Search arXiv⌕ Search

arXiv · 0706.2910

Properties of the descent algebras of type $D$

Abstract

We establish simple combinatorial descriptions of the radical and irreducible representations specifically for the descent algebra of a Coxeter group of type $D$ over any field.

Explore related subjects

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stephanie van Willigenburg. 2007-06-20. Properties of the descent algebras of type $D$. https://arxiv.org/abs/0706.2910

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

Counterexamples to a dimension-free variant of Erdős's diameter conjecture for separated distances

Erdős asked about the least possible diameter of an $n$-point set in Euclidean space when any two pairwise distances differ by at least $1$. We consider the uniform-in-dimension variant of his question, in which the dimension $d$ may grow with $n$ and the error term is independent of $d$, and answer it in the negative by an explicit construction. For every prime power $q$, we construct a set $\mathcal X_n$ of $n=q+1$ points in $\mathbb R^{q^2+q}=\mathbb R^{n^2-n}$ such that the $\binom{n}{2}$ pairwise distances are mutually at least $1$ apart, yet $$\operatorname{diam}(\mathcal X_n)\le \Bigl(1-\frac{1}{π^2}+o(1)\Bigr)n^2.$$ The construction combines a Singer cyclic difference set with a weighted product of $N=\binom n2$ regular $m$-gons, where $m=q^2+q+1$. The weights are chosen so that the resulting distances increase while the gaps between consecutive distances decrease. After normalizing the smallest gap to $1$, all distinct pairwise distances are separated by at least $1$, while the diameter has asymptotic constant $1-1/π^2<1$.

math.CO↗

Amply regular graphs with $μ$ close to half the valency and group divisible designs

In this paper, we classify connected amply regular graphs with diameter $d \geq 4$ and parameters $(v, k, λ, μ)$ satisfying $μ= \frac{k-1}{2}$, where $k\geq 5$ is odd. We prove that such a graph must be exactly one of the following: the $5$-cube, the graph $\K_2 \square Λ$, where $Λ$ is the unique bipartite $(0,2)$-graph on $14$ vertices, or the point--block incidence graph of a group divisible design with the dual property, namely a $GDDDP\left(2, k+1;\, k;\, 0, \frac{k-1}{2}\right)$. For the last family, we give equivalent characterizations in terms of bipartite $Q$-regular graphs and relation graphs of symmetric association schemes with five classes. Furthermore, we present constructions of such amply regular graphs, yielding infinite families of examples derived from Paley graphs, Peisert graphs, and Paley digraphs.

math.CO↗

Near-optimal Turán densities of $r$-graphs on $r+1$ vertices

Let $π(H)$ be the Turán density of an r-uniform hypergraph $H$ and let $H_k^r$ denote the $r$-uniform hypergraph on $r+1$ vertices with exactly $k$ edges, where $1\le k\le r+1$. Sidorenko~(JCT-B, 2024) proved that $π(H_3^r)\ge (1.7215-o(1))r^{-2}$ as $r\to\infty$ and $π(H_k^r)\ge (C_k+o(1))r^{-(1+1/(k-2))}$ for fixed $k$ as $r\to\infty$. Clemen~later improved the first bound to $π(H_3^r)\ge cr^{-2}\sqrt{\log r}$ for some constant $c>0$. In this article, we prove the following results. \begin{itemize} \item For any fixed $\varepsilon>0$, there is a constant $c_\varepsilon>0$ such that $$π(H_3^r)\ge \frac{c_\varepsilon}{r(\log r)^{2+\varepsilon}}.$$ %$π(H_3^r)\ge 1/(r(\log r)^{2+o(1)})$. Together with the known upper bound $π(H_3^r)\le1/r$, this implies $π(H_3^r)=r^{-1+o(1)}$. \item For every $3\le k\le r+1$, let $s=\min\{k-2,r-k+2\}$. Then \begin{equation*} 0\le \frac{k-2}{r}-π(H_k^r) \le \frac{128}{r}\left(\sqrt{s\log\frac{er}{s}}+\log\frac{er}{s}\right). \end{equation*} This estimate yields several asymptotically sharp results for $π(H_k^r)$. For example, $π(H_k^r)=(1+o(1))(k-2)/r$ when $\log(er/(k))=o(k)$. \end{itemize}

math.CO↗