470 research outputs found
Diameter of reduced words
For finite reflection groups of types A and B, we determine the diameter of
the graph whose vertices are reduced words for the longest element and whose
edges are braid relations. This is deduced from a more general theorem that
applies to supersolvable hyperplane arrangements.Comment: Version 4 points out a gap in the proof of Theorem 4.9, filled in
work of T. McConville (arXiv:1411.1305
Critical groups of covering, voltage, and signed graphs
Graph coverings are known to induce surjections of their critical groups.
Here we describe the kernels of these morphisms in terms of data parametrizing
the covering. Regular coverings are parametrized by voltage graphs, and the
above kernel can be identified with a naturally defined voltage graph critical
group. For double covers, the voltage graph is a signed graph, and the theory
takes a particularly pleasant form, leading also to a theory of double covers
of signed graphs.Comment: Version 3 fixes a typo, and adds some details in the proof of Theorem
1.
Hopf Algebras in Combinatorics
These notes -- originating from a one-semester class by their second author
at the University of Minnesota -- survey some of the most important Hopf
algebras appearing in combinatorics. After introducing coalgebras, bialgebras
and Hopf algebras in general, we study the Hopf algebra of symmetric functions,
including Zelevinsky's axiomatic characterization of it as a "positive
self-adjoint Hopf algebra" and its application to the representation theory of
symmetric and (briefly) finite general linear groups. The notes then continue
with the quasisymmetric and the noncommutative symmetric functions, some Hopf
algebras formed from graphs, posets and matroids, and the Malvenuto-Reutenauer
Hopf algebra of permutations. Among the results surveyed are the
Littlewood-Richardson rule and other symmetric function identities,
Zelevinsky's structure theorem for PSHs, the antipode formula for P-partition
enumerators, the Aguiar-Bergeron-Sottile universal property of QSym, the theory
of Lyndon words, the Gessel-Reutenauer bijection, and Hazewinkel's polynomial
freeness of QSym. The notes are written with a graduate student reader in mind,
being mostly self-contained but requiring a good familiarity with multilinear
algebra and -- for the representation-theory applications -- basic group
representation theory.Comment: 282 pages. The version on my website
(http://www.cip.ifi.lmu.de/~grinberg/algebra/HopfComb.pdf) is likely to be
updated more frequently. Solutions to the exercises can be found in
http://www.cip.ifi.lmu.de/~grinberg/algebra/HopfComb-sols.pdf , or as an
ancillary file here. Version v7 has more details, a few more exercises and
fewer errors. Comments are welcome
On configuration spaces and Whitehouse's lifts of the Eulerian representations
S. Whitehouse's lifts of the Eulerian representations of to
are reinterpreted, topologically and ring-theoretically, building on the first
author's work on A. Ocneanu's theory of permutohedral blades.Comment: Exposition improved; proof of Proposition 6 added. To appear in The
Journal of Pure and Applied Algebr
Conjectures on the cohomology of the Grassmannian
We give a series of successively weaker conjectures on the cohomology ring of
the Grassmannian, starting with the Hilbert series of a certain natural
filtration.Comment: 8 page
Differential posets and Smith normal forms
We conjecture a strong property for the up and down maps U and D in an
r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In
particular, this would determine the integral structure of the maps U, D, UD,
DU, including their ranks in any characteristic.
As evidence, we prove the conjecture for the Young-Fibonacci lattice YF
studied by Okada and its r-differential generalizations Z(r), as well as
verifying many of its consequences for Young's lattice Y and the r-differential
Cartesian products Y^r.Comment: 29 pages, 9 figure
Representation stability for cohomology of configuration spaces in
This paper studies representation stability in the sense of Church and Farb
for representations of the symmetric group on the cohomology of the
configuration space of ordered points in . This cohomology is
known to vanish outside of dimensions divisible by ; it is shown here that
the -representation on the cohomology stabilizes sharply at
(resp. ) when is odd (resp. even).
The result comes from analyzing -representations known to control the
cohomology: the Whitney homology of set partition lattices for even, and
the higher Lie representations for odd. A similar analysis shows that the
homology of any rank-selected subposet in the partition lattice stabilizes by
, where is the maximum rank selected.
Further properties of the Whitney homology and more refined stability
statements for -isotypic components are also proven, including conjectures
of J. Wiltshire-Gordon.Comment: Fixed typos, reorganized slightly, and added Remark 3.5 on improved
power-saving boun
Reciprocal domains and Cohen-Macaulay -complexes in
We extend a reciprocity theorem of Stanley about enumeration of integer
points in polyhedral cones when one exchanges strict and weak inequalities. The
proof highlights the roles played by Cohen-Macaulayness and canonical modules.
The extension raises the issue of whether a Cohen-Macaulay complex of dimension
d embedded piecewise-linearly in d-space is necessarily a d-ball. This is
observed to be true for d at most 3, but false for d=4
Shifted set families, degree sequences, and plethysm
We study, in three parts, degree sequences of k-families (or k-uniform
hypergraphs) and shifted k-families. The first part collects for the first time
in one place, various implications such as: Threshold implies Uniquely
Realizable implies Degree-Maximal implies Shifted, which are equivalent
concepts for 2-families (=simple graphs), but strict implications for
k-families with k > 2. The implication that uniquely realizable implies
degree-maximal seems to be new. The second part recalls Merris and Roby's
reformulation of the characterization due to Ruch and Gutman for graphical
degree sequences and shifted 2-families. It then introduces two generalizations
which are characterizations of shifted k-families. The third part recalls the
connection between degree sequences of k-families of size m and the plethysm of
elementary symmetric functions e_m[e_k]. It then uses highest weight theory to
explain how shifted k-families provide the ``top part'' of these plethysm
expansions, along with offering a conjecture about a further relation.Comment: Final version, 26 pages, 3 figure
P-partitions revisited
We compare a traditional and non-traditional view on the subject of
P-partitions, leading to formulas counting linear extensions of certain posets.Comment: 34 pages; we corrected a minor mistake in the definition of the
monomial ordering (Section 6); to appear in J. Commutative Algebr
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