Zonal polynomials

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Polynomials

$n$

$\lambda$

$C_\lambda(x_1,\ldots,x_n)$

x1^2 + 2/3*x1*x2 + x2^2

1, 1:

4/3*x1*x2

x1^3 + 3/5*x1^2*x2 + 3/5*x1*x2^2 + x2^3

2, 1:

12/5*x1^2*x2 + 12/5*x1*x2^2

x1^4 + 4/7*x1^3*x2 + 18/35*x1^2*x2^2 + 4/7*x1*x2^3 + x2^4

3, 1:

24/7*x1^3*x2 + 16/7*x1^2*x2^2 + 24/7*x1*x2^3

2, 2:

16/5*x1^2*x2^2

x1^5 + 5/9*x1^4*x2 + 10/21*x1^3*x2^2 + 10/21*x1^2*x2^3 + 5/9*x1*x2^4 + x2^5

4, 1:

40/9*x1^4*x2 + 8/3*x1^3*x2^2 + 8/3*x1^2*x2^3 + 40/9*x1*x2^4

3, 2:

48/7*x1^3*x2^2 + 48/7*x1^2*x2^3

x1^6 + 6/11*x1^5*x2 + 5/11*x1^4*x2^2 + 100/231*x1^3*x2^3 + 5/11*x1^2*x2^4 + 6/11*x1*x2^5 + x2^6

5, 1:

60/11*x1^5*x2 + 240/77*x1^4*x2^2 + 216/77*x1^3*x2^3 + 240/77*x1^2*x2^4 + 60/11*x1*x2^5

4, 2:

80/7*x1^4*x2^2 + 160/21*x1^3*x2^3 + 80/7*x1^2*x2^4

3, 3:

64/7*x1^3*x2^3

x1^2 + 2/3*x1*x2 + x2^2 + 2/3*x1*x3 + 2/3*x2*x3 + x3^2

1, 1:

4/3*x1*x2 + 4/3*x1*x3 + 4/3*x2*x3

x1^3 + 3/5*x1^2*x2 + 3/5*x1*x2^2 + x2^3 + 3/5*x1^2*x3 + 2/5*x1*x2*x3 + 3/5*x2^2*x3 + 3/5*x1*x3^2 + 3/5*x2*x3^2 + x3^3

2, 1:

12/5*x1^2*x2 + 12/5*x1*x2^2 + 12/5*x1^2*x3 + 18/5*x1*x2*x3 + 12/5*x2^2*x3 + 12/5*x1*x3^2 + 12/5*x2*x3^2

1, 1, 1:

2*x1*x2*x3

3, 1:

24/7*x1^3*x2 + 16/7*x1^2*x2^2 + 24/7*x1*x2^3 + 24/7*x1^3*x3 + 88/21*x1^2*x2*x3 + 88/21*x1*x2^2*x3 + 24/7*x2^3*x3 + 16/7*x1^2*x3^2 + 88/21*x1*x2*x3^2 + 16/7*x2^2*x3^2 + 24/7*x1*x3^3 + 24/7*x2*x3^3

2, 2:

16/5*x1^2*x2^2 + 32/15*x1^2*x2*x3 + 32/15*x1*x2^2*x3 + 16/5*x1^2*x3^2 + 32/15*x1*x2*x3^2 + 16/5*x2^2*x3^2

2, 1, 1:

16/3*x1^2*x2*x3 + 16/3*x1*x2^2*x3 + 16/3*x1*x2*x3^2

4, 1:

3, 2:

3, 1, 1:

10*x1^3*x2*x3 + 20/3*x1^2*x2^2*x3 + 10*x1*x2^3*x3 + 20/3*x1^2*x2*x3^2 + 20/3*x1*x2^2*x3^2 + 10*x1*x2*x3^3

2, 2, 1:

32/3*x1^2*x2^2*x3 + 32/3*x1^2*x2*x3^2 + 32/3*x1*x2^2*x3^2

x1^6 + 6/11*x1^5*x2 + 5/11*x1^4*x2^2 + 100/231*x1^3*x2^3 + 5/11*x1^2*x2^4 + 6/11*x1*x2^5 + x2^6 + 6/11*x1^5*x3 + 10/33*x1^4*x2*x3 + 20/77*x1^3*x2^2*x3 + 20/77*x1^2*x2^3*x3 + 10/33*x1*x2^4*x3 + 6/11*x2^5*x3 + 5/11*x1^4*x3^2 + 20/77*x1^3*x2*x3^2 + 18/77*x1^2*x2^2*x3^2 + 20/77*x1*x2^3*x3^2 + 5/11*x2^4*x3^2 + 100/231*x1^3*x3^3 + 20/77*x1^2*x2*x3^3 + 20/77*x1*x2^2*x3^3 + 100/231*x2^3*x3^3 + 5/11*x1^2*x3^4 + 10/33*x1*x2*x3^4 + 5/11*x2^2*x3^4 + 6/11*x1*x3^5 + 6/11*x2*x3^5 + x3^6

5, 1:

60/11*x1^5*x2 + 240/77*x1^4*x2^2 + 216/77*x1^3*x2^3 + 240/77*x1^2*x2^4 + 60/11*x1*x2^5 + 60/11*x1^5*x3 + 468/77*x1^4*x2*x3 + 1704/385*x1^3*x2^2*x3 + 1704/385*x1^2*x2^3*x3 + 468/77*x1*x2^4*x3 + 60/11*x2^5*x3 + 240/77*x1^4*x3^2 + 1704/385*x1^3*x2*x3^2 + 1296/385*x1^2*x2^2*x3^2 + 1704/385*x1*x2^3*x3^2 + 240/77*x2^4*x3^2 + 216/77*x1^3*x3^3 + 1704/385*x1^2*x2*x3^3 + 1704/385*x1*x2^2*x3^3 + 216/77*x2^3*x3^3 + 240/77*x1^2*x3^4 + 468/77*x1*x2*x3^4 + 240/77*x2^2*x3^4 + 60/11*x1*x3^5 + 60/11*x2*x3^5

4, 2:

80/7*x1^4*x2^2 + 160/21*x1^3*x2^3 + 80/7*x1^2*x2^4 + 160/21*x1^4*x2*x3 + 80/7*x1^3*x2^2*x3 + 80/7*x1^2*x2^3*x3 + 160/21*x1*x2^4*x3 + 80/7*x1^4*x3^2 + 80/7*x1^3*x2*x3^2 + 120/7*x1^2*x2^2*x3^2 + 80/7*x1*x2^3*x3^2 + 80/7*x2^4*x3^2 + 160/21*x1^3*x3^3 + 80/7*x1^2*x2*x3^3 + 80/7*x1*x2^2*x3^3 + 160/21*x2^3*x3^3 + 80/7*x1^2*x3^4 + 160/21*x1*x2*x3^4 + 80/7*x2^2*x3^4

4, 1, 1:

16*x1^4*x2*x3 + 48/5*x1^3*x2^2*x3 + 48/5*x1^2*x2^3*x3 + 16*x1*x2^4*x3 + 48/5*x1^3*x2*x3^2 + 32/5*x1^2*x2^2*x3^2 + 48/5*x1*x2^3*x3^2 + 48/5*x1^2*x2*x3^3 + 48/5*x1*x2^2*x3^3 + 16*x1*x2*x3^4

3, 3:

64/7*x1^3*x2^3 + 192/35*x1^3*x2^2*x3 + 192/35*x1^2*x2^3*x3 + 192/35*x1^3*x2*x3^2 + 128/35*x1^2*x2^2*x3^2 + 192/35*x1*x2^3*x3^2 + 64/7*x1^3*x3^3 + 192/35*x1^2*x2*x3^3 + 192/35*x1*x2^2*x3^3 + 64/7*x2^3*x3^3

3, 2, 1:

144/5*x1^3*x2^2*x3 + 144/5*x1^2*x2^3*x3 + 144/5*x1^3*x2*x3^2 + 216/5*x1^2*x2^2*x3^2 + 144/5*x1*x2^3*x3^2 + 144/5*x1^2*x2*x3^3 + 144/5*x1*x2^2*x3^3

2, 2, 2:

16*x1^2*x2^2*x3^2

x1^2 + 2/3*x1*x2 + x2^2 + 2/3*x1*x3 + 2/3*x2*x3 + x3^2 + 2/3*x1*x4 + 2/3*x2*x4 + 2/3*x3*x4 + x4^2

1, 1:

4/3*x1*x2 + 4/3*x1*x3 + 4/3*x2*x3 + 4/3*x1*x4 + 4/3*x2*x4 + 4/3*x3*x4

2, 1:

1, 1, 1:

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

3, 1:

2, 2:

2, 1, 1:

1, 1, 1, 1:

16/5*x1*x2*x3*x4

x1^5 + 5/9*x1^4*x2 + 10/21*x1^3*x2^2 + 10/21*x1^2*x2^3 + 5/9*x1*x2^4 + x2^5 + 5/9*x1^4*x3 + 20/63*x1^3*x2*x3 + 2/7*x1^2*x2^2*x3 + 20/63*x1*x2^3*x3 + 5/9*x2^4*x3 + 10/21*x1^3*x3^2 + 2/7*x1^2*x2*x3^2 + 2/7*x1*x2^2*x3^2 + 10/21*x2^3*x3^2 + 10/21*x1^2*x3^3 + 20/63*x1*x2*x3^3 + 10/21*x2^2*x3^3 + 5/9*x1*x3^4 + 5/9*x2*x3^4 + x3^5 + 5/9*x1^4*x4 + 20/63*x1^3*x2*x4 + 2/7*x1^2*x2^2*x4 + 20/63*x1*x2^3*x4 + 5/9*x2^4*x4 + 20/63*x1^3*x3*x4 + 4/21*x1^2*x2*x3*x4 + 4/21*x1*x2^2*x3*x4 + 20/63*x2^3*x3*x4 + 2/7*x1^2*x3^2*x4 + 4/21*x1*x2*x3^2*x4 + 2/7*x2^2*x3^2*x4 + 20/63*x1*x3^3*x4 + 20/63*x2*x3^3*x4 + 5/9*x3^4*x4 + 10/21*x1^3*x4^2 + 2/7*x1^2*x2*x4^2 + 2/7*x1*x2^2*x4^2 + 10/21*x2^3*x4^2 + 2/7*x1^2*x3*x4^2 + 4/21*x1*x2*x3*x4^2 + 2/7*x2^2*x3*x4^2 + 2/7*x1*x3^2*x4^2 + 2/7*x2*x3^2*x4^2 + 10/21*x3^3*x4^2 + 10/21*x1^2*x4^3 + 20/63*x1*x2*x4^3 + 10/21*x2^2*x4^3 + 20/63*x1*x3*x4^3 + 20/63*x2*x3*x4^3 + 10/21*x3^2*x4^3 + 5/9*x1*x4^4 + 5/9*x2*x4^4 + 5/9*x3*x4^4 + x4^5

4, 1:

40/9*x1^4*x2 + 8/3*x1^3*x2^2 + 8/3*x1^2*x2^3 + 40/9*x1*x2^4 + 40/9*x1^4*x3 + 46/9*x1^3*x2*x3 + 4*x1^2*x2^2*x3 + 46/9*x1*x2^3*x3 + 40/9*x2^4*x3 + 8/3*x1^3*x3^2 + 4*x1^2*x2*x3^2 + 4*x1*x2^2*x3^2 + 8/3*x2^3*x3^2 + 8/3*x1^2*x3^3 + 46/9*x1*x2*x3^3 + 8/3*x2^2*x3^3 + 40/9*x1*x3^4 + 40/9*x2*x3^4 + 40/9*x1^4*x4 + 46/9*x1^3*x2*x4 + 4*x1^2*x2^2*x4 + 46/9*x1*x2^3*x4 + 40/9*x2^4*x4 + 46/9*x1^3*x3*x4 + 14/3*x1^2*x2*x3*x4 + 14/3*x1*x2^2*x3*x4 + 46/9*x2^3*x3*x4 + 4*x1^2*x3^2*x4 + 14/3*x1*x2*x3^2*x4 + 4*x2^2*x3^2*x4 + 46/9*x1*x3^3*x4 + 46/9*x2*x3^3*x4 + 40/9*x3^4*x4 + 8/3*x1^3*x4^2 + 4*x1^2*x2*x4^2 + 4*x1*x2^2*x4^2 + 8/3*x2^3*x4^2 + 4*x1^2*x3*x4^2 + 14/3*x1*x2*x3*x4^2 + 4*x2^2*x3*x4^2 + 4*x1*x3^2*x4^2 + 4*x2*x3^2*x4^2 + 8/3*x3^3*x4^2 + 8/3*x1^2*x4^3 + 46/9*x1*x2*x4^3 + 8/3*x2^2*x4^3 + 46/9*x1*x3*x4^3 + 46/9*x2*x3*x4^3 + 8/3*x3^2*x4^3 + 40/9*x1*x4^4 + 40/9*x2*x4^4 + 40/9*x3*x4^4

3, 2:

48/7*x1^3*x2^2 + 48/7*x1^2*x2^3 + 32/7*x1^3*x2*x3 + 176/21*x1^2*x2^2*x3 + 32/7*x1*x2^3*x3 + 48/7*x1^3*x3^2 + 176/21*x1^2*x2*x3^2 + 176/21*x1*x2^2*x3^2 + 48/7*x2^3*x3^2 + 48/7*x1^2*x3^3 + 32/7*x1*x2*x3^3 + 48/7*x2^2*x3^3 + 32/7*x1^3*x2*x4 + 176/21*x1^2*x2^2*x4 + 32/7*x1*x2^3*x4 + 32/7*x1^3*x3*x4 + 64/7*x1^2*x2*x3*x4 + 64/7*x1*x2^2*x3*x4 + 32/7*x2^3*x3*x4 + 176/21*x1^2*x3^2*x4 + 64/7*x1*x2*x3^2*x4 + 176/21*x2^2*x3^2*x4 + 32/7*x1*x3^3*x4 + 32/7*x2*x3^3*x4 + 48/7*x1^3*x4^2 + 176/21*x1^2*x2*x4^2 + 176/21*x1*x2^2*x4^2 + 48/7*x2^3*x4^2 + 176/21*x1^2*x3*x4^2 + 64/7*x1*x2*x3*x4^2 + 176/21*x2^2*x3*x4^2 + 176/21*x1*x3^2*x4^2 + 176/21*x2*x3^2*x4^2 + 48/7*x3^3*x4^2 + 48/7*x1^2*x4^3 + 32/7*x1*x2*x4^3 + 48/7*x2^2*x4^3 + 32/7*x1*x3*x4^3 + 32/7*x2*x3*x4^3 + 48/7*x3^2*x4^3

3, 1, 1:

10*x1^3*x2*x3 + 20/3*x1^2*x2^2*x3 + 10*x1*x2^3*x3 + 20/3*x1^2*x2*x3^2 + 20/3*x1*x2^2*x3^2 + 10*x1*x2*x3^3 + 10*x1^3*x2*x4 + 20/3*x1^2*x2^2*x4 + 10*x1*x2^3*x4 + 10*x1^3*x3*x4 + 130/7*x1^2*x2*x3*x4 + 130/7*x1*x2^2*x3*x4 + 10*x2^3*x3*x4 + 20/3*x1^2*x3^2*x4 + 130/7*x1*x2*x3^2*x4 + 20/3*x2^2*x3^2*x4 + 10*x1*x3^3*x4 + 10*x2*x3^3*x4 + 20/3*x1^2*x2*x4^2 + 20/3*x1*x2^2*x4^2 + 20/3*x1^2*x3*x4^2 + 130/7*x1*x2*x3*x4^2 + 20/3*x2^2*x3*x4^2 + 20/3*x1*x3^2*x4^2 + 20/3*x2*x3^2*x4^2 + 10*x1*x2*x4^3 + 10*x1*x3*x4^3 + 10*x2*x3*x4^3

2, 2, 1:

32/3*x1^2*x2^2*x3 + 32/3*x1^2*x2*x3^2 + 32/3*x1*x2^2*x3^2 + 32/3*x1^2*x2^2*x4 + 16*x1^2*x2*x3*x4 + 16*x1*x2^2*x3*x4 + 32/3*x1^2*x3^2*x4 + 16*x1*x2*x3^2*x4 + 32/3*x2^2*x3^2*x4 + 32/3*x1^2*x2*x4^2 + 32/3*x1*x2^2*x4^2 + 32/3*x1^2*x3*x4^2 + 16*x1*x2*x3*x4^2 + 32/3*x2^2*x3*x4^2 + 32/3*x1*x3^2*x4^2 + 32/3*x2*x3^2*x4^2

2, 1, 1, 1:

80/7*x1^2*x2*x3*x4 + 80/7*x1*x2^2*x3*x4 + 80/7*x1*x2*x3^2*x4 + 80/7*x1*x2*x3*x4^2

x1^2 + 2/3*x1*x2 + x2^2 + 2/3*x1*x3 + 2/3*x2*x3 + x3^2 + 2/3*x1*x4 + 2/3*x2*x4 + 2/3*x3*x4 + x4^2 + 2/3*x1*x5 + 2/3*x2*x5 + 2/3*x3*x5 + 2/3*x4*x5 + x5^2

1, 1:

4/3*x1*x2 + 4/3*x1*x3 + 4/3*x2*x3 + 4/3*x1*x4 + 4/3*x2*x4 + 4/3*x3*x4 + 4/3*x1*x5 + 4/3*x2*x5 + 4/3*x3*x5 + 4/3*x4*x5

x1^3 + 3/5*x1^2*x2 + 3/5*x1*x2^2 + x2^3 + 3/5*x1^2*x3 + 2/5*x1*x2*x3 + 3/5*x2^2*x3 + 3/5*x1*x3^2 + 3/5*x2*x3^2 + x3^3 + 3/5*x1^2*x4 + 2/5*x1*x2*x4 + 3/5*x2^2*x4 + 2/5*x1*x3*x4 + 2/5*x2*x3*x4 + 3/5*x3^2*x4 + 3/5*x1*x4^2 + 3/5*x2*x4^2 + 3/5*x3*x4^2 + x4^3 + 3/5*x1^2*x5 + 2/5*x1*x2*x5 + 3/5*x2^2*x5 + 2/5*x1*x3*x5 + 2/5*x2*x3*x5 + 3/5*x3^2*x5 + 2/5*x1*x4*x5 + 2/5*x2*x4*x5 + 2/5*x3*x4*x5 + 3/5*x4^2*x5 + 3/5*x1*x5^2 + 3/5*x2*x5^2 + 3/5*x3*x5^2 + 3/5*x4*x5^2 + x5^3

2, 1:

12/5*x1^2*x2 + 12/5*x1*x2^2 + 12/5*x1^2*x3 + 18/5*x1*x2*x3 + 12/5*x2^2*x3 + 12/5*x1*x3^2 + 12/5*x2*x3^2 + 12/5*x1^2*x4 + 18/5*x1*x2*x4 + 12/5*x2^2*x4 + 18/5*x1*x3*x4 + 18/5*x2*x3*x4 + 12/5*x3^2*x4 + 12/5*x1*x4^2 + 12/5*x2*x4^2 + 12/5*x3*x4^2 + 12/5*x1^2*x5 + 18/5*x1*x2*x5 + 12/5*x2^2*x5 + 18/5*x1*x3*x5 + 18/5*x2*x3*x5 + 12/5*x3^2*x5 + 18/5*x1*x4*x5 + 18/5*x2*x4*x5 + 18/5*x3*x4*x5 + 12/5*x4^2*x5 + 12/5*x1*x5^2 + 12/5*x2*x5^2 + 12/5*x3*x5^2 + 12/5*x4*x5^2

1, 1, 1:

2*x1*x2*x3 + 2*x1*x2*x4 + 2*x1*x3*x4 + 2*x2*x3*x4 + 2*x1*x2*x5 + 2*x1*x3*x5 + 2*x2*x3*x5 + 2*x1*x4*x5 + 2*x2*x4*x5 + 2*x3*x4*x5

x1^4 + 4/7*x1^3*x2 + 18/35*x1^2*x2^2 + 4/7*x1*x2^3 + x2^4 + 4/7*x1^3*x3 + 12/35*x1^2*x2*x3 + 12/35*x1*x2^2*x3 + 4/7*x2^3*x3 + 18/35*x1^2*x3^2 + 12/35*x1*x2*x3^2 + 18/35*x2^2*x3^2 + 4/7*x1*x3^3 + 4/7*x2*x3^3 + x3^4 + 4/7*x1^3*x4 + 12/35*x1^2*x2*x4 + 12/35*x1*x2^2*x4 + 4/7*x2^3*x4 + 12/35*x1^2*x3*x4 + 8/35*x1*x2*x3*x4 + 12/35*x2^2*x3*x4 + 12/35*x1*x3^2*x4 + 12/35*x2*x3^2*x4 + 4/7*x3^3*x4 + 18/35*x1^2*x4^2 + 12/35*x1*x2*x4^2 + 18/35*x2^2*x4^2 + 12/35*x1*x3*x4^2 + 12/35*x2*x3*x4^2 + 18/35*x3^2*x4^2 + 4/7*x1*x4^3 + 4/7*x2*x4^3 + 4/7*x3*x4^3 + x4^4 + 4/7*x1^3*x5 + 12/35*x1^2*x2*x5 + 12/35*x1*x2^2*x5 + 4/7*x2^3*x5 + 12/35*x1^2*x3*x5 + 8/35*x1*x2*x3*x5 + 12/35*x2^2*x3*x5 + 12/35*x1*x3^2*x5 + 12/35*x2*x3^2*x5 + 4/7*x3^3*x5 + 12/35*x1^2*x4*x5 + 8/35*x1*x2*x4*x5 + 12/35*x2^2*x4*x5 + 8/35*x1*x3*x4*x5 + 8/35*x2*x3*x4*x5 + 12/35*x3^2*x4*x5 + 12/35*x1*x4^2*x5 + 12/35*x2*x4^2*x5 + 12/35*x3*x4^2*x5 + 4/7*x4^3*x5 + 18/35*x1^2*x5^2 + 12/35*x1*x2*x5^2 + 18/35*x2^2*x5^2 + 12/35*x1*x3*x5^2 + 12/35*x2*x3*x5^2 + 18/35*x3^2*x5^2 + 12/35*x1*x4*x5^2 + 12/35*x2*x4*x5^2 + 12/35*x3*x4*x5^2 + 18/35*x4^2*x5^2 + 4/7*x1*x5^3 + 4/7*x2*x5^3 + 4/7*x3*x5^3 + 4/7*x4*x5^3 + x5^4

3, 1:

24/7*x1^3*x2 + 16/7*x1^2*x2^2 + 24/7*x1*x2^3 + 24/7*x1^3*x3 + 88/21*x1^2*x2*x3 + 88/21*x1*x2^2*x3 + 24/7*x2^3*x3 + 16/7*x1^2*x3^2 + 88/21*x1*x2*x3^2 + 16/7*x2^2*x3^2 + 24/7*x1*x3^3 + 24/7*x2*x3^3 + 24/7*x1^3*x4 + 88/21*x1^2*x2*x4 + 88/21*x1*x2^2*x4 + 24/7*x2^3*x4 + 88/21*x1^2*x3*x4 + 32/7*x1*x2*x3*x4 + 88/21*x2^2*x3*x4 + 88/21*x1*x3^2*x4 + 88/21*x2*x3^2*x4 + 24/7*x3^3*x4 + 16/7*x1^2*x4^2 + 88/21*x1*x2*x4^2 + 16/7*x2^2*x4^2 + 88/21*x1*x3*x4^2 + 88/21*x2*x3*x4^2 + 16/7*x3^2*x4^2 + 24/7*x1*x4^3 + 24/7*x2*x4^3 + 24/7*x3*x4^3 + 24/7*x1^3*x5 + 88/21*x1^2*x2*x5 + 88/21*x1*x2^2*x5 + 24/7*x2^3*x5 + 88/21*x1^2*x3*x5 + 32/7*x1*x2*x3*x5 + 88/21*x2^2*x3*x5 + 88/21*x1*x3^2*x5 + 88/21*x2*x3^2*x5 + 24/7*x3^3*x5 + 88/21*x1^2*x4*x5 + 32/7*x1*x2*x4*x5 + 88/21*x2^2*x4*x5 + 32/7*x1*x3*x4*x5 + 32/7*x2*x3*x4*x5 + 88/21*x3^2*x4*x5 + 88/21*x1*x4^2*x5 + 88/21*x2*x4^2*x5 + 88/21*x3*x4^2*x5 + 24/7*x4^3*x5 + 16/7*x1^2*x5^2 + 88/21*x1*x2*x5^2 + 16/7*x2^2*x5^2 + 88/21*x1*x3*x5^2 + 88/21*x2*x3*x5^2 + 16/7*x3^2*x5^2 + 88/21*x1*x4*x5^2 + 88/21*x2*x4*x5^2 + 88/21*x3*x4*x5^2 + 16/7*x4^2*x5^2 + 24/7*x1*x5^3 + 24/7*x2*x5^3 + 24/7*x3*x5^3 + 24/7*x4*x5^3

2, 2:

16/5*x1^2*x2^2 + 32/15*x1^2*x2*x3 + 32/15*x1*x2^2*x3 + 16/5*x1^2*x3^2 + 32/15*x1*x2*x3^2 + 16/5*x2^2*x3^2 + 32/15*x1^2*x2*x4 + 32/15*x1*x2^2*x4 + 32/15*x1^2*x3*x4 + 16/5*x1*x2*x3*x4 + 32/15*x2^2*x3*x4 + 32/15*x1*x3^2*x4 + 32/15*x2*x3^2*x4 + 16/5*x1^2*x4^2 + 32/15*x1*x2*x4^2 + 16/5*x2^2*x4^2 + 32/15*x1*x3*x4^2 + 32/15*x2*x3*x4^2 + 16/5*x3^2*x4^2 + 32/15*x1^2*x2*x5 + 32/15*x1*x2^2*x5 + 32/15*x1^2*x3*x5 + 16/5*x1*x2*x3*x5 + 32/15*x2^2*x3*x5 + 32/15*x1*x3^2*x5 + 32/15*x2*x3^2*x5 + 32/15*x1^2*x4*x5 + 16/5*x1*x2*x4*x5 + 32/15*x2^2*x4*x5 + 16/5*x1*x3*x4*x5 + 16/5*x2*x3*x4*x5 + 32/15*x3^2*x4*x5 + 32/15*x1*x4^2*x5 + 32/15*x2*x4^2*x5 + 32/15*x3*x4^2*x5 + 16/5*x1^2*x5^2 + 32/15*x1*x2*x5^2 + 16/5*x2^2*x5^2 + 32/15*x1*x3*x5^2 + 32/15*x2*x3*x5^2 + 16/5*x3^2*x5^2 + 32/15*x1*x4*x5^2 + 32/15*x2*x4*x5^2 + 32/15*x3*x4*x5^2 + 16/5*x4^2*x5^2

2, 1, 1:

16/3*x1^2*x2*x3 + 16/3*x1*x2^2*x3 + 16/3*x1*x2*x3^2 + 16/3*x1^2*x2*x4 + 16/3*x1*x2^2*x4 + 16/3*x1^2*x3*x4 + 64/5*x1*x2*x3*x4 + 16/3*x2^2*x3*x4 + 16/3*x1*x3^2*x4 + 16/3*x2*x3^2*x4 + 16/3*x1*x2*x4^2 + 16/3*x1*x3*x4^2 + 16/3*x2*x3*x4^2 + 16/3*x1^2*x2*x5 + 16/3*x1*x2^2*x5 + 16/3*x1^2*x3*x5 + 64/5*x1*x2*x3*x5 + 16/3*x2^2*x3*x5 + 16/3*x1*x3^2*x5 + 16/3*x2*x3^2*x5 + 16/3*x1^2*x4*x5 + 64/5*x1*x2*x4*x5 + 16/3*x2^2*x4*x5 + 64/5*x1*x3*x4*x5 + 64/5*x2*x3*x4*x5 + 16/3*x3^2*x4*x5 + 16/3*x1*x4^2*x5 + 16/3*x2*x4^2*x5 + 16/3*x3*x4^2*x5 + 16/3*x1*x2*x5^2 + 16/3*x1*x3*x5^2 + 16/3*x2*x3*x5^2 + 16/3*x1*x4*x5^2 + 16/3*x2*x4*x5^2 + 16/3*x3*x4*x5^2

1, 1, 1, 1:

16/5*x1*x2*x3*x4 + 16/5*x1*x2*x3*x5 + 16/5*x1*x2*x4*x5 + 16/5*x1*x3*x4*x5 + 16/5*x2*x3*x4*x5

Definition

The real zonal polynomial $C_\lambda(x_1,\ldots,x_n)$ is $Z_\lambda(\operatorname{diag}(x_1,\ldots,x_n))$, where $Z_\lambda$ is defined by the orthogonal-group integral and identity-matrix normalization in [3], equations (35.4.2) and (35.4.3).

Parameters

$n$

— number of alphabet variables (a positive integer)

$\lambda$

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

Formulas

(1)

$C_\lambda(x)=\dfrac{2^d d!}{j_\lambda}J_\lambda(x;2)$, where $J_\lambda$ is the integral Jack polynomial and $j_\lambda=\prod_{s\in\lambda}(2a(s)+l(s)+1)(2a(s)+l(s)+2)$ [2].

(2)

$\displaystyle\sum_{\lambda\vdash d,\,\ell(\lambda)\leq n}C_\lambda(x_1,\ldots,x_n)=(x_1+\cdots+x_n)^d$ for $d\geq0$ [3].

(3)

$\mathcal D_n C_\lambda=\left(\sum_{i=1}^{\ell}\lambda_i(\lambda_i+n-i-1)\right)C_\lambda$, where $\mathcal D_n=\sum_i x_i^2\partial_{x_i}^2+\sum_{i<j}\dfrac{x_i^2\partial_{x_i}-x_j^2\partial_{x_j}}{x_i-x_j}$. Each apparent denominator cancels when the operator acts on a symmetric polynomial [1].

(4)

$C_{(2)}=(p_1^2+2p_2)/3$ and $C_{(1,1)}=2(p_1^2-p_2)/3$, where $p_r=\sum_i x_i^r$ is the power sum symmetric polynomial.

(5)

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

(6)

$C_\varnothing=1$, $C_{(1)}(x_1,\ldots,x_n)=x_1+\cdots+x_n$, and $C_{(d)}(x_1)=x_1^d$ for $d\geq1$.

(7)

$C_\lambda(x_1,\ldots,x_{n-1},0)=C_\lambda(x_1,\ldots,x_{n-1})$ when $\ell(\lambda)<n$; 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 $d=\sum_i\lambda_i$ and length $\ell\leq n$. The conjugate partition is $\lambda_j^\prime=\#\{i:\lambda_i\geq j\}$. For a box $s=(i,j)$, with rows and columns indexed from $1$, its arm and leg lengths are $a(s)=\lambda_i-j$ and $l(s)=\lambda_j^\prime-i$.

(9)

The alphabet variables $x_1,\ldots,x_n$ may be regarded as the eigenvalues of a real symmetric matrix. The notation $C_\lambda$ here agrees with $Z_\lambda$ in [3]; SageMath calls the monic Jack $P$ specialization at parameter $2$ its zonal basis [4], which generally has a different scalar normalization.

(10)

Each entry is a symmetric homogeneous polynomial of degree $d$ in the alphabet variables. For fixed $n$ and $d$, the polynomials indexed by the partitions of $d$ of length at most $n$ form a basis of the symmetric homogeneous polynomials over $\mathbb Q$ [1].

Programs

(P1)

Sage

import numberdb.sage as numberdb
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
from math import factorial

# One entry in C normalization, with n alphabet variables.
n, lam = 3, (3, 2)
J = SymmetricFunctions(QQ).jack(t=QQ(2)).J()
j = QQ(1)
for i, row in enumerate(lam, 1):
    for k in range(1, row + 1):
        arm = row - k
        leg = sum(a >= k for a in lam) - i
        j *= (2*arm + leg + 1)*(2*arm + leg + 2)
R = PolynomialRing(QQ, n, names=['x%d' % i for i in range(1, n+1)])
f = R(str(J[list(lam)].expand(n, alphabet=list(R.variable_names()))))
print(QQ(2**sum(lam)*factorial(sum(lam)))/j*f)

References

[1]

L. Jiu and C. Koutschan, Calculation and Properties of Zonal Polynomials, Mathematics in Computer Science 14 (2020), 623–640, Sections 2–3, Example 3.5 and Remark 8.1. (arXiv) (doi)

[2]

P. Koev and A. Edelman, The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument, preprint (2005), Section 1 and equations (3.1)–(3.2). (arXiv)

Links

[3]

DLMF: Partitions and zonal polynomials

[4]

SageMath: Jack symmetric functions

Similar tables

Jack polynomials — give the real zonal family at parameter $2$ after the hook-product rescaling

monomial symmetric polynomials — give the triangular basis used to construct zonal polynomials

Schur polynomials — are the monic Jack specialization at parameter $1$, while real zonal polynomials use parameter $2$

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it holds every partition with $\ell(\lambda)\leq n$ and $2\leq|\lambda|\leq6$ for $n=2,3$, $2\leq|\lambda|\leq5$ for $n=4$, and $2\leq|\lambda|\leq4$ for $n=5$, giving 61 entries. These complete degree slices cover rows, columns and interior shapes across small matrix dimensions; the noncanonical bounds are chosen for readability, with a longest entry of 1157 characters. The next omitted slices for $n=4,5$ reach 1705 and 2254 characters respectively. Degrees 0 and 1 and the one-variable case are given in (6); the attached argument-selection.md records the coverage rationale)

How they were obtained:

All coefficients are exact rational numbers. The generator constructs monic triangular eigenpolynomials of $\mathcal D_n$ in the monomial symmetric basis, then decomposes $(x_1+\cdots+x_n)^d$ in the resulting eigenbasis to fix the $C$ normalization. No floating-point arithmetic is used.

Every polynomial is independently compared coefficient by coefficient with SageMath's integral Jack $J$ implementation and the hook-product conversion [2]. The checks also compare every evaluation at $(1,\ldots,1)$ with DLMF (35.4.2), every two-variable polynomial with the Haar integral (35.4.3) evaluated using exact trigonometric moments, and every degree-four polynomial with the coefficient table in [1], Example 3.5. Known quadratic values and elementary trigonometric moments serve as controls. Symmetry, homogeneity, the differential eigenidentity, trace-power sums, column formulas, boundary values and alphabet stability are checked before submission and on the entries read back from the database.