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.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 arXiv:
 arXiv:1307.4766
 Bibcode:
 2013arXiv1307.4766C
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Combinatorics