Lie groups of controlled characters of combinatorial Hopf algebras

AbstractThe dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriformstructures on the combinatorial Hopf algebras FQSym andWQSym.

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A Uniform Generalization of Some Combinatorial Hopf Algebras

Algebras and Representation Theory ◽

10.1007/s10468-016-9648-x ◽

2016 ◽

Vol 20 (2) ◽

pp. 379-431 ◽

Cited By ~ 1

Author(s):

Jia Huang

Keyword(s):

Hopf Algebras ◽

Combinatorial Hopf Algebras

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Antipode Formulas for some Combinatorial Hopf Algebras

The Electronic Journal of Combinatorics ◽

10.37236/5949 ◽

2016 ◽

Vol 23 (4) ◽

Cited By ~ 2

Author(s):

Rebecca Patrias

Keyword(s):

Hopf Algebra ◽

Hopf Algebras ◽

Symmetric Functions ◽

Explicit Formulas ◽

Quasisymmetric Functions ◽

Noncommutative Symmetric Functions ◽

Combinatorial Hopf Algebras ◽

Combinatorial Formulas ◽

K Theory

Motivated by work of Buch on set-valued tableaux in relation to the K-theory of the Grassmannian, Lam and Pylyavskyy studied six combinatorial Hopf algebras that can be thought of as K-theoretic analogues of the Hopf algebras of symmetric functions, quasisymmetric functions, noncommutative symmetric functions, and of the Malvenuto-Reutenauer Hopf algebra of permutations. They described the bialgebra structure in all cases that were not yet known but left open the question of finding explicit formulas for the antipode maps. We give combinatorial formulas for the antipode map for the K-theoretic analogues of the symmetric functions, quasisymmetric functions, and noncommutative symmetric functions.

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Combinatorial Hopf algebras, noncommutative Hall–Littlewood functions, and permutation tableaux

Advances in Mathematics ◽

10.1016/j.aim.2010.01.006 ◽

2010 ◽

Vol 224 (4) ◽

pp. 1311-1348 ◽

Cited By ~ 9

Author(s):

J.-C. Novelli ◽

J.-Y. Thibon ◽

L.K. Williams

Keyword(s):

Hopf Algebras ◽

Combinatorial Hopf Algebras

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Polynomial realizations of some combinatorial Hopf algebras

Journal of Noncommutative Geometry ◽

10.4171/jncg/151 ◽

2014 ◽

Vol 8 (1) ◽

pp. 141-162 ◽

Cited By ~ 1

Author(s):

Loïc Foissy ◽

Jean-Christophe Novelli ◽

Jean-Yves Thibon

Keyword(s):

Hopf Algebras ◽

Combinatorial Hopf Algebras

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INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS

International Journal of Modern Physics A ◽

10.1142/s0217751x92002805 ◽

1992 ◽

Vol 07 (25) ◽

pp. 6175-6213 ◽

Cited By ~ 30

Author(s):

T. TJIN

Keyword(s):

Lie Groups ◽

Quantum Groups ◽

Hopf Algebras ◽

Baxter Equation ◽

Product Decomposition ◽

R Matrix ◽

Conformal Field ◽

Finite Dimensional ◽

Lie Groups And Algebras ◽

Poisson Lie Groups

We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

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