Hall–Littlewood polynomials $P_\lambda(x_1,\dots,x_n;t)$

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Polynomials

$n$

$\lambda$

$P_\lambda(x_1,\dots,x_n;t)$

x1^2

x1^3

x1^4

x1 + x2

-x1*x2*t + x1^2 + x1*x2 + x2^2

1, 1:

x1*x2

-x1^2*x2*t - x1*x2^2*t + x1^3 + x1^2*x2 + x1*x2^2 + x2^3

2, 1:

x1^2*x2 + x1*x2^2

-x1^3*x2*t - x1^2*x2^2*t - x1*x2^3*t + x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4

3, 1:

-x1^2*x2^2*t + x1^3*x2 + x1^2*x2^2 + x1*x2^3

2, 2:

x1^2*x2^2

x1 + x2 + x3

-x1*x2*t - x1*x3*t - x2*x3*t + x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2

1, 1:

x1*x2 + x1*x3 + x2*x3

x1*x2*x3*t^2 - x1^2*x2*t - x1*x2^2*t - x1^2*x3*t - 2*x1*x2*x3*t - x2^2*x3*t - x1*x3^2*t - x2*x3^2*t + x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3

2, 1:

-x1*x2*x3*t^2 - x1*x2*x3*t + x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2

1, 1, 1:

x1*x2*x3

x1^2*x2*x3*t^2 + x1*x2^2*x3*t^2 + x1*x2*x3^2*t^2 - x1^3*x2*t - x1^2*x2^2*t - x1*x2^3*t - x1^3*x3*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x2^3*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t - x1*x3^3*t - x2*x3^3*t + x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4

3, 1:

-x1^2*x2^2*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3

2, 2:

-x1^2*x2*x3*t - x1*x2^2*x3*t - x1*x2*x3^2*t + x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2

2, 1, 1:

x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2

x1 + x2 + x3 + x4

-x1*x2*t - x1*x3*t - x2*x3*t - x1*x4*t - x2*x4*t - x3*x4*t + x1^2 + x1*x2 + x2^2 + x1*x3 + x2*x3 + x3^2 + x1*x4 + x2*x4 + x3*x4 + x4^2

1, 1:

x1*x2 + x1*x3 + x2*x3 + x1*x4 + x2*x4 + x3*x4

x1*x2*x3*t^2 + x1*x2*x4*t^2 + x1*x3*x4*t^2 + x2*x3*x4*t^2 - x1^2*x2*t - x1*x2^2*t - x1^2*x3*t - 2*x1*x2*x3*t - x2^2*x3*t - x1*x3^2*t - x2*x3^2*t - x1^2*x4*t - 2*x1*x2*x4*t - x2^2*x4*t - 2*x1*x3*x4*t - 2*x2*x3*x4*t - x3^2*x4*t - x1*x4^2*t - x2*x4^2*t - x3*x4^2*t + x1^3 + x1^2*x2 + x1*x2^2 + x2^3 + x1^2*x3 + x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x3^3 + x1^2*x4 + x1*x2*x4 + x2^2*x4 + x1*x3*x4 + x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2 + x4^3

2, 1:

-x1*x2*x3*t^2 - x1*x2*x4*t^2 - x1*x3*x4*t^2 - x2*x3*x4*t^2 - x1*x2*x3*t - x1*x2*x4*t - x1*x3*x4*t - x2*x3*x4*t + x1^2*x2 + x1*x2^2 + x1^2*x3 + 2*x1*x2*x3 + x2^2*x3 + x1*x3^2 + x2*x3^2 + x1^2*x4 + 2*x1*x2*x4 + x2^2*x4 + 2*x1*x3*x4 + 2*x2*x3*x4 + x3^2*x4 + x1*x4^2 + x2*x4^2 + x3*x4^2

1, 1, 1:

x1*x2*x3 + x1*x2*x4 + x1*x3*x4 + x2*x3*x4

-x1*x2*x3*x4*t^3 + x1^2*x2*x3*t^2 + x1*x2^2*x3*t^2 + x1*x2*x3^2*t^2 + x1^2*x2*x4*t^2 + x1*x2^2*x4*t^2 + x1^2*x3*x4*t^2 + 3*x1*x2*x3*x4*t^2 + x2^2*x3*x4*t^2 + x1*x3^2*x4*t^2 + x2*x3^2*x4*t^2 + x1*x2*x4^2*t^2 + x1*x3*x4^2*t^2 + x2*x3*x4^2*t^2 - x1^3*x2*t - x1^2*x2^2*t - x1*x2^3*t - x1^3*x3*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x2^3*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t - x1*x3^3*t - x2*x3^3*t - x1^3*x4*t - 2*x1^2*x2*x4*t - 2*x1*x2^2*x4*t - x2^3*x4*t - 2*x1^2*x3*x4*t - 3*x1*x2*x3*x4*t - 2*x2^2*x3*x4*t - 2*x1*x3^2*x4*t - 2*x2*x3^2*x4*t - x3^3*x4*t - x1^2*x4^2*t - 2*x1*x2*x4^2*t - x2^2*x4^2*t - 2*x1*x3*x4^2*t - 2*x2*x3*x4^2*t - x3^2*x4^2*t - x1*x4^3*t - x2*x4^3*t - x3*x4^3*t + x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4 + x1^3*x3 + x1^2*x2*x3 + x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x3^4 + x1^3*x4 + x1^2*x2*x4 + x1*x2^2*x4 + x2^3*x4 + x1^2*x3*x4 + x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3 + x4^4

3, 1:

x1*x2*x3*x4*t^3 + x1*x2*x3*x4*t^2 - x1^2*x2^2*t - 2*x1^2*x2*x3*t - 2*x1*x2^2*x3*t - x1^2*x3^2*t - 2*x1*x2*x3^2*t - x2^2*x3^2*t - 2*x1^2*x2*x4*t - 2*x1*x2^2*x4*t - 2*x1^2*x3*x4*t - 5*x1*x2*x3*x4*t - 2*x2^2*x3*x4*t - 2*x1*x3^2*x4*t - 2*x2*x3^2*x4*t - x1^2*x4^2*t - 2*x1*x2*x4^2*t - x2^2*x4^2*t - 2*x1*x3*x4^2*t - 2*x2*x3*x4^2*t - x3^2*x4^2*t + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x1^3*x3 + 2*x1^2*x2*x3 + 2*x1*x2^2*x3 + x2^3*x3 + x1^2*x3^2 + 2*x1*x2*x3^2 + x2^2*x3^2 + x1*x3^3 + x2*x3^3 + x1^3*x4 + 2*x1^2*x2*x4 + 2*x1*x2^2*x4 + x2^3*x4 + 2*x1^2*x3*x4 + 3*x1*x2*x3*x4 + 2*x2^2*x3*x4 + 2*x1*x3^2*x4 + 2*x2*x3^2*x4 + x3^3*x4 + x1^2*x4^2 + 2*x1*x2*x4^2 + x2^2*x4^2 + 2*x1*x3*x4^2 + 2*x2*x3*x4^2 + x3^2*x4^2 + x1*x4^3 + x2*x4^3 + x3*x4^3

2, 2:

x1*x2*x3*x4*t^3 - x1^2*x2*x3*t - x1*x2^2*x3*t - x1*x2*x3^2*t - x1^2*x2*x4*t - x1*x2^2*x4*t - x1^2*x3*x4*t - 3*x1*x2*x3*x4*t - x2^2*x3*x4*t - x1*x3^2*x4*t - x2*x3^2*x4*t - x1*x2*x4^2*t - x1*x3*x4^2*t - x2*x3*x4^2*t + x1^2*x2^2 + x1^2*x2*x3 + x1*x2^2*x3 + x1^2*x3^2 + x1*x2*x3^2 + x2^2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + 2*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1^2*x4^2 + x1*x2*x4^2 + x2^2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2 + x3^2*x4^2

2, 1, 1:

-x1*x2*x3*x4*t^3 - x1*x2*x3*x4*t^2 - x1*x2*x3*x4*t + x1^2*x2*x3 + x1*x2^2*x3 + x1*x2*x3^2 + x1^2*x2*x4 + x1*x2^2*x4 + x1^2*x3*x4 + 3*x1*x2*x3*x4 + x2^2*x3*x4 + x1*x3^2*x4 + x2*x3^2*x4 + x1*x2*x4^2 + x1*x3*x4^2 + x2*x3*x4^2

1, 1, 1, 1:

x1*x2*x3*x4

Definition

The Hall–Littlewood polynomial $P_\lambda(x_1,\dots,x_n;t)$ [2] is Macdonald's $P$-normalised Hall–Littlewood symmetric polynomial [1], given by (1) for a partition $\lambda$ with at most $n$ parts.

Parameters

$n$

— number of variables (a positive integer)

$\lambda$

— shape (a partition with at most $n$ parts)

Formulas

(1)

$P_\lambda(x_1,\dots,x_n;t)=v_\lambda(t)^{-1}\sum_{w\in S_n} w\left(x_1^{\lambda_1}\cdots x_n^{\lambda_n} \prod_{1\leq i<j\leq n}\frac{x_i-tx_j}{x_i-x_j}\right)$, where $v_\lambda(t)=\prod_{r\geq0}\prod_{j=1}^{m_r(\lambda)}(1-t^j)/(1-t)$, $m_r(\lambda)$ is the number of parts of $\lambda$ equal to $r$, $\ell(\lambda)$ is the number of nonzero parts, and $m_0(\lambda)=n-\ell(\lambda)$ counts the zeros added by padding.

(2)

$P_\lambda(x;0)=s_\lambda(x)$, the Schur polynomial.

(3)

$P_\lambda(x;1)=m_\lambda(x)$, the monomial symmetric polynomial.

(4)

$s_\lambda(x)=\sum_\mu K_{\lambda\mu}(t)P_\mu(x;t)$, where the sum is over partitions $\mu$ of $|\lambda|$ with at most $n$ parts and $K_{\lambda\mu}(t)$ is the Kostka-Foulkes polynomial.

Comments

(5)

The partition is written as its parts, largest first, separated by commas: $2,1,1$ is the partition $(2,1,1)$ of $4$.

(6)

$t$ is an indeterminate, and $\lambda$ is padded with zeros to length $n$ in (1). In this $P$ normalisation, the coefficient of the monomial symmetric polynomial $m_\lambda$ is $1$, and the other monomial terms are indexed by partitions $\mu$ lower than $\lambda$ in dominance order. The $Q_\lambda$ normalisation multiplies $P_\lambda$ by a factor depending on $\lambda$ and $t$. The transformed Hall–Littlewood polynomial $Q'_\lambda(x;t)=Q_\lambda[x/(1-t);t]$ is not a scalar multiple of $P_\lambda$ in general.

Programs

(P1)

Sage

from sage.all import QQ, PolynomialRing, SymmetricFunctions

T = PolynomialRing(QQ, 't')
P = SymmetricFunctions(T.fraction_field()).hall_littlewood().P()
grouped = P([3, 2]).expand(4, alphabet=['x1', 'x2', 'x3', 'x4'])

# Sage groups coefficients in t; this writes the same polynomial
# expanded over QQ[x1, x2, x3, x4, t], as the table does.
R = PolynomialRing(QQ, ['x1', 'x2', 'x3', 'x4', 't'])
print(R(str(grouped)))

References

[1]

I. G. Macdonald, Symmetric Functions and Hall Polynomials, second ed., The Clarendon Press, Oxford University Press, New York, 1995. (MR)

Links

[2]

Wikipedia: Hall-Littlewood polynomials

[3]

SageMath: Hall-Littlewood symmetric functions

Similar tables

Schur polynomials — are the $t=0$ specialization

monomial symmetric polynomials — are the $t=1$ specialization

Kostka-Foulkes polynomials — are the transition coefficients from Schur polynomials to this Hall–Littlewood $P$ basis

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every $P_\lambda$ with $1\leq n\leq4$, $1\leq|\lambda|\leq4$, and $\ell(\lambda)\leq n$; extending the same rectangle to $n=5$ would make $P_{(4)}$ 2493 characters long, while extending to $|\lambda|=5$ at $n=4$ would make $P_{(5)}$ 2114 characters long)

How they were obtained:

Every entry is an exact polynomial with integer coefficients. The generator computes Macdonald's symmetrization formula in the fraction field of $\mathbb{Z}[x_1,\dots,x_n,t]$ and refuses an entry unless the denominator cancels. Before the values were written, every entry was compared with Sage's Hall-Littlewood $P$ basis [3].

The same values were checked at $t=0$ against the Schur polynomials, at $t=1$ against the monomial symmetric polynomials, and in the transition identity with the Kostka-Foulkes polynomials.