Combinatorial Hopf algebras and \(K\)-homology of Grassmannians.

*(English)*Zbl 1134.16017The authors study six combinatorial Hopf algebras, which fit into a diagram generalizing a diagram describing the relations between four well-known combinatorial Hopf algebras: \(Sym\) of symmetric functions [R. P. Stanley, Enumerative combinatorics. Vol. 2. Cambridge: Cambridge Univ. Press (1999; Zbl 0928.05001)], \(NSym\) of noncommutative symmetric functions [I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh and J.-Y. Thibon, Adv. Math. 112, No. 2, 218–348 (1995; Zbl 0831.05063)], \(QSym\) of quasisymmetric functions [I. M. Gessel, Contemp. Math. 34, 289–301 (1984; Zbl 0562.05007)] and \(MR\), the Malvenuto-Reutenauer Hopf algebra of permutations [C. Malvenuto and C. Reutenauer, J. Algebra 177, No. 3, 967–982 (1995; Zbl 0838.05100)].

The diagram for these four Hopf algebras is \[ \begin{matrix} &Sym&\twoheadleftarrow &NSym&\twoheadrightarrow &MR\\ &|& &|& &|\\ &Sym&\twoheadrightarrow &QSym&\twoheadleftarrow &MR\end{matrix} \] where \(\twoheadleftarrow\) denotes a surjection, \(\twoheadrightarrow\) an injection, and \(|\) duality of Hopf algebras. The six Hopf algebras studied are denoted \(mSym\), \(mQSym\), \(mMR\) and \(MSym\), \(MNSym\) and \(MM\)R. They fit in the diagram \[ \begin{matrix} &MSym&\twoheadleftarrow &MNSym&\twoheadrightarrow &MMR\\ &|& &|& &|\\ &mSym&\twoheadrightarrow &mQsym&\twoheadleftarrow &mMR\end{matrix} \] We cannot give the definitions of these six Hopf algebras here, but will make a few comments. In the three prefixed by \(m\), the classical bases are the lowest degree components of the new basis, and products in these new bases are infinite (except in \(mSym\)), and both the product and coproduct contain classical terms plus terms of higher degree. In the three prefixed by \(M\), the classical bases are the highest degree components of the new bases, and products and coproducts are both finite and contain classical terms plus terms of lower degree.

To explain the title, A. Lascoux and M.-P. Schützenberger introduced Grothendieck polynomials as representatives of K-theory classes of structure sheaves of Schubert varieties [C. R. Acad. Sci., Paris, Sér. I 295, 629–633 (1982; Zbl 0542.14030)]. S. Fomin and A. N. Kirillov introduced stable Grothendieck polynomials, which are symmetric power-series obtained as limits of Grothendieck polynomials [Discrete Math. 153, No. 1–3, 123–143 (1996; Zbl 0852.05078)]. A. S. Buch gave a combinatorial expression for stable Grothendieck polynomials as generating series of set-valued tableaux [Acta Math. 189, No. 1, 37–78 (2002; Zbl 1090.14015)]. He showed that the stable Grothendieck polynomials play the role of Schur functions in the \(K\)-theory of Grassmannians, and studied a bialgebra spanned by the stable Grothendieck polynomials. \(mSym\) is the completion of this bialgebra.

Besides studying the Hopf algebra structure of these six Hopf algebras, the main results include a theory of set-valued \(P\)-partitions in the context of \(mQSym\); and three new families of symmetric functions in the context of \(MSym\) and \(mSym\). These symmetric functions are weight-generating functions of weak set-valued tableaux, valued-set tableaux and reverse plane partitions. The last of these are the dual stable Grothendieck polynomials.

The diagram for these four Hopf algebras is \[ \begin{matrix} &Sym&\twoheadleftarrow &NSym&\twoheadrightarrow &MR\\ &|& &|& &|\\ &Sym&\twoheadrightarrow &QSym&\twoheadleftarrow &MR\end{matrix} \] where \(\twoheadleftarrow\) denotes a surjection, \(\twoheadrightarrow\) an injection, and \(|\) duality of Hopf algebras. The six Hopf algebras studied are denoted \(mSym\), \(mQSym\), \(mMR\) and \(MSym\), \(MNSym\) and \(MM\)R. They fit in the diagram \[ \begin{matrix} &MSym&\twoheadleftarrow &MNSym&\twoheadrightarrow &MMR\\ &|& &|& &|\\ &mSym&\twoheadrightarrow &mQsym&\twoheadleftarrow &mMR\end{matrix} \] We cannot give the definitions of these six Hopf algebras here, but will make a few comments. In the three prefixed by \(m\), the classical bases are the lowest degree components of the new basis, and products in these new bases are infinite (except in \(mSym\)), and both the product and coproduct contain classical terms plus terms of higher degree. In the three prefixed by \(M\), the classical bases are the highest degree components of the new bases, and products and coproducts are both finite and contain classical terms plus terms of lower degree.

To explain the title, A. Lascoux and M.-P. Schützenberger introduced Grothendieck polynomials as representatives of K-theory classes of structure sheaves of Schubert varieties [C. R. Acad. Sci., Paris, Sér. I 295, 629–633 (1982; Zbl 0542.14030)]. S. Fomin and A. N. Kirillov introduced stable Grothendieck polynomials, which are symmetric power-series obtained as limits of Grothendieck polynomials [Discrete Math. 153, No. 1–3, 123–143 (1996; Zbl 0852.05078)]. A. S. Buch gave a combinatorial expression for stable Grothendieck polynomials as generating series of set-valued tableaux [Acta Math. 189, No. 1, 37–78 (2002; Zbl 1090.14015)]. He showed that the stable Grothendieck polynomials play the role of Schur functions in the \(K\)-theory of Grassmannians, and studied a bialgebra spanned by the stable Grothendieck polynomials. \(mSym\) is the completion of this bialgebra.

Besides studying the Hopf algebra structure of these six Hopf algebras, the main results include a theory of set-valued \(P\)-partitions in the context of \(mQSym\); and three new families of symmetric functions in the context of \(MSym\) and \(mSym\). These symmetric functions are weight-generating functions of weak set-valued tableaux, valued-set tableaux and reverse plane partitions. The last of these are the dual stable Grothendieck polynomials.

Reviewer: Earl J. Taft (New Brunswick)

##### MSC:

16T05 | Hopf algebras and their applications |

16T30 | Connections of Hopf algebras with combinatorics |

05E05 | Symmetric functions and generalizations |

14M15 | Grassmannians, Schubert varieties, flag manifolds |