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The (1-E)-transform in combinatorial Hopf algebras

Authors :
Hivert, F.
Luque, J. -G.
Novelli, J. -C.
Thibon, J. -Y.
Source :
J. Algebraic Combin. 33 (2011), 277-312
Publication Year :
2009

Abstract

We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a transformation of alphabets, this is the (1-E)-transform, where E is the exponential alphabet, whose elementary symmetric functions are e_n=1/n!. In the case of noncommutative symmetric functions, we recover Schocker's idempotents for derangement numbers [Discr. Math. 269 (2003), 239]. From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon-Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.<br />Comment: 33 pages

Subjects

Subjects :
Mathematics - Combinatorics

Details

Database :
arXiv
Journal :
J. Algebraic Combin. 33 (2011), 277-312
Publication Type :
Report
Accession number :
edsarx.0912.0184
Document Type :
Working Paper