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The \((1-\mathbb{E})\)-transform in combinatorial Hopf algebras. (English) Zbl 1228.05290
Let sym be the commutative algebra of symmetric functions in countably infinite many variables \(x_1, x_2, \dots,\) Sym the non-commutative symmetric functions, FQSym the free quasi-symmetric functions, WQSym the word quasi-symmetric functions and 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 Sym, by imitating the construction of the peak algebra of N. Bergeron, F. Hivert and 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 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 WQSym leads to similar subalgebras of the Solomon-Tits algebras. For FQSym, the study of the transformatiom comes down to a solution of the Tsetlin library in the uniform case. PQSym has an internal product extending that of \(\mathbf{WQSym}^*\). Nothing new arises for PQSym, nor for the Catalan algebra CQSym [see J.-C. Novelli and J.-Y. Thibon, “Hopf algebras and dendriform structures arising from parking functions,” Fundam. Math. 193, No. 3, 189–241 (2007; Zbl 1127.16033)].

MSC:
05E05 Symmetric functions and generalizations
16T30 Connections of Hopf algebras with combinatorics
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