Let $\mathcal{U}(N)$ be the unitary group.

It is well known that $$ \int_{\mathcal{U}(N)} U_{ij} U^\dagger_{nm} \,dU=\delta_{im}\delta_{jn}\frac{1}{N},$$ where $dU$ is the Haar measure.

More complicated averages can also be computed, such as $$\int_{\mathcal{U}(N)} |U_{11}|^2|U_{12}|^2 \,dU = \frac{1}{N(N+1)}.$$

Now, let $R_\lambda$ be an irreducible representation of $\mathcal{U}(N)$ which is different from the fundamental one. Then, the main orthogonality still holds, $$ \int_{\mathcal{U}(N)} [R_\lambda(U)]_{ij}[R_\lambda(U^\dagger)]_{nm} \, dU = \delta_{im} \delta_{jn} \frac{1}{d_\lambda(N)},$$ with the denominator replaced by the dimension of the irrep.

My question is: are there calculations of such integrals involving more matrix elements? Like $$\int_{\mathcal{U}(N)} \Bigl|[R_\lambda(U)]_{11}\Bigr|^2\Bigl|[R_\lambda(U)]_{12}\Bigr|^2 dU=\text{?}$$

  • $\begingroup$ We have the Schur orthogonality relations that say $\int_G \langle\pi_1(g)v_1, w_1\rangle\overline{\langle\pi_2(g)v_2, w_2\rangle}\mathrm dg = \begin{cases} 0, & \pi_1 \simeq \pi_2 \\ \langle v_1, v_2\rangle\overline{\langle w_1, w_2\rangle}, & \pi_1 = \pi_2. \end{cases}$. If I understand your notation correctly, this answers your question. $\endgroup$ – LSpice Feb 8 '19 at 2:53
  • $\begingroup$ @LSpice No, your integral involves only two matrix elements. I ask for a generalization. $\endgroup$ – Marcel Feb 8 '19 at 15:50

One can reduce your more complicated example which is of the form $$ \int_{U(n)} \langle a, R_{\lambda}(U)b\rangle \langle c, R_{\lambda}(U)d\rangle \times ({\rm complex\ conjugate}) \ dU $$ for some vectors $a,b,c,d$ to the simpler case with two matrix elements as follows.

Use $$ \langle a, R_{\lambda}(U)b\rangle \langle c, R_{\lambda}(U)d\rangle = \langle a\otimes c, (R_{\lambda}(U)\otimes R_{\lambda}(U)) b\otimes d\rangle\ , $$ and then the decomposition into irreducibles of $R_{\lambda}\otimes R_{\lambda}$.

The catch though is that this requires not only the Littlewood-Richardson rule (which says what irreducibles occur and with what multiplicity), but a very explicit form of it which involves the Clebsch-Gordan coefficients (or analogues of Wigner's $3jm$ symbols) for $U(n)$. As far as I know, there are no good formulas for them in full generality. Some references: this article by Elvang, Cvitanović and Kennedy or the multivolume series on "Representation of Lie Groups and Special Functions" by Vilenkin and Klimyk.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.