Many HH-copies in graphs with a forbidden tree

For graphs $H$ and $F$, let $\operatorname{ex}(n, H, F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalises the well-studied Tur\'an numbers of graphs, was initiated recently by Alon and Shikhelman. We show that if $F$ is a tree then $\operatorname{ex}(n, H, F) = \Theta(n^r)$ for some integer $r = r(H, F)$, thus answering one of their questions.Comment: 9 pages, 1 figur

Quantum Zonal Spherical Functions and Macdonald Polynomials

A unified theory of quantum symmetric pairs is applied to q-special functions. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions. Here a distinguished family of such functions, invariant under the Weyl group associated to the restricted roots, is shown to be a family of Macdonald polynomials, as conjectured by Koornwinder and Macdonald. Our results place earlier work for Lie algebras of classical type in a general context and extend to the exceptional cases.Comment: Minor revisions, changes to section

Radon Numbers for Trees

Many interesting problems are obtained by attempting to generalize classical results on convexity in Euclidean spaces to other convexity spaces, in particular to convexity spaces on graphs. In this paper we consider $P_3$-convexity on graphs. A set $U$ of vertices in a graph $G$ is $P_3$-convex if every vertex not in $U$ has at most one neighbour in $U$. More specifically, we consider Radon numbers for $P_3$-convexity in trees. Tverberg's theorem states that every set of $(k-1)(d+1)-1$ points in $\mathbb{R}^d$ can be partitioned into $k$ sets with intersecting convex hulls. As a special case of Eckhoff's conjecture, we show that a similar result holds for $P_3$-convexity in trees. A set $U$ of vertices in a graph $G$ is called free, if no vertex of $G$ has more than one neighbour in $U$. We prove an inequality relating the Radon number for $P_3$-convexity in trees with the size of a maximal free set.Comment: 17 pages, 13 figure

Effective Detection of Nonsplit Module Extensions

Let n be a positive integer, and let R be a finitely presented (but not necessarily finite dimensional) associative algebra over a computable field. We examine algorithmic tests for deciding (1) if every n-dimensional representation of R is semisimple, and (2) if there exist nonsplit extensions of non-isomorphic irreducible R-modules whose dimensions sum to no greater than n. Our basic strategy is to reduce each of the considered representation theoretic decision problems to the problem of deciding whether a particular set of commutative polynomials has a common zero. Standard methods of computational algebraic geometry can then be applied (in principle).Comment: AMS-TeX; 13 pages; no figures. Revised version. To appear in Journal of Pure and Applied Algebr