[NT seminar] A Selberg-type integral from an irrational question

The Hankel determinants built on moment sequences and their connection with transfinite diameter show their usefulness in proving irrationality of meaningful mathematical constants (think of the values of logarithm and $\pi$). They can be also viewed as particular generalisations of the celebrated Selberg integral, a closed-form period evaluation possessing a symmetry of root system. In my talk I will take a detour and discuss a general case of such Selberg-type integrals which a priory has nothing to do with irrationality. What are those creatures?! In our ongoing joint work with Ole Warnaar, we get an answer for a particular (interesting) instance - and it is surprisingly irrational, though a general case remains mysterious.