 Clebsch-Gordan splitting of a defining module and its cubic symmetric power

ID: clebsch-gordan-splitting-of-a-defining-module-and-its-cubic-symmetric-power

For the defining sl2 Lie algebra module with basis

x, y

, put

A_{k} = x \otimes x^{3 - k} y^{k}

and

B_{k} = y \otimes x^{3 - k} y^{k}

e A_{k} = k A_{k - 1}

and

e B_{k} = A_{k} + k B_{k - 1}

f A_{k} = B_{k} + (3 - k) A_{k + 1}

and

f B_{k} = (3 - k) B_{k + 1}

A_{0}

and

B_{0} - A_{1}

generate irreducible submodules of highest weights four and two. Bases are respectively

(A_{0}, 3 A_{1} + B_{0}, A_{2} + B_{1}, A_{3} + 3 B_{2}, B_{3})

and

(B_{0} - A_{1}, B_{1} - A_{2}, B_{2} - A_{3})

. Their vectors are independent in every weight space, proving the displayed direct sum without only comparing dimensions.

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