Proposition. A positive coefficient expansion on SU(2) [fdip-0082]
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  • November 1, 2026

Let \(f\) be the defining two-dimensional representation of \(SU(2)\) and \(\beta >0\). Then \[k(g)=e^{\beta \chi _f(g)}\] satisfies the transfer kernel conditions. More precisely, if \(m_\rho (n)\) is the multiplicity of \(\rho \) in \(f^{\otimes n}\), then \[\begin {aligned} c_\rho &=\sum _{n=0}^\infty \frac {\beta ^n}{n!}m_\rho (n)>0,\\ a_\rho &=\frac {c_\rho }{d_\rho },\qquad \sum _\rho d_\rho ^2a_\rho =e^{2\beta }. \end {aligned}\] This is the tensor-power argument used in [luscher1977transfer, Proposition 1, p. 288], with strict occurrence and absolute convergence justified below.

Proof. 

  • November 1, 2026

The matrix description of SU(2) writes its elements as \[g=\begin {pmatrix}z&w\\-\overline w&\overline z\end {pmatrix}, \qquad |z|^2+|w|^2=1.\] Hence \(\chi _f(g)=2\operatorname {Re}z\) is real, \(\chi _f(g^{-1})=\chi _f(g)\), and \(k\) is continuous, central and strictly positive. Every tensor power is unitary; invariant orthogonal complements decompose it into irreducibles. Therefore \[\chi _f(g)^n=\sum _\rho m_\rho (n)\chi _\rho (g), \qquad \sum _\rho m_\rho (n)d_\rho =2^n.\] Expanding the exponential and using nonnegative multiplicities gives \[\sum _\rho d_\rho c_\rho =\sum _{n=0}^\infty \frac {\beta ^n2^n}{n!}=e^{2\beta }.\] Since \(|\chi _\rho (g)|\le d_\rho \), this also justifies rearranging the series uniformly and proves the claimed expansion. It remains to show that no \(c_\rho \) is zero.

Consider the algebra of finite linear combinations of matrix coefficients of the powers \(f^{\otimes n}\), including \(n=0\). Products are coefficients of tensor products, so this is a unital algebra of continuous functions. The entries of \(g\) distinguish its points. Their complex conjugates are again entries up to a sign, by the displayed matrix form, so the algebra is closed under complex conjugation. The complex Stone–Weierstrass theorem makes it uniformly dense in \(C(SU(2),\mathbb C)\). Here that analytic theorem is used as a cited statement: a conjugation-closed unital subalgebra of continuous complex functions on a compact space is dense if it separates points. Its exact formal statement is ContinuousMap.starSubalgebra_topologicalClosure_eq_top_of_separatesPoints. The reference supplies this density theorem, not the representation argument that follows.

If an irreducible \(\rho \) occurred in no tensor power, Schur orthogonality from the Peter–Weyl theorem would make all its matrix coefficients orthogonal to this algebra. Choose a nonzero such coefficient \(u\). Uniform density gives algebra functions converging uniformly, hence in normalized Haar \(L^2\), to \(u\). Orthogonality would then give \(\|u\|_2^2=0\), a contradiction. Thus \(m_\rho (n)>0\) for some \(n\), and \(\beta >0\) makes \(c_\rho >0\). Dividing by \(d_\rho \) proves all remaining coefficient assertions.