Matrix Units in the Symmetric Group Algebra, and Unitary Integration

Abstract: In this paper, we establish an explicit isomorphism between the symmetric group algebra and the path algebra of the Young graph. Specifically, we construct a family of matrix units in the group algebra. As a main application of this construction, we obtain new formulas, alternative to Weingarten calculus, for the integral of a polynomial over the unitary group with respect to the Haar measure. In particular, we obtain a closed formula for the law of moments of the first k rows of the unitary group.