Jack polynomials

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Polynomials

$n$

$\lambda$

$J_\lambda(x_1,\ldots,x_n;\alpha)$

x1^2*a + x2^2*a + x1^2 + 2*x1*x2 + x2^2

1, 1:

2*x1*x2

2*x1^3*a^2 + 2*x2^3*a^2 + 3*x1^3*a + 3*x1^2*x2*a + 3*x1*x2^2*a + 3*x2^3*a + x1^3 + 3*x1^2*x2 + 3*x1*x2^2 + x2^3

2, 1:

x1^2*x2*a + x1*x2^2*a + 2*x1^2*x2 + 2*x1*x2^2

6*x1^4*a^3 + 6*x2^4*a^3 + 11*x1^4*a^2 + 8*x1^3*x2*a^2 + 6*x1^2*x2^2*a^2 + 8*x1*x2^3*a^2 + 11*x2^4*a^2 + 6*x1^4*a + 12*x1^3*x2*a + 12*x1^2*x2^2*a + 12*x1*x2^3*a + 6*x2^4*a + x1^4 + 4*x1^3*x2 + 6*x1^2*x2^2 + 4*x1*x2^3 + x2^4

3, 1:

2*x1^3*x2*a^2 + 2*x1*x2^3*a^2 + 4*x1^3*x2*a + 4*x1^2*x2^2*a + 4*x1*x2^3*a + 2*x1^3*x2 + 4*x1^2*x2^2 + 2*x1*x2^3

2, 2:

2*x1^2*x2^2*a^2 + 6*x1^2*x2^2*a + 4*x1^2*x2^2

24*x1^5*a^4 + 24*x2^5*a^4 + 50*x1^5*a^3 + 30*x1^4*x2*a^3 + 20*x1^3*x2^2*a^3 + 20*x1^2*x2^3*a^3 + 30*x1*x2^4*a^3 + 50*x2^5*a^3 + 35*x1^5*a^2 + 55*x1^4*x2*a^2 + 50*x1^3*x2^2*a^2 + 50*x1^2*x2^3*a^2 + 55*x1*x2^4*a^2 + 35*x2^5*a^2 + 10*x1^5*a + 30*x1^4*x2*a + 40*x1^3*x2^2*a + 40*x1^2*x2^3*a + 30*x1*x2^4*a + 10*x2^5*a + x1^5 + 5*x1^4*x2 + 10*x1^3*x2^2 + 10*x1^2*x2^3 + 5*x1*x2^4 + x2^5

4, 1:

6*x1^4*x2*a^3 + 6*x1*x2^4*a^3 + 13*x1^4*x2*a^2 + 9*x1^3*x2^2*a^2 + 9*x1^2*x2^3*a^2 + 13*x1*x2^4*a^2 + 9*x1^4*x2*a + 15*x1^3*x2^2*a + 15*x1^2*x2^3*a + 9*x1*x2^4*a + 2*x1^4*x2 + 6*x1^3*x2^2 + 6*x1^2*x2^3 + 2*x1*x2^4

3, 2:

2*x1^3*x2^2*a^3 + 2*x1^2*x2^3*a^3 + 8*x1^3*x2^2*a^2 + 8*x1^2*x2^3*a^2 + 10*x1^3*x2^2*a + 10*x1^2*x2^3*a + 4*x1^3*x2^2 + 4*x1^2*x2^3

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

1, 1:

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

2*x1^3*a^2 + 2*x2^3*a^2 + 2*x3^3*a^2 + 3*x1^3*a + 3*x1^2*x2*a + 3*x1*x2^2*a + 3*x2^3*a + 3*x1^2*x3*a + 3*x2^2*x3*a + 3*x1*x3^2*a + 3*x2*x3^2*a + 3*x3^3*a + x1^3 + 3*x1^2*x2 + 3*x1*x2^2 + x2^3 + 3*x1^2*x3 + 6*x1*x2*x3 + 3*x2^2*x3 + 3*x1*x3^2 + 3*x2*x3^2 + x3^3

2, 1:

x1^2*x2*a + x1*x2^2*a + x1^2*x3*a + x2^2*x3*a + x1*x3^2*a + x2*x3^2*a + 2*x1^2*x2 + 2*x1*x2^2 + 2*x1^2*x3 + 6*x1*x2*x3 + 2*x2^2*x3 + 2*x1*x3^2 + 2*x2*x3^2

1, 1, 1:

6*x1*x2*x3

6*x1^4*a^3 + 6*x2^4*a^3 + 6*x3^4*a^3 + 11*x1^4*a^2 + 8*x1^3*x2*a^2 + 6*x1^2*x2^2*a^2 + 8*x1*x2^3*a^2 + 11*x2^4*a^2 + 8*x1^3*x3*a^2 + 8*x2^3*x3*a^2 + 6*x1^2*x3^2*a^2 + 6*x2^2*x3^2*a^2 + 8*x1*x3^3*a^2 + 8*x2*x3^3*a^2 + 11*x3^4*a^2 + 6*x1^4*a + 12*x1^3*x2*a + 12*x1^2*x2^2*a + 12*x1*x2^3*a + 6*x2^4*a + 12*x1^3*x3*a + 12*x1^2*x2*x3*a + 12*x1*x2^2*x3*a + 12*x2^3*x3*a + 12*x1^2*x3^2*a + 12*x1*x2*x3^2*a + 12*x2^2*x3^2*a + 12*x1*x3^3*a + 12*x2*x3^3*a + 6*x3^4*a + x1^4 + 4*x1^3*x2 + 6*x1^2*x2^2 + 4*x1*x2^3 + x2^4 + 4*x1^3*x3 + 12*x1^2*x2*x3 + 12*x1*x2^2*x3 + 4*x2^3*x3 + 6*x1^2*x3^2 + 12*x1*x2*x3^2 + 6*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3 + x3^4

3, 1:

2*x1^3*x2*a^2 + 2*x1*x2^3*a^2 + 2*x1^3*x3*a^2 + 2*x2^3*x3*a^2 + 2*x1*x3^3*a^2 + 2*x2*x3^3*a^2 + 4*x1^3*x2*a + 4*x1^2*x2^2*a + 4*x1*x2^3*a + 4*x1^3*x3*a + 6*x1^2*x2*x3*a + 6*x1*x2^2*x3*a + 4*x2^3*x3*a + 4*x1^2*x3^2*a + 6*x1*x2*x3^2*a + 4*x2^2*x3^2*a + 4*x1*x3^3*a + 4*x2*x3^3*a + 2*x1^3*x2 + 4*x1^2*x2^2 + 2*x1*x2^3 + 2*x1^3*x3 + 10*x1^2*x2*x3 + 10*x1*x2^2*x3 + 2*x2^3*x3 + 4*x1^2*x3^2 + 10*x1*x2*x3^2 + 4*x2^2*x3^2 + 2*x1*x3^3 + 2*x2*x3^3

2, 2:

2*x1^2*x2^2*a^2 + 2*x1^2*x3^2*a^2 + 2*x2^2*x3^2*a^2 + 6*x1^2*x2^2*a + 4*x1^2*x2*x3*a + 4*x1*x2^2*x3*a + 6*x1^2*x3^2*a + 4*x1*x2*x3^2*a + 6*x2^2*x3^2*a + 4*x1^2*x2^2 + 8*x1^2*x2*x3 + 8*x1*x2^2*x3 + 4*x1^2*x3^2 + 8*x1*x2*x3^2 + 4*x2^2*x3^2

2, 1, 1:

2*x1^2*x2*x3*a + 2*x1*x2^2*x3*a + 2*x1*x2*x3^2*a + 6*x1^2*x2*x3 + 6*x1*x2^2*x3 + 6*x1*x2*x3^2

24*x1^5*a^4 + 24*x2^5*a^4 + 24*x3^5*a^4 + 50*x1^5*a^3 + 30*x1^4*x2*a^3 + 20*x1^3*x2^2*a^3 + 20*x1^2*x2^3*a^3 + 30*x1*x2^4*a^3 + 50*x2^5*a^3 + 30*x1^4*x3*a^3 + 30*x2^4*x3*a^3 + 20*x1^3*x3^2*a^3 + 20*x2^3*x3^2*a^3 + 20*x1^2*x3^3*a^3 + 20*x2^2*x3^3*a^3 + 30*x1*x3^4*a^3 + 30*x2*x3^4*a^3 + 50*x3^5*a^3 + 35*x1^5*a^2 + 55*x1^4*x2*a^2 + 50*x1^3*x2^2*a^2 + 50*x1^2*x2^3*a^2 + 55*x1*x2^4*a^2 + 35*x2^5*a^2 + 55*x1^4*x3*a^2 + 40*x1^3*x2*x3*a^2 + 30*x1^2*x2^2*x3*a^2 + 40*x1*x2^3*x3*a^2 + 55*x2^4*x3*a^2 + 50*x1^3*x3^2*a^2 + 30*x1^2*x2*x3^2*a^2 + 30*x1*x2^2*x3^2*a^2 + 50*x2^3*x3^2*a^2 + 50*x1^2*x3^3*a^2 + 40*x1*x2*x3^3*a^2 + 50*x2^2*x3^3*a^2 + 55*x1*x3^4*a^2 + 55*x2*x3^4*a^2 + 35*x3^5*a^2 + 10*x1^5*a + 30*x1^4*x2*a + 40*x1^3*x2^2*a + 40*x1^2*x2^3*a + 30*x1*x2^4*a + 10*x2^5*a + 30*x1^4*x3*a + 60*x1^3*x2*x3*a + 60*x1^2*x2^2*x3*a + 60*x1*x2^3*x3*a + 30*x2^4*x3*a + 40*x1^3*x3^2*a + 60*x1^2*x2*x3^2*a + 60*x1*x2^2*x3^2*a + 40*x2^3*x3^2*a + 40*x1^2*x3^3*a + 60*x1*x2*x3^3*a + 40*x2^2*x3^3*a + 30*x1*x3^4*a + 30*x2*x3^4*a + 10*x3^5*a + x1^5 + 5*x1^4*x2 + 10*x1^3*x2^2 + 10*x1^2*x2^3 + 5*x1*x2^4 + x2^5 + 5*x1^4*x3 + 20*x1^3*x2*x3 + 30*x1^2*x2^2*x3 + 20*x1*x2^3*x3 + 5*x2^4*x3 + 10*x1^3*x3^2 + 30*x1^2*x2*x3^2 + 30*x1*x2^2*x3^2 + 10*x2^3*x3^2 + 10*x1^2*x3^3 + 20*x1*x2*x3^3 + 10*x2^2*x3^3 + 5*x1*x3^4 + 5*x2*x3^4 + x3^5

4, 1:

6*x1^4*x2*a^3 + 6*x1*x2^4*a^3 + 6*x1^4*x3*a^3 + 6*x2^4*x3*a^3 + 6*x1*x3^4*a^3 + 6*x2*x3^4*a^3 + 13*x1^4*x2*a^2 + 9*x1^3*x2^2*a^2 + 9*x1^2*x2^3*a^2 + 13*x1*x2^4*a^2 + 13*x1^4*x3*a^2 + 16*x1^3*x2*x3*a^2 + 6*x1^2*x2^2*x3*a^2 + 16*x1*x2^3*x3*a^2 + 13*x2^4*x3*a^2 + 9*x1^3*x3^2*a^2 + 6*x1^2*x2*x3^2*a^2 + 6*x1*x2^2*x3^2*a^2 + 9*x2^3*x3^2*a^2 + 9*x1^2*x3^3*a^2 + 16*x1*x2*x3^3*a^2 + 9*x2^2*x3^3*a^2 + 13*x1*x3^4*a^2 + 13*x2*x3^4*a^2 + 9*x1^4*x2*a + 15*x1^3*x2^2*a + 15*x1^2*x2^3*a + 9*x1*x2^4*a + 9*x1^4*x3*a + 30*x1^3*x2*x3*a + 30*x1^2*x2^2*x3*a + 30*x1*x2^3*x3*a + 9*x2^4*x3*a + 15*x1^3*x3^2*a + 30*x1^2*x2*x3^2*a + 30*x1*x2^2*x3^2*a + 15*x2^3*x3^2*a + 15*x1^2*x3^3*a + 30*x1*x2*x3^3*a + 15*x2^2*x3^3*a + 9*x1*x3^4*a + 9*x2*x3^4*a + 2*x1^4*x2 + 6*x1^3*x2^2 + 6*x1^2*x2^3 + 2*x1*x2^4 + 2*x1^4*x3 + 14*x1^3*x2*x3 + 24*x1^2*x2^2*x3 + 14*x1*x2^3*x3 + 2*x2^4*x3 + 6*x1^3*x3^2 + 24*x1^2*x2*x3^2 + 24*x1*x2^2*x3^2 + 6*x2^3*x3^2 + 6*x1^2*x3^3 + 14*x1*x2*x3^3 + 6*x2^2*x3^3 + 2*x1*x3^4 + 2*x2*x3^4

3, 2:

2*x1^3*x2^2*a^3 + 2*x1^2*x2^3*a^3 + 2*x1^3*x3^2*a^3 + 2*x2^3*x3^2*a^3 + 2*x1^2*x3^3*a^3 + 2*x2^2*x3^3*a^3 + 8*x1^3*x2^2*a^2 + 8*x1^2*x2^3*a^2 + 4*x1^3*x2*x3*a^2 + 6*x1^2*x2^2*x3*a^2 + 4*x1*x2^3*x3*a^2 + 8*x1^3*x3^2*a^2 + 6*x1^2*x2*x3^2*a^2 + 6*x1*x2^2*x3^2*a^2 + 8*x2^3*x3^2*a^2 + 8*x1^2*x3^3*a^2 + 4*x1*x2*x3^3*a^2 + 8*x2^2*x3^3*a^2 + 10*x1^3*x2^2*a + 10*x1^2*x2^3*a + 12*x1^3*x2*x3*a + 22*x1^2*x2^2*x3*a + 12*x1*x2^3*x3*a + 10*x1^3*x3^2*a + 22*x1^2*x2*x3^2*a + 22*x1*x2^2*x3^2*a + 10*x2^3*x3^2*a + 10*x1^2*x3^3*a + 12*x1*x2*x3^3*a + 10*x2^2*x3^3*a + 4*x1^3*x2^2 + 4*x1^2*x2^3 + 8*x1^3*x2*x3 + 20*x1^2*x2^2*x3 + 8*x1*x2^3*x3 + 4*x1^3*x3^2 + 20*x1^2*x2*x3^2 + 20*x1*x2^2*x3^2 + 4*x2^3*x3^2 + 4*x1^2*x3^3 + 8*x1*x2*x3^3 + 4*x2^2*x3^3

3, 1, 1:

4*x1^3*x2*x3*a^2 + 4*x1*x2^3*x3*a^2 + 4*x1*x2*x3^3*a^2 + 10*x1^3*x2*x3*a + 8*x1^2*x2^2*x3*a + 10*x1*x2^3*x3*a + 8*x1^2*x2*x3^2*a + 8*x1*x2^2*x3^2*a + 10*x1*x2*x3^3*a + 6*x1^3*x2*x3 + 12*x1^2*x2^2*x3 + 6*x1*x2^3*x3 + 12*x1^2*x2*x3^2 + 12*x1*x2^2*x3^2 + 6*x1*x2*x3^3

2, 2, 1:

2*x1^2*x2^2*x3*a^2 + 2*x1^2*x2*x3^2*a^2 + 2*x1*x2^2*x3^2*a^2 + 10*x1^2*x2^2*x3*a + 10*x1^2*x2*x3^2*a + 10*x1*x2^2*x3^2*a + 12*x1^2*x2^2*x3 + 12*x1^2*x2*x3^2 + 12*x1*x2^2*x3^2

x1^2*a + x2^2*a + x3^2*a + x4^2*a + x1^2 + 2*x1*x2 + x2^2 + 2*x1*x3 + 2*x2*x3 + x3^2 + 2*x1*x4 + 2*x2*x4 + 2*x3*x4 + x4^2

1, 1:

2*x1*x2 + 2*x1*x3 + 2*x2*x3 + 2*x1*x4 + 2*x2*x4 + 2*x3*x4

2*x1^3*a^2 + 2*x2^3*a^2 + 2*x3^3*a^2 + 2*x4^3*a^2 + 3*x1^3*a + 3*x1^2*x2*a + 3*x1*x2^2*a + 3*x2^3*a + 3*x1^2*x3*a + 3*x2^2*x3*a + 3*x1*x3^2*a + 3*x2*x3^2*a + 3*x3^3*a + 3*x1^2*x4*a + 3*x2^2*x4*a + 3*x3^2*x4*a + 3*x1*x4^2*a + 3*x2*x4^2*a + 3*x3*x4^2*a + 3*x4^3*a + x1^3 + 3*x1^2*x2 + 3*x1*x2^2 + x2^3 + 3*x1^2*x3 + 6*x1*x2*x3 + 3*x2^2*x3 + 3*x1*x3^2 + 3*x2*x3^2 + x3^3 + 3*x1^2*x4 + 6*x1*x2*x4 + 3*x2^2*x4 + 6*x1*x3*x4 + 6*x2*x3*x4 + 3*x3^2*x4 + 3*x1*x4^2 + 3*x2*x4^2 + 3*x3*x4^2 + x4^3

2, 1:

x1^2*x2*a + x1*x2^2*a + x1^2*x3*a + x2^2*x3*a + x1*x3^2*a + x2*x3^2*a + x1^2*x4*a + x2^2*x4*a + x3^2*x4*a + x1*x4^2*a + x2*x4^2*a + x3*x4^2*a + 2*x1^2*x2 + 2*x1*x2^2 + 2*x1^2*x3 + 6*x1*x2*x3 + 2*x2^2*x3 + 2*x1*x3^2 + 2*x2*x3^2 + 2*x1^2*x4 + 6*x1*x2*x4 + 2*x2^2*x4 + 6*x1*x3*x4 + 6*x2*x3*x4 + 2*x3^2*x4 + 2*x1*x4^2 + 2*x2*x4^2 + 2*x3*x4^2

1, 1, 1:

6*x1*x2*x3 + 6*x1*x2*x4 + 6*x1*x3*x4 + 6*x2*x3*x4

6*x1^4*a^3 + 6*x2^4*a^3 + 6*x3^4*a^3 + 6*x4^4*a^3 + 11*x1^4*a^2 + 8*x1^3*x2*a^2 + 6*x1^2*x2^2*a^2 + 8*x1*x2^3*a^2 + 11*x2^4*a^2 + 8*x1^3*x3*a^2 + 8*x2^3*x3*a^2 + 6*x1^2*x3^2*a^2 + 6*x2^2*x3^2*a^2 + 8*x1*x3^3*a^2 + 8*x2*x3^3*a^2 + 11*x3^4*a^2 + 8*x1^3*x4*a^2 + 8*x2^3*x4*a^2 + 8*x3^3*x4*a^2 + 6*x1^2*x4^2*a^2 + 6*x2^2*x4^2*a^2 + 6*x3^2*x4^2*a^2 + 8*x1*x4^3*a^2 + 8*x2*x4^3*a^2 + 8*x3*x4^3*a^2 + 11*x4^4*a^2 + 6*x1^4*a + 12*x1^3*x2*a + 12*x1^2*x2^2*a + 12*x1*x2^3*a + 6*x2^4*a + 12*x1^3*x3*a + 12*x1^2*x2*x3*a + 12*x1*x2^2*x3*a + 12*x2^3*x3*a + 12*x1^2*x3^2*a + 12*x1*x2*x3^2*a + 12*x2^2*x3^2*a + 12*x1*x3^3*a + 12*x2*x3^3*a + 6*x3^4*a + 12*x1^3*x4*a + 12*x1^2*x2*x4*a + 12*x1*x2^2*x4*a + 12*x2^3*x4*a + 12*x1^2*x3*x4*a + 12*x2^2*x3*x4*a + 12*x1*x3^2*x4*a + 12*x2*x3^2*x4*a + 12*x3^3*x4*a + 12*x1^2*x4^2*a + 12*x1*x2*x4^2*a + 12*x2^2*x4^2*a + 12*x1*x3*x4^2*a + 12*x2*x3*x4^2*a + 12*x3^2*x4^2*a + 12*x1*x4^3*a + 12*x2*x4^3*a + 12*x3*x4^3*a + 6*x4^4*a + x1^4 + 4*x1^3*x2 + 6*x1^2*x2^2 + 4*x1*x2^3 + x2^4 + 4*x1^3*x3 + 12*x1^2*x2*x3 + 12*x1*x2^2*x3 + 4*x2^3*x3 + 6*x1^2*x3^2 + 12*x1*x2*x3^2 + 6*x2^2*x3^2 + 4*x1*x3^3 + 4*x2*x3^3 + x3^4 + 4*x1^3*x4 + 12*x1^2*x2*x4 + 12*x1*x2^2*x4 + 4*x2^3*x4 + 12*x1^2*x3*x4 + 24*x1*x2*x3*x4 + 12*x2^2*x3*x4 + 12*x1*x3^2*x4 + 12*x2*x3^2*x4 + 4*x3^3*x4 + 6*x1^2*x4^2 + 12*x1*x2*x4^2 + 6*x2^2*x4^2 + 12*x1*x3*x4^2 + 12*x2*x3*x4^2 + 6*x3^2*x4^2 + 4*x1*x4^3 + 4*x2*x4^3 + 4*x3*x4^3 + x4^4

3, 1:

2*x1^3*x2*a^2 + 2*x1*x2^3*a^2 + 2*x1^3*x3*a^2 + 2*x2^3*x3*a^2 + 2*x1*x3^3*a^2 + 2*x2*x3^3*a^2 + 2*x1^3*x4*a^2 + 2*x2^3*x4*a^2 + 2*x3^3*x4*a^2 + 2*x1*x4^3*a^2 + 2*x2*x4^3*a^2 + 2*x3*x4^3*a^2 + 4*x1^3*x2*a + 4*x1^2*x2^2*a + 4*x1*x2^3*a + 4*x1^3*x3*a + 6*x1^2*x2*x3*a + 6*x1*x2^2*x3*a + 4*x2^3*x3*a + 4*x1^2*x3^2*a + 6*x1*x2*x3^2*a + 4*x2^2*x3^2*a + 4*x1*x3^3*a + 4*x2*x3^3*a + 4*x1^3*x4*a + 6*x1^2*x2*x4*a + 6*x1*x2^2*x4*a + 4*x2^3*x4*a + 6*x1^2*x3*x4*a + 6*x2^2*x3*x4*a + 6*x1*x3^2*x4*a + 6*x2*x3^2*x4*a + 4*x3^3*x4*a + 4*x1^2*x4^2*a + 6*x1*x2*x4^2*a + 4*x2^2*x4^2*a + 6*x1*x3*x4^2*a + 6*x2*x3*x4^2*a + 4*x3^2*x4^2*a + 4*x1*x4^3*a + 4*x2*x4^3*a + 4*x3*x4^3*a + 2*x1^3*x2 + 4*x1^2*x2^2 + 2*x1*x2^3 + 2*x1^3*x3 + 10*x1^2*x2*x3 + 10*x1*x2^2*x3 + 2*x2^3*x3 + 4*x1^2*x3^2 + 10*x1*x2*x3^2 + 4*x2^2*x3^2 + 2*x1*x3^3 + 2*x2*x3^3 + 2*x1^3*x4 + 10*x1^2*x2*x4 + 10*x1*x2^2*x4 + 2*x2^3*x4 + 10*x1^2*x3*x4 + 24*x1*x2*x3*x4 + 10*x2^2*x3*x4 + 10*x1*x3^2*x4 + 10*x2*x3^2*x4 + 2*x3^3*x4 + 4*x1^2*x4^2 + 10*x1*x2*x4^2 + 4*x2^2*x4^2 + 10*x1*x3*x4^2 + 10*x2*x3*x4^2 + 4*x3^2*x4^2 + 2*x1*x4^3 + 2*x2*x4^3 + 2*x3*x4^3

2, 2:

2*x1^2*x2^2*a^2 + 2*x1^2*x3^2*a^2 + 2*x2^2*x3^2*a^2 + 2*x1^2*x4^2*a^2 + 2*x2^2*x4^2*a^2 + 2*x3^2*x4^2*a^2 + 6*x1^2*x2^2*a + 4*x1^2*x2*x3*a + 4*x1*x2^2*x3*a + 6*x1^2*x3^2*a + 4*x1*x2*x3^2*a + 6*x2^2*x3^2*a + 4*x1^2*x2*x4*a + 4*x1*x2^2*x4*a + 4*x1^2*x3*x4*a + 4*x2^2*x3*x4*a + 4*x1*x3^2*x4*a + 4*x2*x3^2*x4*a + 6*x1^2*x4^2*a + 4*x1*x2*x4^2*a + 6*x2^2*x4^2*a + 4*x1*x3*x4^2*a + 4*x2*x3*x4^2*a + 6*x3^2*x4^2*a + 4*x1^2*x2^2 + 8*x1^2*x2*x3 + 8*x1*x2^2*x3 + 4*x1^2*x3^2 + 8*x1*x2*x3^2 + 4*x2^2*x3^2 + 8*x1^2*x2*x4 + 8*x1*x2^2*x4 + 8*x1^2*x3*x4 + 24*x1*x2*x3*x4 + 8*x2^2*x3*x4 + 8*x1*x3^2*x4 + 8*x2*x3^2*x4 + 4*x1^2*x4^2 + 8*x1*x2*x4^2 + 4*x2^2*x4^2 + 8*x1*x3*x4^2 + 8*x2*x3*x4^2 + 4*x3^2*x4^2

2, 1, 1:

2*x1^2*x2*x3*a + 2*x1*x2^2*x3*a + 2*x1*x2*x3^2*a + 2*x1^2*x2*x4*a + 2*x1*x2^2*x4*a + 2*x1^2*x3*x4*a + 2*x2^2*x3*x4*a + 2*x1*x3^2*x4*a + 2*x2*x3^2*x4*a + 2*x1*x2*x4^2*a + 2*x1*x3*x4^2*a + 2*x2*x3*x4^2*a + 6*x1^2*x2*x3 + 6*x1*x2^2*x3 + 6*x1*x2*x3^2 + 6*x1^2*x2*x4 + 6*x1*x2^2*x4 + 6*x1^2*x3*x4 + 24*x1*x2*x3*x4 + 6*x2^2*x3*x4 + 6*x1*x3^2*x4 + 6*x2*x3^2*x4 + 6*x1*x2*x4^2 + 6*x1*x3*x4^2 + 6*x2*x3*x4^2

1, 1, 1, 1:

24*x1*x2*x3*x4

Definition

The integral Jack polynomial $J_\lambda(x_1,\ldots,x_n;\alpha)$ [1] is the specialization to $n$ alphabet variables of $c_\lambda(\alpha)P_\lambda^{(\alpha)}$, with $P_\lambda^{(\alpha)}$ characterized by (1) and $c_\lambda$ given by (3).

Parameters

$n$

— number of alphabet variables (a positive integer)

$\lambda$

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

Formulas

(1)

$P_\lambda^{(\alpha)}=m_\lambda+\sum_{\mu\lhd\lambda}u_{\lambda\mu}(\alpha)m_\mu$, with $\langle P_\lambda^{(\alpha)},P_\nu^{(\alpha)}\rangle_\alpha=0$ 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 $k$, after padding with zeros [1].

(2)

$\langle p_\mu,p_\nu\rangle_\alpha=\delta_{\mu\nu}z_\mu\alpha^{\ell(\mu)}$, where $p_\mu=\prod_i p_{\mu_i}$, $p_r=\sum_j x_j^r$ is the power sum symmetric function, and $z_\mu=\prod_{r\geq1}r^{m_r(\mu)}m_r(\mu)!$ with $m_r(\mu)=\#\{i:\mu_i=r\}$ [1].

(3)

$J_\lambda(x;\alpha)=c_\lambda(\alpha)P_\lambda^{(\alpha)}(x)$, where $c_\lambda(\alpha)=\prod_{s\in\lambda}(\alpha A(s)+L(s)+1)$ [1].

(4)

$J_\lambda(x;1)=H_\lambda s_\lambda(x)$, where $H_\lambda=\prod_{s\in\lambda}(A(s)+L(s)+1)$ and $s_\lambda$ is the Schur polynomial [2].

(5)

$J_{(1^d)}(x_1,\ldots,x_n;\alpha)=d!e_d(x_1,\ldots,x_n)$ for $1\leq d\leq n$, where $e_d$ is the elementary symmetric polynomial [2].

(6)

$J_\varnothing=1$, $J_{(1)}(x_1,\ldots,x_n;\alpha)=\sum_{i=1}^n x_i$, and $J_{(d)}(x_1;\alpha)=x_1^d\prod_{j=0}^{d-1}(1+j\alpha)$ for $d\geq1$ [2].

(7)

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

Comments

(8)

A partition $\lambda=(\lambda_1,\ldots,\lambda_\ell)$ has positive parts in nonincreasing order, weight $d=|\lambda|=\sum_i\lambda_i$, and length $\ell=\ell(\lambda)$. Its entry identity lists the parts separated by commas, without zero padding. Write $\lambda^\prime_j=\#\{i:\lambda_i\geq j\}$ for the conjugate partition. For a box $s=(i,j)$ in row $i$ and column $j$, both indexed from $1$, the arm and leg lengths are $A(s)=\lambda_i-j$ and $L(s)=\lambda^\prime_j-i$.

(9)

The scalar product and triangularity are defined in the ring of symmetric functions over $\mathbb Q(\alpha)$ before the variables $x_{n+1},x_{n+2},\ldots$ are set to zero. The factor $c_\lambda(\alpha)$ selects the integral $J$ normalization [1]; it is generally different from the monic $P$ normalization. Each entry is fully expanded over $\mathbb Z[x_1,\ldots,x_n,a]$, where $a$ denotes the formal indeterminate $\alpha$.

(10)

For each fixed weight $d$ and positive integer $n$, the polynomials indexed by partitions of $d$ with length at most $n$ form a basis over $\mathbb Q(\alpha)$ for the symmetric homogeneous polynomials of degree $d$. Homogeneity refers to the alphabet variables $x_1,\ldots,x_n$, excluding $\alpha$ [2].

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
from sage.combinat.sf.sf import SymmetricFunctions

A = PolynomialRing(QQ, 'a')
J = SymmetricFunctions(A.fraction_field()).jack(t=A.gen()).J()
R = PolynomialRing(ZZ, ['x1', 'x2', 'x3', 'a'])
print(R(str(J[3, 2].expand(3, alphabet=['x1', 'x2', 'x3']))))

References

[1]

H. Nakajima, Jack polynomials and Hilbert schemes of points on surfaces, preprint, 1996, Section 2(i), Theorem 2.3 and equation (2.4). (arXiv)

[2]

R. P. Stanley, Some combinatorial properties of Jack symmetric functions, Advances in Mathematics 77 (1989), 76–115, Theorem 1.1, Proposition 1.2, equation (5), and Proposition 2.2. (doi)

Links

[3]

SageMath: Jack symmetric functions

Similar tables

Schur polynomials — give the specialization $\alpha=1$ after division by the ordinary hook product

monomial symmetric polynomials — give the triangular basis used in the Jack characterization

Hall–Littlewood polynomials — form another partition-indexed deformation of the Schur basis

Data properties

Entries are of type: integral polynomial

How well the digits are known: exact

Table is complete: no (it holds all partitions with $\ell(\lambda)\leq n$ and $2\leq|\lambda|\leq5$ for $n=2,3$, and $2\leq|\lambda|\leq4$ for $n=4$, giving 34 entries; these complete degree slices cover rows, columns and interior shapes across small alphabets, with noncanonical bounds chosen for readability. The longest entry has 1412 characters, whereas the omitted four-variable degree-5 slice reaches 3341. The cases of weights 0 and 1 and of one alphabet variable are given in (6))

How they were obtained:

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

Every expansion is compared with SageMath's Jack $J$ implementation [3]. Independent polynomial arithmetic checks the Laplace–Beltrami eigenidentity and the hook-normalized Schur specialization using an alternant determinant. The checks also cover symmetry, alphabet degree, dominance support, leading coefficient, one-row and one-column formulas, and specialization of the last alphabet variable to zero. Twenty-six overlapping Schur specializations are compared with the stored Schur table. The attached checks are applied both before submission and to the values read back from the database.