Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2013 (vol 66 pp. 205)
Generalized Frobenius Algebras and Hopf Algebras "Co-Frobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
left-right symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasi-co-Frobenius (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) co-Frobenius, 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)co-Frobenius coalgebra $C$. We
give many examples of co-Frobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
q-homological algebra, topology or graph theory, showing the
importance of the concept.
Keywords:coalgebra, Hopf algebra, integral, Frobenius, QcF, co-Frobenius Categories:16T15, 18G35, 16T05, 20N99, 18D10, 05E10 |
2. CJM 2009 (vol 61 pp. 1092)
Minimal Transitive Factorizations of Permutations into Cycles We introduce a new approach to an enumerative problem
closely linked with the geometry of branched coverings,
that is, we study the number $H_{\alpha}(i_2,i_3,\dots)$ of ways a
given permutation (with cycles described by the partition $\a$) can be
decomposed into a product of exactly $i_2$ 2-cycles, $i_3$ 3-cycles,
\emph{etc.}, with certain minimality and transitivity conditions imposed on the factors. The method is to
encode such factorizations as planar maps with certain \emph{descent structure} and apply a new combinatorial
decomposition to make their enumeration more manageable. We apply our technique to determine
$H_{\alpha}(i_2,i_3,\dots)$ when $\a$ has one or two parts, extending earlier work of Goulden and Jackson.
We also show how these methods are readily modified to count \emph{inequivalent} factorizations, where
equivalence is defined by permitting commutations of adjacent disjoint factors. Our technique permits us to
generalize recent work of Goulden, Jackson, and Latour, while allowing for a considerable simplification of
their analysis.
Categories:05A15, 05E10 |
3. CJM 2008 (vol 60 pp. 266)
Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables We introduce a natural Hopf algebra structure on the space of noncommutative
symmetric functions.
The bases for this algebra are indexed
by set partitions. We show that there exists a natural inclusion of the Hopf
algebra of noncommutative symmetric functions
in this larger space. We also consider this algebra as a subspace of
noncommutative polynomials and use it to
understand the structure of the spaces of harmonics and coinvariants
with respect to this collection of noncommutative polynomials and conclude
two analogues of Chevalley's theorem in the noncommutative setting.
Categories:16W30, 05A18;, 05E10 |
4. CJM 2004 (vol 56 pp. 871)
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 |
5. CJM 1998 (vol 50 pp. 525)
Nilpotent orbit varieties and the atomic decomposition of the $q$-Kostka polynomials We study the coordinate rings~$k[\Cmubar\cap\hbox{\Frakvii t}]$ of
scheme-theoretic
intersections of nilpotent orbit closures with the diagonal matrices.
Here $\mu'$ gives the Jordan block structure of the nilpotent matrix.
de Concini and Procesi~\cite{deConcini&Procesi} proved a conjecture of
Kraft~\cite{Kraft} that these rings are isomorphic to the cohomology
rings of the varieties constructed by
Springer~\cite{Springer76,Springer78}. The famous $q$-Kostka
polynomial~$\Klmt(q)$ is the Hilbert series for the
multiplicity of the irreducible symmetric group representation indexed
by~$\lambda$ in the ring $k[\Cmubar\cap\hbox{\Frakvii t}]$.
\LS~\cite{L&S:Plaxique,Lascoux} gave combinatorially a decomposition
of~$\Klmt(q)$ as a sum of ``atomic'' polynomials with
non-negative integer coefficients, and Lascoux proposed a
corresponding decomposition in the cohomology model.
Our work provides a geometric interpretation of the atomic
decomposition. The Frobenius-splitting results of Mehta and van der
Kallen~\cite{Mehta&vanderKallen} imply a direct-sum decomposition of
the ideals of nilpotent orbit closures, arising from the inclusions of
the corresponding sets. We carry out the restriction to the diagonal
using a recent theorem of Broer~\cite{Broer}. This gives a direct-sum
decomposition of the ideals yielding the $k[\Cmubar\cap
\hbox{\Frakvii t}]$, and a new proof of the atomic decomposition of
the $q$-Kostka polynomials.
Keywords:$q$-Kostka polynomials, atomic decomposition, nilpotent conjugacy classes, nilpotent orbit varieties Categories:05E10, 14M99, 20G05, 05E15 |
6. CJM 1997 (vol 49 pp. 1281)
Pieri's formula via explicit rational equivalence Pieri's formula describes the intersection product of a Schubert
cycle by a special Schubert cycle on a Grassmannian.
We present a new geometric proof,
exhibiting an explicit chain of rational equivalences
from a suitable sum of distinct Schubert cycles
to the intersection of a Schubert cycle with a special
Schubert cycle. The geometry of these rational equivalences
indicates a link to a combinatorial proof of Pieri's formula using
Schensted insertion.
Keywords:Pieri's formula, rational equivalence, Grassmannian, Schensted insertion Categories:14M15, 05E10 |
7. CJM 1997 (vol 49 pp. 263)
Determinantal forms for symplectic and orthogonal Schur functions Symplectic and orthogonal Schur functions can be defined
combinatorially in a manner similar to the classical Schur functions.
This paper demonstrates that they can also be expressed as determinants.
These determinants are generated using planar decompositions of tableaux
into strips and the equivalence of these determinants to symplectic or
orthogonal Schur functions is established by Gessel-Viennot lattice path
techniques. Results for rational (also called {\it composite}) Schur functions
are also obtained.
Categories:05E05, 05E10, 20C33 |