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Commutative combinatorial Hopf algebras. (English) Zbl 1181.16031
Let \(K\) be a field of characteristic zero, \(R=K[x_{ij}]\) for \(i,j\) bigger than or equal to one, \(J\) the ideal of \(R\) generated by all \(x_{ij}x_{ik}\). For \(f\) an endofunction of \([n]=\{1,2,\dots,n\}\), i.e., \(f\) maps \([n]\) to \([n]\), a homogeneous polynomial \(M_f\) of degree \(n\) in the \(x_{ij}\) is defined. The \(M_f\), as \(n\) and \(f\) vary, span a subalgebra EQSym of \(R/J\).
The product \(M_fM_g\) is a linear combination of the \(M_h\) with integral coefficients defined in terms of shifted concatenations, which leads to a product of \(f\) and \(g\). This in turn leads to a comultiplication on each \(M_h\), so that EQSym is a commutative noncocommutative graded Hopf algebra. Its graded dual ESym is free over those elements \(S^f\) of the dual basis to the \(M_f\) for which \(f\) is connected, i.e., \(f\) is not a nontrivial shifted concatenation. If one only uses the \(M_s\) for \(s\) a permutation of some \([n]\), then one gets a Hopf subalgebra SQSym of EQSym, whose graded dual SSym is free over the \(S_s\) for \(s\) connected.
It turns out that SSym is isomorphic to the Grossman-Larson Hopf algebra of heap ordered trees [R. Grossman and R. G. Larson, J. Algebra 126, No. 1, 184-210 (1989; Zbl 0717.16029)] and also to the Hopf algebra of permutations of F. Patras and C. Reutenauer [Mosc. Math. J. 4, No. 1, 199-216 (2004; Zbl 1103.16026)]. Some subalgebras of SQSym are discussed, including symmetric functions in noncommuting variables (dual), quasi-symmetric functions, and ordinary symmetric functions. SQSym is noncommutative as an algebra. It can be realized as a Hopf algebra within \(K\langle a_{ij}\rangle\), polynomials in noncommuting \(a_{ij}\) for \(i,j\) bigger than or equal to one, whose operations can be described in terms of the cycle structure of permutations. The paper concludes with some attempts to relate to the spirit of quantum groups.

MSC:
16T30 Connections of Hopf algebras with combinatorics
05E05 Symmetric functions and generalizations
05C05 Trees
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