Macdonald polynomials

edit · history · discussion · files · long url · bases for symmetric polynomials combinatorics algebra polynomial

Polynomials

$n$

$\lambda$

$J_\lambda(x_1,\ldots,x_n;q,t)$

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

1, 1:

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

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

2, 1:

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

x1^4*q^6*t^4 + x1^3*x2*q^6*t^4 + x1^2*x2^2*q^6*t^4 + x1*x2^3*q^6*t^4 + x2^4*q^6*t^4 - x1^4*q^6*t^3 - 2*x1^3*x2*q^6*t^3 - 2*x1^2*x2^2*q^6*t^3 - 2*x1*x2^3*q^6*t^3 - x2^4*q^6*t^3 + x1^3*x2*q^5*t^4 + x1^2*x2^2*q^5*t^4 + x1*x2^3*q^5*t^4 + x1^3*x2*q^6*t^2 + x1^2*x2^2*q^6*t^2 + x1*x2^3*q^6*t^2 - x1^4*q^5*t^3 - 3*x1^3*x2*q^5*t^3 - 4*x1^2*x2^2*q^5*t^3 - 3*x1*x2^3*q^5*t^3 - x2^4*q^5*t^3 + x1^3*x2*q^4*t^4 + 2*x1^2*x2^2*q^4*t^4 + x1*x2^3*q^4*t^4 + x1^4*q^5*t^2 + 3*x1^3*x2*q^5*t^2 + 5*x1^2*x2^2*q^5*t^2 + 3*x1*x2^3*q^5*t^2 + x2^4*q^5*t^2 - x1^4*q^4*t^3 - 4*x1^3*x2*q^4*t^3 - 6*x1^2*x2^2*q^4*t^3 - 4*x1*x2^3*q^4*t^3 - x2^4*q^4*t^3 + x1^3*x2*q^3*t^4 + x1^2*x2^2*q^3*t^4 + x1*x2^3*q^3*t^4 - x1^3*x2*q^5*t - 2*x1^2*x2^2*q^5*t - x1*x2^3*q^5*t + x1^4*q^4*t^2 + 5*x1^3*x2*q^4*t^2 + 7*x1^2*x2^2*q^4*t^2 + 5*x1*x2^3*q^4*t^2 + x2^4*q^4*t^2 - x1^4*q^3*t^3 - 4*x1^3*x2*q^3*t^3 - 6*x1^2*x2^2*q^3*t^3 - 4*x1*x2^3*q^3*t^3 - x2^4*q^3*t^3 + x1^2*x2^2*q^2*t^4 - 2*x1^3*x2*q^4*t - 4*x1^2*x2^2*q^4*t - 2*x1*x2^3*q^4*t + 2*x1^4*q^3*t^2 + 6*x1^3*x2*q^3*t^2 + 10*x1^2*x2^2*q^3*t^2 + 6*x1*x2^3*q^3*t^2 + 2*x2^4*q^3*t^2 - 2*x1^3*x2*q^2*t^3 - 4*x1^2*x2^2*q^2*t^3 - 2*x1*x2^3*q^2*t^3 + x1^2*x2^2*q^4 - x1^4*q^3*t - 4*x1^3*x2*q^3*t - 6*x1^2*x2^2*q^3*t - 4*x1*x2^3*q^3*t - x2^4*q^3*t + x1^4*q^2*t^2 + 5*x1^3*x2*q^2*t^2 + 7*x1^2*x2^2*q^2*t^2 + 5*x1*x2^3*q^2*t^2 + x2^4*q^2*t^2 - x1^3*x2*q*t^3 - 2*x1^2*x2^2*q*t^3 - x1*x2^3*q*t^3 + x1^3*x2*q^3 + x1^2*x2^2*q^3 + x1*x2^3*q^3 - x1^4*q^2*t - 4*x1^3*x2*q^2*t - 6*x1^2*x2^2*q^2*t - 4*x1*x2^3*q^2*t - x2^4*q^2*t + x1^4*q*t^2 + 3*x1^3*x2*q*t^2 + 5*x1^2*x2^2*q*t^2 + 3*x1*x2^3*q*t^2 + x2^4*q*t^2 + x1^3*x2*q^2 + 2*x1^2*x2^2*q^2 + x1*x2^3*q^2 - x1^4*q*t - 3*x1^3*x2*q*t - 4*x1^2*x2^2*q*t - 3*x1*x2^3*q*t - x2^4*q*t + x1^3*x2*t^2 + x1^2*x2^2*t^2 + x1*x2^3*t^2 + x1^3*x2*q + x1^2*x2^2*q + x1*x2^3*q - x1^4*t - 2*x1^3*x2*t - 2*x1^2*x2^2*t - 2*x1*x2^3*t - x2^4*t + x1^4 + x1^3*x2 + x1^2*x2^2 + x1*x2^3 + x2^4

3, 1:

x1^3*x2*q^3*t^5 + x1^2*x2^2*q^3*t^5 + x1*x2^3*q^3*t^5 - 2*x1^3*x2*q^3*t^4 - 3*x1^2*x2^2*q^3*t^4 - 2*x1*x2^3*q^3*t^4 + x1^2*x2^2*q^2*t^5 + x1^3*x2*q^3*t^3 + 3*x1^2*x2^2*q^3*t^3 + x1*x2^3*q^3*t^3 - x1^3*x2*q^2*t^4 - 3*x1^2*x2^2*q^2*t^4 - x1*x2^3*q^2*t^4 - x1^2*x2^2*q^3*t^2 + 2*x1^3*x2*q^2*t^3 + 3*x1^2*x2^2*q^2*t^3 + 2*x1*x2^3*q^2*t^3 - x1^3*x2*q^2*t^2 - x1^2*x2^2*q^2*t^2 - x1*x2^3*q^2*t^2 - x1^3*x2*q*t^3 - x1^2*x2^2*q*t^3 - x1*x2^3*q*t^3 + 2*x1^3*x2*q*t^2 + 3*x1^2*x2^2*q*t^2 + 2*x1*x2^3*q*t^2 - x1^2*x2^2*t^3 - x1^3*x2*q*t - 3*x1^2*x2^2*q*t - x1*x2^3*q*t + x1^3*x2*t^2 + 3*x1^2*x2^2*t^2 + x1*x2^3*t^2 + x1^2*x2^2*q - 2*x1^3*x2*t - 3*x1^2*x2^2*t - 2*x1*x2^3*t + x1^3*x2 + x1^2*x2^2 + x1*x2^3

2, 2:

x1^2*x2^2*q^2*t^6 - x1^2*x2^2*q^2*t^5 - x1^2*x2^2*q^2*t^4 - x1^2*x2^2*q*t^5 + x1^2*x2^2*q^2*t^3 + 2*x1^2*x2^2*q*t^3 + x1^2*x2^2*t^3 - x1^2*x2^2*q*t - x1^2*x2^2*t^2 - x1^2*x2^2*t + x1^2*x2^2

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

1, 1:

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

-x1^3*q^3*t^3 - x1^2*x2*q^3*t^3 - x1*x2^2*q^3*t^3 - x2^3*q^3*t^3 - x1^2*x3*q^3*t^3 - x1*x2*x3*q^3*t^3 - x2^2*x3*q^3*t^3 - x1*x3^2*q^3*t^3 - x2*x3^2*q^3*t^3 - x3^3*q^3*t^3 + x1^3*q^3*t^2 + 2*x1^2*x2*q^3*t^2 + 2*x1*x2^2*q^3*t^2 + x2^3*q^3*t^2 + 2*x1^2*x3*q^3*t^2 + 3*x1*x2*x3*q^3*t^2 + 2*x2^2*x3*q^3*t^2 + 2*x1*x3^2*q^3*t^2 + 2*x2*x3^2*q^3*t^2 + x3^3*q^3*t^2 - x1^2*x2*q^2*t^3 - x1*x2^2*q^2*t^3 - x1^2*x3*q^2*t^3 - 2*x1*x2*x3*q^2*t^3 - x2^2*x3*q^2*t^3 - x1*x3^2*q^2*t^3 - x2*x3^2*q^2*t^3 - x1^2*x2*q^3*t - x1*x2^2*q^3*t - x1^2*x3*q^3*t - 3*x1*x2*x3*q^3*t - x2^2*x3*q^3*t - x1*x3^2*q^3*t - x2*x3^2*q^3*t + x1^3*q^2*t^2 + 3*x1^2*x2*q^2*t^2 + 3*x1*x2^2*q^2*t^2 + x2^3*q^2*t^2 + 3*x1^2*x3*q^2*t^2 + 6*x1*x2*x3*q^2*t^2 + 3*x2^2*x3*q^2*t^2 + 3*x1*x3^2*q^2*t^2 + 3*x2*x3^2*q^2*t^2 + x3^3*q^2*t^2 - x1^2*x2*q*t^3 - x1*x2^2*q*t^3 - x1^2*x3*q*t^3 - 2*x1*x2*x3*q*t^3 - x2^2*x3*q*t^3 - x1*x3^2*q*t^3 - x2*x3^2*q*t^3 + x1*x2*x3*q^3 - x1^3*q^2*t - 3*x1^2*x2*q^2*t - 3*x1*x2^2*q^2*t - x2^3*q^2*t - 3*x1^2*x3*q^2*t - 6*x1*x2*x3*q^2*t - 3*x2^2*x3*q^2*t - 3*x1*x3^2*q^2*t - 3*x2*x3^2*q^2*t - x3^3*q^2*t + x1^3*q*t^2 + 3*x1^2*x2*q*t^2 + 3*x1*x2^2*q*t^2 + x2^3*q*t^2 + 3*x1^2*x3*q*t^2 + 6*x1*x2*x3*q*t^2 + 3*x2^2*x3*q*t^2 + 3*x1*x3^2*q*t^2 + 3*x2*x3^2*q*t^2 + x3^3*q*t^2 - x1*x2*x3*t^3 + x1^2*x2*q^2 + x1*x2^2*q^2 + x1^2*x3*q^2 + 2*x1*x2*x3*q^2 + x2^2*x3*q^2 + x1*x3^2*q^2 + x2*x3^2*q^2 - x1^3*q*t - 3*x1^2*x2*q*t - 3*x1*x2^2*q*t - x2^3*q*t - 3*x1^2*x3*q*t - 6*x1*x2*x3*q*t - 3*x2^2*x3*q*t - 3*x1*x3^2*q*t - 3*x2*x3^2*q*t - x3^3*q*t + x1^2*x2*t^2 + x1*x2^2*t^2 + x1^2*x3*t^2 + 3*x1*x2*x3*t^2 + x2^2*x3*t^2 + x1*x3^2*t^2 + x2*x3^2*t^2 + x1^2*x2*q + x1*x2^2*q + x1^2*x3*q + 2*x1*x2*x3*q + x2^2*x3*q + x1*x3^2*q + x2*x3^2*q - x1^3*t - 2*x1^2*x2*t - 2*x1*x2^2*t - x2^3*t - 2*x1^2*x3*t - 3*x1*x2*x3*t - 2*x2^2*x3*t - 2*x1*x3^2*t - 2*x2*x3^2*t - x3^3*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:

-x1^2*x2*q*t^4 - x1*x2^2*q*t^4 - x1^2*x3*q*t^4 - 2*x1*x2*x3*q*t^4 - x2^2*x3*q*t^4 - x1*x3^2*q*t^4 - x2*x3^2*q*t^4 + 2*x1^2*x2*q*t^3 + 2*x1*x2^2*q*t^3 + 2*x1^2*x3*q*t^3 + 5*x1*x2*x3*q*t^3 + 2*x2^2*x3*q*t^3 + 2*x1*x3^2*q*t^3 + 2*x2*x3^2*q*t^3 - x1*x2*x3*t^4 - x1^2*x2*q*t^2 - x1*x2^2*q*t^2 - x1^2*x3*q*t^2 - 3*x1*x2*x3*q*t^2 - x2^2*x3*q*t^2 - x1*x3^2*q*t^2 - x2*x3^2*q*t^2 + x1*x2*x3*t^3 - x1*x2*x3*q*t + x1^2*x2*t^2 + x1*x2^2*t^2 + x1^2*x3*t^2 + 3*x1*x2*x3*t^2 + x2^2*x3*t^2 + x1*x3^2*t^2 + x2*x3^2*t^2 + x1*x2*x3*q - 2*x1^2*x2*t - 2*x1*x2^2*t - 2*x1^2*x3*t - 5*x1*x2*x3*t - 2*x2^2*x3*t - 2*x1*x3^2*t - 2*x2*x3^2*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*t^6 + x1*x2*x3*t^5 + x1*x2*x3*t^4 - x1*x2*x3*t^2 - x1*x2*x3*t + x1*x2*x3

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

Definition

The integral Macdonald polynomial $J_\lambda(x_1,\ldots,x_n;q,t)$ [1] is the specialization to $n$ alphabet variables of $c_\lambda(q,t)P_\lambda(x;q,t)$, where $P_\lambda$ is characterized by (1) and $c_\lambda$ is the factor in (3).

Parameters

$n$

— number of alphabet variables (a positive integer)

$\lambda$

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

Formulas

(1)

$P_\lambda=m_\lambda+\sum_{\mu\lhd\lambda}u_{\lambda\mu}(q,t)m_\mu$, with $\langle P_\lambda,P_\nu\rangle_{q,t}=0$ for $\lambda\ne\nu$. Here $m_\lambda$ is the monomial symmetric function, and $\mu\lhd\lambda$ means $|\mu|=|\lambda|$, $\mu\ne\lambda$, and $\sum_{i=1}^k\mu_i\leq\sum_{i=1}^k\lambda_i$ for every $k$, using zero padding [1].

(2)

$\langle p_\mu,p_\nu\rangle_{q,t}=\delta_{\mu\nu}z_\mu\prod_{i=1}^{\ell(\mu)}(1-q^{\mu_i})/(1-t^{\mu_i})$, where $p_\mu=\prod_i p_{\mu_i}$, $p_r=\sum_jx_j^r$ is the power sum symmetric function, and $z_\mu=\prod_{r\geq1}r^{m_r(\mu)}m_r(\mu)!$ with $m_r(\mu)=\#\{i:\mu_i=r\}$.

(3)

$J_\lambda(x;q,t)=c_\lambda(q,t)P_\lambda(x;q,t)$, where $c_\lambda(q,t)=\prod_{s\in\lambda}(1-q^{a(s)}t^{l(s)+1})$.

(4)

$D_{q,t}^{(n)}J_\lambda=(\sum_{i=1}^n q^{\lambda_i}t^{n-i})J_\lambda$, where $\lambda_i=0$ for $i>\ell(\lambda)$ and $D_{q,t}^{(n)}f=\sum_{i=1}^n\bigl(\prod_{j\ne i}(tx_i-x_j)/(x_i-x_j)\bigr)f(x_1,\ldots,qx_i,\ldots,x_n)$. The rational expression defines an operator on symmetric polynomials by cancellation of the apparent poles [1].

(5)

$J_\lambda(x;q,q)=\bigl(\prod_{s\in\lambda}(1-q^{a(s)+l(s)+1})\bigr)s_\lambda(x)$, where $s_\lambda$ is the Schur polynomial.

(6)

$J_\lambda(x;0,t)=b_\lambda(t)P_\lambda^{\mathrm{HL}}(x;t)$, where $P_\lambda^{\mathrm{HL}}$ is the Hall–Littlewood polynomial in the $P$ normalization and $b_\lambda(t)=\prod_{r\geq1}\prod_{j=1}^{m_r(\lambda)}(1-t^j)$.

(7)

$J_{(1^d)}(x_1,\ldots,x_n;q,t)=\bigl(\prod_{j=1}^d(1-t^j)\bigr)e_d(x_1,\ldots,x_n)$ for $1\leq d\leq n$, where $e_d$ is the elementary symmetric polynomial.

(8)

$J_\varnothing=1$, $J_{(1)}(x_1,\ldots,x_n;q,t)=(1-t)\sum_{i=1}^n x_i$, and $J_{(d)}(x_1;q,t)=x_1^d\prod_{j=0}^{d-1}(1-q^jt)$ for $d\geq1$.

(9)

$J_\lambda(x_1,\ldots,x_{n-1},0;q,t)=J_\lambda(x_1,\ldots,x_{n-1};q,t)$ for $\ell(\lambda)<n$; the specialization is zero when $\ell(\lambda)=n$.

Comments

(10)

A partition $\lambda=(\lambda_1,\ldots,\lambda_r)$ has positive parts in nonincreasing order, weight $d=|\lambda|=\sum_i\lambda_i$, and length $r=\ell(\lambda)$. Its entry identity lists the parts separated by commas, without zero padding. For a box $s=(i,j)$, with row and column indices starting at $1$, set $a(s)=\lambda_i-j$ and $l(s)=\lambda^\prime_j-i$, where $\lambda^\prime_j=\#\{i:\lambda_i\geq j\}$.

(11)

The scalar product and triangularity are defined in the ring of symmetric functions over $\mathbb Q(q,t)$ before the alphabet variables $x_{n+1},x_{n+2},\ldots$ are set to zero. Each entry is fully expanded over $\mathbb Z[x_1,\ldots,x_n,q,t]$. The symbols $q,t$ are formal indeterminates in that order. The normalization is the integral $J$ form, rather than the monic $P$ form or the modified Macdonald polynomial $\widetilde H_\lambda$ [1].

(12)

For each fixed weight $d$ and positive integer $n$, the polynomials with $\ell(\lambda)\leq n$ form a basis over $\mathbb Q(q,t)$ of the symmetric homogeneous polynomials of degree $d$. Degree refers to the alphabet variables $x_1,\ldots,x_n$, excluding $q,t$ [1].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.symbolic.ring import SR
import sage.rings.polynomial.laurent_polynomial_ring
from sage.combinat.sf.sf import SymmetricFunctions

QT = PolynomialRing(QQ, ['q', 't'])
q, t = QT.gens()
J = SymmetricFunctions(QT.fraction_field()).macdonald(q=q, t=t).J()
R = PolynomialRing(ZZ, ['x1', 'x2', 'x3', 'q', 't'])
print(R(str(J[2, 1].expand(3, alphabet=['x1', 'x2', 'x3']))))

Links

[1]

Mark Haiman: Macdonald polynomials and geometry, Section 2, equation (2.16)

[2]

SageMath: Macdonald polynomials, J and P bases

Similar tables

Schur polynomials — give the $q=t$ specialization after division by the hook factor

Hall–Littlewood polynomials — give the $q=0$ specialization after division by $b_\lambda(t)$

monomial symmetric polynomials — give the triangular basis in the Macdonald characterization

Macdonald–Kostka polynomials — record Schur expansion coefficients for the modified Macdonald normalization

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds every partition with $\ell(\lambda)\leq n$ in degrees $2\leq|\lambda|\leq4$ for $n=2$, degrees $2\leq|\lambda|\leq3$ for $n=3$, and degree $|\lambda|=2$ for $n=4$, giving 14 entries. These complete degree slices cover all admissible shapes across small alphabets, with noncanonical bounds chosen for readability. The longest entry has 1916 characters; the omitted three-variable degree-4 and four-variable degree-3 slices reach 7185 and 4198 characters. The four-variable quadratic slice adds alphabet breadth within the six-variable limit including $q,t$. Boundary cases of weights 0 and 1 and of one alphabet variable are given in (8))

How they were obtained:

All coefficients are exact integers. The attached generator converts monomial symmetric functions to power sums by set-partition inclusion–exclusion, performs exact Gram–Schmidt orthogonalization over $\mathbb Q(q,t)$, multiplies by $c_\lambda(q,t)$, and specializes the alphabet. Coercion to $\mathbb Z[x_1,\ldots,x_n,q,t]$ rejects residual denominators.

Every entry is compared with the independent SageMath Macdonald $J$ implementation, which uses determinantal formulas [2]. Exact polynomial arithmetic checks the difference-operator eigenidentity after multiplication by the Vandermonde. Controls include the constant polynomial and the linear power sum, and an incorrect quadratic is required to fail. Symmetry, alphabet degree, dominance support, leading coefficient, column formulas, boundary formulas and alphabet stability are also checked.

All 14 $q=t$ specializations agree with an independently computed Schur alternant and with the stored Schur table. All 14 $q=0$ specializations agree with the stored Hall–Littlewood table after multiplication by $b_\lambda(t)$. The attached checks run before submission and on the values read back from the draft.