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

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

On configuration spaces and Whitehouse's lifts of the Eulerian representations

S. Whitehouse's lifts of the Eulerian representations of $S_n$ to $S_{n+1}$ 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

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

Representation stability for cohomology of configuration spaces in Rd\mathbf{R}^d

This paper studies representation stability in the sense of Church and Farb for representations of the symmetric group $S_n$ on the cohomology of the configuration space of $n$ ordered points in $\mathbf{R}^d$. This cohomology is known to vanish outside of dimensions divisible by $d-1$; it is shown here that the $S_n$-representation on the $i(d-1)^{st}$ cohomology stabilizes sharply at $n=3i$ (resp. $n=3i+1$) when $d$ is odd (resp. even). The result comes from analyzing $S_n$-representations known to control the cohomology: the Whitney homology of set partition lattices for $d$ even, and the higher Lie representations for $d$ odd. A similar analysis shows that the homology of any rank-selected subposet in the partition lattice stabilizes by $n\geq 4i$, where $i$ is the maximum rank selected. Further properties of the Whitney homology and more refined stability statements for $S_n$-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 dd-complexes in RdR^d

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