

LoadBasisMatrices[] loads previously stored transition matrices.
To store transition matrices, use PrecomputeBasisMatrices[]

LRCoefficient

LRCoefficient[lam, mu, nu] gives the Littlewood-Richardson coefficient of Schur function S_nu in S_lam*S_mu. The arguments are partitions.

Background: Littlewood–Richardson coefficients

LRExpand

LRExpand[expr, x] expands products and powers of Schur symmetric functions in expr using the Littlewood-Richardson rule. The alphabet x defaults to None.

Background: Littlewood–Richardson coefficients

LyndonSymmetric

LyndonSymmetric[lam, x] returns the Lyndon symmetric function indexed by lam. The alphabet x defaults to None.

MacdonaldHSymbol

MacdonaldHSymbol[lam, x] represents a modified Macdonald H basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Modified Macdonald polynomials

MacdonaldHSymmetric

MacdonaldHSymmetric[lam, q, t, x] returns the modified Macdonald symmetric function H~_lam(x; q, t) in Haglund’s convention, so that H~_{2} = s_2 + q s_11, H~_{1,1} = s_2 + t s_11, and H~_mu = t^n(mu) J_mu[X/(1 - 1/t); q, 1/t]. The alphabet x defaults to None.
MacdonaldHSymmetric[{lam, mu}, q, t, x] gives the skew version.

Background: Modified Macdonald polynomials

MacdonaldJSymmetric

MacdonaldJSymmetric[lam, q, t, x] returns the integral-form Macdonald J symmetric function indexed by partition lam with parameters q and t. The alphabet x defaults to None.

Background: Macdonald P polynomials

MacdonaldPSymmetric

MacdonaldPSymmetric[lam, q, t, x] returns the Macdonald P symmetric function indexed by partition lam with parameters q and t. The alphabet x defaults to None.

Background: Macdonald P polynomials

MExpand

MExpand[expr] expands products and powers of monomial symmetric functions in expr.

MonomialSymbol

MonomialSymbol[lam, x] represents the monomial symmetric-function basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

MonomialSymmetric

MonomialSymmetric[lam, x] returns the monomial symmetric function indexed by the partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

NablaOperator

NablaOperator[g, q, t] applies the nabla operator to symmetric function g with parameters q and t.

OmegaInvolution

OmegaInvolution[expr, x] applies the omega involution to expr. The alphabet x defaults to None. Caution: It only acts on the common symmetric functions.

OrthogonalSchurSymmetric

OrthogonalSchurSymmetric[lam, x] returns the orthogonal Schur symmetric function indexed by partition lam in alphabet x.

PetrieSymmetric

PetrieSymmetric[k, m, x] returns the degree-m part of the kth Petrie symmetric function in alphabet x. The alphabet x defaults to None.

Background: Petrie symmetric functions

Plethysm

Plethysm[f, g, xx] computes the plethysm of symmetric functions f and g, acting on all alphabets in g. The alphabet xx of f defaults to None.

Background: Plethysm

PolynomialToSymmetricFunction

PolynomialToSymmetricFunction[poly, x, basisSymbol, n] converts a symmetric polynomial in x[1], …, x[n] to the requested basis (MonomialSymbol, SchurSymbol, ElementaryESymbol, CompleteHSymbol, PowerSumSymbol or ForgottenSymbol). The basis defaults to SchurSymbol and n is inferred from the variables present when omitted. Only partitions with at most n parts can occur.

PositiveCoefficientsQ

PositiveCoefficientsQ[expr, basis, extraSymbols] returns True if all coefficients of basis terms in expr are nonnegative numbers. The basis defaults to MonomialSymbol and extraSymbols defaults to {}.

PowerSumSymbol

PowerSumSymbol[lam, x] represents the power-sum symmetric-function basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

PowerSumSymmetric

PowerSumSymmetric[lam, x] returns the power-sum symmetric function indexed by the partition lam in alphabet x. The alphabet x defaults to None.

Background: Standard symmetric functions

PrecomputeBasisMatrices

PrecomputeBasisMatrices[Dimensions->n] computes the transition matrices for
all the most common elementary symmetric functions indexed by partitions of size at most n.
These matrices are stored as a local object (in your home folder).
Use LoadBasisMatrices[] to load these matrices.
Note that time and space used grows rapidly with n!

PrincipalSpecialization

PrincipalSpecialization[poly, q, k, x] gives the principal specialization of poly. The parameter k gives the number of variables and defaults to Infinity; the alphabet x defaults to None.

SchursPSymmetric

SchursPSymmetric[lam, x] returns the Schur P symmetric function indexed by strict partition lam. The alphabet x defaults to None.

Background: Schur polynomials

SchursQSymmetric

SchursQSymmetric[lam]

Background: Schur polynomials

SchurSymbol

SchurSymbol[lam, x] represents the Schur symmetric-function basis element indexed by partition lam in alphabet x. The alphabet x defaults to None.

Background: Schur polynomials

SchurSymmetric

SchurSymmetric[lam, x] returns the Schur symmetric function indexed by the partition lam in alphabet x. The alphabet x defaults to None.

Background: Schur polynomials

SkewKostkaCoefficient

SkewKostkaCoefficient[lam, mu, w] returns the skew Kostka coefficient for skew shape lam/mu and weight w. The weight w is sorted to a partition.

Background: Kostka coefficients, Kostka–Foulkes polynomials and charge

SkewMacdonaldESymmetric

SkewMacdonaldESymmetric[{lam, mu}, q, x] returns the skew Macdonald E symmetric function of shape lam/mu with parameter q in alphabet x.

SkewSchurSymmetric

SkewSchurSymmetric[lam, x] returns the Schur symmetric function of partition lam in alphabet x. The alphabet x defaults to None.
SkewSchurSymmetric[{lam, mu}, x] returns the skew Schur symmetric function of shape lam/mu in alphabet x. The alphabet x defaults to None.

Background: Schur polynomials

SnModuleCharacters

SnModuleCharacters[polynomialBasis, x] takes a list of polynomials in x[1]…x[n] and returns the Sn-character under the action where Sn acts on the variable indices

SymFuncTransMat

SymFuncTransMat[fromBasis, toBasis, d] returns the transition matrix
in degree d. Here, fromBasis and toBasis are functions which given a partition, returns the monomial expansion.

SymmetricFunctionDegree

SymmetricFunctionDegree[expr] returns the degree of the symmetric function.

SymmetricFunctionToPolynomial

SymmetricFunctionToPolynomial[expr, x, n, y] expresses expr as a polynomial in n variables x[1] through x[n], using alphabet y in expr. The alphabet x and variable count n default to None and the degree of expr, respectively; y defaults to None.

SymmetricMonomialList

SymmetricMonomialList[expr] returns the list of symmetric-function monomials occurring in expr.

SymplecticSchurSymmetric

SymplecticSchurSymmetric[lam, x] returns the symplectic Schur symmetric function indexed by partition lam in alphabet x.

ToCompleteHBasis

ToCompleteHBasis[poly, x] converts poly to the complete homogeneous basis. The alphabet x defaults to None.

ToElementaryEBasis

ToElementaryEBasis[poly, x] converts poly to the elementary basis. The alphabet x defaults to None.

ToHallLittlewoodPBasis

ToHallLittlewoodPBasis[poly, t, x, mh] converts poly to the Hall-Littlewood P basis with parameter t. The alphabet x defaults to None and the basis symbol mh defaults to HallLittlewoodPSymbol.

ToMacdonaldHBasis

ToMacdonaldHBasis[poly, q, t, x, mh] converts poly to the modified Macdonald H basis with parameters q and t. The alphabet x defaults to None and the basis symbol mh defaults to MacdonaldHSymbol.

ToMonomialBasis

ToMonomialBasis[poly, x] converts poly to the monomial basis. The alphabet x defaults to None.

toOtherSymmetricBasis

toOtherSymmetricBasis[{basisFunc, basisSymbol}, poly, x] converts poly to the basis specified by basisFunc and basisSymbol. The alphabet x defaults to None; a list of alphabets is also accepted.

ToPowerSumBasis

ToPowerSumBasis[poly, x] converts poly to the power-sum basis. The alphabet x defaults to None.

ToPowerSumZBasis

ToPowerSumZBasis[poly, x] converts poly to the power-sum basis with each power-sum term multiplied by its z-coefficient. The alphabet x defaults to None.

ToSchurBasis

ToSchurBasis[poly, x] converts poly to the Schur basis. The alphabet x defaults to None.