On polynomial integrals over the orthogonal group

Teodor Banica, Benoit Collins, Jean-Marc Schlenker

Abstract: We consider integrals of type ∫Onu11a1⋯u1nanu21b1⋯u2nbndu<math><msub is="true"><mo is="true">∫</mo><msub is="true"><mi is="true">O</mi><mi is="true">n</mi></msub></msub><msubsup is="true"><mi is="true">u</mi><mn is="true">11</mn><msub is="true"><mi is="true">a</mi><mn is="true">1</mn></msub></msubsup><mo is="true">⋯</mo><msubsup is="true"><mi is="true">u</mi><mrow is="true"><mn is="true">1</mn><mi is="true">n</mi></mrow><msub is="true"><mi is="true">a</mi><mi is="true">n</mi></msub></msubsup><msubsup is="true"><mi is="true">u</mi><mn is="true">21</mn><msub is="true"><mi is="true">b</mi><mn is="true">1</mn></msub></msubsup><mo is="true">⋯</mo><msubsup is="true"><mi is="true">u</mi><mrow is="true"><mn is="true">2</mn><mi is="true">n</mi></mrow><msub is="true"><mi is="true">b</mi><mi is="true">n</mi></msub></msubsup><mspace width="0.2em" is="true"></mspace><mi mathvariant="normal" is="true">d</mi><mi is="true">u</mi></math>, with respect to the Haar measure on the orthogonal group. We establish several remarkable invariance properties satisfied by such integrals, by using combinatorial methods. We present as well a general formula for such integrals, as a sum of products of factorials.