Foulkes characters, Eulerian idempotents, and an amazing matrix

Authors:
Persi Diaconis;Jason Fulman
Affiliations:
Department of Mathematics, Stanford University, Stanford, USA;Department of Mathematics, University of Southern California, Los Angeles, USA
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2012

Citing 5
Cited 1

Enumerative combinatorics

Enumerative combinatorics
Ribbon Schur functions

European Journal of Combinatorics
Counting permutations with given cycle structure and descent set

Journal of Combinatorial Theory Series A
Applications of Symmetric Functions to Cycle and Increasing Subsequence Structure after Shuffles

Journal of Algebraic Combinatorics: An International Journal
Alternating permutations and symmetric functions

Journal of Combinatorial Theory Series A

Block characters of the symmetric groups

Journal of Algebraic Combinatorics: An International Journal

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Abstract

John Holte (Am. Math. Mon. 104:138---149, 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. 42:545---556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788---803, 2009; Adv. Appl. Math. 43:176---196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788---803, 2009; Adv. Appl. Math. 43:176---196, 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.