Foulkes characters, Eulerian idempotents, and an amazing matrix
John Holte (Am. Math. Mon. 104:138149, 1997) introduced a family of amazing matrices which give the transition probabilities of carries when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra (Brenti and Welker, Adv. Appl. Math. 42:545556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788803, 2009; Adv. Appl. Math. 43:176196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788803, 2009; Adv. Appl. Math. 43:176196, 2009). We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday (Cyclic Homology, 1992) in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.
foulkes character eulerian idempotent carry symmetric group