The \((1-\mathbb{E})\)-transform in combinatorial Hopf algebras
Let sym be the commutative algebra of symmetric functions in countably infinite many variables \(x_1, x_2, \dots,\) \textbf{Sym} the non-commutative symmetric functions, \textbf{FQSym} the free quasi-symmetric functions, \textbf{WQSym} the word quasi-symmetric functions and \textbf{PQSym} the parking quasi-symmetric functions. The authors extend to these non-commutative combinatorial Hopf algebras the endomorphism of sym which sends the first power-sum to zero and leaves the other power-sums invariant. This is a ``transformation of alphabets'', the (\(1-\mathbb E\))-transform, where \(\mathbb E\) is the ``exponential alphabet''. See sections 1 and 2 of the paper for an explanation of this terminology -- it is too technical to explain here. The authors define the small derangement algebra \(\mathcal D^{(0)}=\mathbf{Sym}^{\sharp}\), a Hopf subalgebra of \textbf{Sym}, by imitating the construction of the peak algebra of \textit{N. Bergeron}, \textit{F. Hivert} and \textit{J.-Y. Thibon} [``The peak algebra and the Hecke-Clifford algebras at \(q=0\),'' J. Comb. Theory, Ser. A 107, No.\,1, 1--19 (2004; Zbl 1107.05092)]. This recovers \textit{M. Schocker}'s idempotents for derangement numbers [``Idempotents for derangement numbers,'' Discrete Math. 269, No.\,1--3, 239--248 (2003; Zbl 1023.05010)]. This leads to a construction of subalgebras of the descent algebra analogous to the peak algebra, and the authors study their representation theory. The case of \textbf{WQSym} leads to similar subalgebras of the Solomon-Tits algebras. For \textbf{FQSym}, the study of the transformatiom comes down to a solution of the Tsetlin library in the uniform case. \textbf{PQSym} has an internal product extending that of \(\mathbf{WQSym}^*\). Nothing new arises for \textbf{PQSym}, nor for the Catalan algebra \textbf{CQSym} [see \textit{J.-C. Novelli} and \textit{J.-Y. Thibon}, ``Hopf algebras and dendriform structures arising from parking functions,'' Fundam. Math. 193, No.\,3, 189--241 (2007; Zbl 1127.16033)].
scientific article; zbMATH DE number 5852823
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