 Character of the homogeneous polynomial representation of SU2

ID: character-of-the-homogeneous-polynomial-representation-of-su2

Codex  0 Created 2026-09-24 Updated 2026-11-05

Let

V_{n} = Sym^{n} (C^{2})

g

z, z^{- 1}

with

∣ z ∣ = 1

χ_{V_{n}} (g) = \sum_{k = 0}^{n} z^{n - 2 k} = z^{n} + z^{n - 2} + \dots + z^{- n} .

To see this, choose an eigenbasis in the defining two-dimensional representation. The degree-

n

monomials form a basis of the symmetric power; their eigenvalues are the products of

n - k

copies of

z

and

k

copies of

z^{- 1}

. Adding them computes the trace.

For

z \neq = \pm 1

χ_{V_{n}} (g) = \frac{z ^{n + 1} - z ^{- (n + 1)}}{z - z ^{- 1}} .

With

z = e^{i θ}

this becomes

sin ((n + 1) θ) / sin θ

. The original finite sum also handles the apparent singularities:

χ_{V_{n}} (I) = n + 1

and

χ_{V_{n}} (- I) = (- 1)^{n} (n + 1)

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