1. CJM 2016 (vol 69 pp. 21)
 Grinberg, Darij

Dual Creation Operators and a Dendriform Algebra Structure on the Quasisymmetric Functions
The dual immaculate functions are a basis of the ring $\operatorname*{QSym}$
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 be
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 have been 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 Mike Zabrocki which provides
an
alternative construction for the dual immaculate functions in
terms of certain
"vertex operators". The proof uses a dendriform structure on
the ring
$\operatorname*{QSym}$; we discuss the relation of this structure
to known
dendriform structures on the combinatorial Hopf algebras
$\operatorname*{FQSym}$ and $\operatorname*{WQSym}$.
Keywords:combinatorial Hopf algebras, quasisymmetric functions, dendriform algebras, immaculate functions, Young tableaux Category:05E05 

2. CJM 2013 (vol 66 pp. 205)
 Iovanov, Miodrag Cristian

Generalized Frobenius Algebras and Hopf Algebras
"CoFrobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
leftright symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasicoFrobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the
sense that certain coproduct or product powers of these objects are
isomorphic. Fundamental results of Hopf algebras, such as the
equivalent characterizations of Hopf algebras with nonzero integrals
as left (or right) coFrobenius, QcF, semiperfect or with nonzero
rational dual, as well as the uniqueness of integrals and a short
proof of the bijectivity of the antipode for such Hopf algebras all
follow as a consequence of these results. This gives a purely
representation theoretic approach to many of the basic fundamental
results in the theory of Hopf algebras. Furthermore, we introduce a
general concept of Frobenius algebra, which makes sense for infinite
dimensional and for topological algebras, and specializes to the
classical notion in the finite case. This will be a topological
algebra $A$ that is isomorphic to its complete topological dual
$A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only
if $A$ is the dual $C^*$ of a (quasi)coFrobenius coalgebra $C$. We
give many examples of coFrobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
qhomological algebra, topology or graph theory, showing the
importance of the concept.
Keywords:coalgebra, Hopf algebra, integral, Frobenius, QcF, coFrobenius Categories:16T15, 18G35, 16T05, 20N99, 18D10, 05E10 

3. CJM 2012 (vol 65 pp. 241)
 Aguiar, Marcelo; Lauve, Aaron

Lagrange's Theorem for Hopf Monoids in Species
Following Radford's proof of Lagrange's theorem for pointed Hopf algebras,
we prove Lagrange's theorem for Hopf monoids in the category of
connected species.
As a corollary, we obtain necessary conditions for a given subspecies
$\mathbf k$ of a Hopf monoid $\mathbf h$ to be a Hopf submonoid: the quotient of
any one of the generating series of $\mathbf h$ by the corresponding
generating series of $\mathbf k$ must have nonnegative coefficients. Other
corollaries include a necessary condition for a sequence of
nonnegative integers to be the
dimension sequence of a Hopf monoid
in the form of certain polynomial inequalities, and of
a settheoretic Hopf monoid in the form of certain linear inequalities.
The latter express that the binomial transform of the sequence must be nonnegative.
Keywords:Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, PoincarÃ©BirkhoffWitt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement Categories:05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35 

4. CJM 2004 (vol 56 pp. 871)
 Schocker, Manfred

Lie Elements and Knuth Relations
A coplactic class in the symmetric group $\Sym_n$ consists of all
permutations in $\Sym_n$ with a given Schensted $Q$symbol, and may
be described in terms of local relations introduced by Knuth. Any
Lie element in the group algebra of $\Sym_n$ which is constant on
coplactic classes is already constant on descent classes. As a
consequence, the intersection of the Lie convolution algebra
introduced by Patras and Reutenauer and the coplactic algebra
introduced by Poirier and Reutenauer is the direct sum of all
Solomon descent algebras.
Keywords:symmetric group, descent set, coplactic relation, Hopf algebra,, convolution product Categories:17B01, 05E10, 20C30, 16W30 
