Harmonic polynomial of Cartan type in 3 variables.

dimension of harmonic polynomial of degree 1 = 3 :

x_{1} x_{2} x_{3}

dimension of harmonic polynomial of degree 2 = 3 :

x_{1}x_{2} x_{1}x_{3} x_{2}x_{3}

dimension of harmonic polynomial of degree 3 = 4 :

(x_{1} − x_{2}) x_{1}x_{2} (x_{1} − x_{3}) x_{1}x_{3} (x_{2} − x_{3}) x_{2}x_{3} x_{1}x_{2}x_{3}

dimension of harmonic polynomial of degree 4 = 2 :

(x_{1} − x_{2}) x_{1}x_{2}x_{3} (x_{1} − x_{3}) x_{1}x_{2}x_{3}

dimension of harmonic polynomial of degree 5 = 2 :

(x_{1} − x_{2}) (x_{1} + x_{2} − 3 x_{3}) x_{1}x_{2}x_{3} (x_{1} − x_{3}) (x_{1} − 3 x_{2} + x_{3}) x_{1}x_{2}x_{3}

dimension of harmonic polynomial of degree 6 = 1 :

(x_{2} − x_{3}) (x_{1} − x_{3}) (x_{1} − x_{2}) x_{1}x_{2}x_{3} Harmonic polynomial of Cartan type in 4 variables.

dimension of harmonic polynomial of degree 1 = 4 :

x_{1} x_{2} x_{3} x_{4}

dimension of harmonic polynomial of degree 2 = 6 :

x_{3}x_{4} x_{1}x_{3} x_{1}x_{2} x_{2}x_{3} x_{1}x_{4} x_{2}x_{4}

dimension of harmonic polynomial of degree 3 = 10 :

(x_{1} − x_{2}) x_{1}x_{2} (x_{1} − x_{3}) x_{1}x_{3} (x_{2} − x_{3}) x_{2}x_{3} (x_{1} − x_{4}) x_{1}x_{4} (x_{2} − x_{4}) x_{2}x_{4} (x_{3} − x_{4}) x_{3}x_{4} x_{1}x_{2}x_{3} x_{1}x_{2}x_{4} x_{1}x_{3}x_{4} x_{2}x_{3}x_{4}

dimension of harmonic polynomial of degree 4 = 9 :

(x_{2} − x_{4}) x_{2}x_{3}x_{4} (x_{1} − x_{2}) x_{1}x_{2}x_{3} (x_{1} − x_{2}) x_{1}x_{2}x_{4} (x_{1} − x_{3}) x_{1}x_{3}x_{4} (x_{2} − x_{3}) x_{2}x_{3}x_{4} (x_{1} − x_{4}) x_{1}x_{2}x_{4} (x_{1} − x_{3}) x_{1}x_{2}x_{3} (x_{1} − x_{4}) x_{1}x_{3}x_{4} x_{1}x_{2}x_{3}x_{4} dimension of harmonic polynomial of degree 5 = 11 :

(x_{1} − x_{2}) x_{1}x_{2}x_{3}x_{4} (x_{1} − x_{3}) x_{1}x_{2}x_{3}x_{4} (x_{1} − x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{1} − x_{3}) (x_{1} + x_{3} − 3 x_{4}) x_{1}x_{3}x_{4} (x_{2} − x_{3}) (x_{2} + x_{3} − 3 x_{4}) x_{2}x_{3}x_{4} (x_{1} − x_{4}) (x_{1} − 3 x_{3} + x_{4}) x_{1}x_{3}x_{4} (x_{2} − x_{4}) (x_{2} − 3 x_{3} + x_{4}) x_{2}x_{3}x_{4} (x_{1} − x_{4}) (x_{1} − 3 x_{2} + x_{4}) x_{1}x_{2}x_{4} (x_{1} − x_{2}) (x_{1} + x_{2} − 3 x_{3}) x_{1}x_{2}x_{3} (x_{1} − x_{3}) (x_{1} − 3 x_{2} + x_{3}) x_{1}x_{2}x_{3} (x_{1} − x_{2}) (x_{1} + x_{2} − 3 x_{4}) x_{1}x_{2}x_{4}

dimension of harmonic polynomial of degree 6 = 9 :

(x_{2} − x_{3}) (x_{1} − x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{2} − x_{4}) (x_{1} − x_{3}) x_{1}x_{2}x_{3}x_{4} (x_{1} − x_{4}) (x_{1} − 3 x_{2} + x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{2} − x_{4}) (x_{2} − 3 x_{3} + x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{3} − x_{4}) (3 x_{2} − x_{3} − x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{2} − x_{3}) (x_{1} − x_{3}) (x_{1} − x_{2}) x_{1}x_{2}x_{3} (x_{3} − x_{4}) (x_{2} − x_{4}) (x_{2} − x_{3}) x_{2}x_{3}x_{4} (x_{2} − x_{4}) (x_{1} − x_{4}) (x_{1} − x_{2}) x_{1}x_{2}x_{4} (x_{3} − x_{4}) (x_{1} − x_{4}) (x_{1} − x_{3}) x_{1}x_{3}x_{4}

dimension of harmonic polynomial of degree 7 = 6 :

(x_{2} − x_{4}) (3 x_{1} − x_{2} − x_{4}) (x_{1} + x_{2} − 5 x_{3} + x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{1} − x_{4}) (x_{1} − 3 x_{2} + x_{4}) (x_{1} + x_{2} − 5 x_{3} + x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{3} − x_{4}) (3 x_{1} − x_{3} − x_{4}) (x_{1} − 5 x_{2} + x_{3} + x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{2} − x_{4}) (x_{1} − x_{4}) (x_{1} − x_{2}) x_{1}x_{2}x_{3}x_{4} (x_{3} − x_{4}) (x_{1} − x_{4}) (x_{1} − x_{3}) x_{1}x_{2}x_{3}x_{4} (x_{3} − x_{4}) (x_{2} − x_{4}) (x_{2} − x_{3}) x_{1}x_{2}x_{3}x_{4}

dimension of harmonic polynomial of degree 8 = 5 :

(x_{2} − x_{3}) (x_{1} − x_{3}) (x_{1} − x_{2}) (x_{1} + x_{2} + x_{3} − 6 x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{3} − x_{4}) (x_{2} − x_{4}) (x_{2} − x_{3}) (6 x_{1} − x_{2} − x_{3} − x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{3} − x_{4}) (x_{1} − x_{4}) (x_{1} − x_{3}) (x_{1} − 6 x_{2} + x_{3} + x_{4}) x_{1}x_{2}x_{3}x_{4} (x_{2} − x_{4}) (x_{1} − x_{3}) (5 x_{1}x_{2} − 15 x_{1}x_{3} + 5 x_{1}x_{4} − 3 x_{2}^{2} + 5 x_{2}x_{3} + 5 x_{3}x_{4} − 3 x_{4}^{2}) x_{1}x_{2}x_{3}x_{4} (x_{2} − x_{3}) (x_{1} − x_{4}) (5 x_{1}x_{2} + 5 x_{1}x_{3} − 15 x_{1}x_{4} − 3 x_{2}^{2} + 5 x_{2}x_{4} − 3 x_{3}^{2} + 5 x_{3}x_{4}) x_{1}x_{2}x_{3}x_{4} dimension of harmonic polynomial of degree 9 = 3 :

(x_{2} − x_{3}) (x_{1} − x_{3}) (x_{1} − x_{2}) (x_{1}^{2} − 14 x_{1}x_{2} − 14 x_{1}x_{3} + 35 x_{1}x_{4} + x_{2}^{2} − 14 x_{2}x_{3}

+35 x_{2}x_{4} + x_{3}^{2} + 35 x_{3}x_{4} − 70 x_{4}^{2}) x_{1}x_{2}x_{3}x_{4}

(x_{3} − x_{4}) (x_{1} − x_{4}) (x_{1} − x_{3}) (x_{1}^{2} + 35 x_{1}x_{2} − 14 x_{1}x_{3} − 14 x_{1}x_{4} − 70 x_{2}^{2} + 35 x_{2}x_{3}

+35 x_{2}x_{4} + x_{3}^{2} − 14 x_{3}x_{4} + x_{4}^{2}) x_{1}x_{2}x_{3}x_{4}

(x_{3} − x_{4}) (x_{2} − x_{4}) (x_{2} − x_{3}) (70 x_{1}^{2} − 35 x_{1}x_{2} − 35 x_{1}x_{3} − 35 x_{1}x_{4} − x_{2}^{2}

+14 x_{2}x_{3} + 14 x_{2}x_{4} − x_{3}^{2} + 14 x_{3}x_{4} − x_{4}^{2}) x_{1}x_{2}x_{3}x_{4}

dimension of harmonic polynomial of degree 10 = 1 : (x_{3} − x_{4}) (x_{2} − x_{4}) (x_{2} − x_{3}) (x_{1} − x_{4}) (x_{1} − x_{3}) (x_{1} − x_{2}) x_{1}x_{2}x_{3}x_{4}