In this note, we give a classification of the maximal order Abelian subgroups of finite irreducible Coxeter groups. We also prove a Weyl group analog of Cartan’s theorem that all maximal tori in a connected compact Lie group are conjugate.

Root metrics and plant height for 256 excavated saplings and small trees of 27 species, including sown plants, were used to describe belowground structure and assess factors that influence shoot growth in a tropical dry forest (TDF) in Zambia. Models were developed to (i) estimate taproot depth from incomplete excavations and (ii) coarse lateral root biomass from proximal diameter data. The majority of the species studied are slow-growing and had a median height of <200 cm at the age of 16 years. Root development advanced sequentially from taproot elongation to thickening to coarse lateral root development. Shrubs in shallow soil had short taproots with a lower wood density. Plant age explained <10% of the variance in shoot height. Root variables explained the majority of the variance in shoot height. More research is needed to improve our knowledge about how belowground structures influence shoot growth and tree recruitment in TDFs of southern Africa.

Let $U_q'({\mathfrak {g}})$ be a quantum affine algebra with an indeterminate $q$, and let $\mathscr {C}_{\mathfrak {g}}$ be the category of finite-dimensional integrable $U_q'({\mathfrak {g}})$-modules. We write $\mathscr {C}_{\mathfrak {g}}^0$ for the monoidal subcategory of $\mathscr {C}_{\mathfrak {g}}$ introduced by Hernandez and Leclerc. In this paper, we associate a simply laced finite-type root system to each quantum affine algebra $U_q'({\mathfrak {g}})$ in a natural way and show that the block decompositions of $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$ are parameterized by the lattices associated with the root system. We first define a certain abelian group $\mathcal {W}$ (respectively $\mathcal {W} _0$) arising from simple modules of $\mathscr {C}_{\mathfrak {g}}$ (respectively $\mathscr {C}_{\mathfrak {g}}^0$) by using the invariant $\Lambda ^\infty$ introduced in previous work by the authors. The groups $\mathcal {W}$ and $\mathcal {W} _0$ have subsets $\Delta$ and $\Delta _0$ determined by the fundamental representations in $\mathscr {C}_{\mathfrak {g}}$ and $\mathscr {C}_{\mathfrak {g}}^0$, respectively. We prove that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ is an irreducible simply laced root system of finite type and that the pair $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} , \Delta )$ is isomorphic to the direct sum of infinite copies of $( \mathbb {R} \otimes _\mathbb {\mspace {1mu}Z\mspace {1mu}} \mathcal {W} _0, \Delta _0)$ as a root system.

This paper examines a systematic method of constructing a pair of (inter-related) root systems for arbitrary Coxeter groups from a class of nonstandard geometric representations. This method can be employed to construct generalizations of root systems for a large family of linear groups generated by involutions. We then give a characterization of Coxeter groups, among these groups, in terms of such paired root systems. Furthermore, we use this method to construct and study the paired root systems for reflection subgroups of Coxeter groups.

Thomas Karp Pavlychenko's pioneering investigations on the root systems of crops and weeds and the results of his research on weed biology and ecology at the University of Saskatchewan in the 1930's and 1940's provided the foundation for many of the principles of modern weed science. His classical studies in which he quantified the distribution and length of roots of many plant species continue to be regarded as being of unsurpassed caliber. In addition to his work in the areas of crop ecology and plant competition, Pavlychenko is also noted for his many contributions relating to the cultural and chemical control of weeds. In this paper, a brief chronicle of Pavlychenko's life and major accomplishments during his remarkable career at the University of Saskatchewan and later with the American Chemical and Paint Company is presented.

We study the absolute continuity of the convolution ${\it\delta}_{e^{X}}^{\natural }\star {\it\delta}_{e^{Y}}^{\natural }$ of two orbital measures on the symmetric spaces $\mathbf{SO}_{0}(p,p)/\mathbf{SO}(p)\times \mathbf{SO}(p)$, $\mathbf{SU}(p,p)/\mathbf{S}(\mathbf{U}(p)\times \mathbf{U}(p))$ and $\mathbf{Sp}(p,p)/\mathbf{Sp}(p)\times \mathbf{Sp}(p)$. We prove sharp conditions on $X$, $Y\in \mathfrak{a}$ for the existence of the density of the convolution measure. This measure intervenes in the product formula for the spherical functions.

We define in an axiomatic fashion a Coxeter datum for an arbitrary Coxeter group $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear pairing without any integrality or nondegeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these nonorthogonal root systems and the classical root systems. Further, we study the equivalent of the Tits cone in these nonorthogonal representations.

We present an explicit construction of the Macdonald polynomials and of the Heckman–Opdam generalized Jacobi polynomials associated with root systems.

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions $F(t_1, …, t_n)$ of WDVV equations of associativity polynomial in $t_1, …, t_n-1$, exp $t_n$.

Topology is an important component of the architecture of whole root systems. Unfortunately, most commonly applied indices used for characterizing topology are poorly correlated with one another and thus reflect different aspects of topology. In order to understand better how different methods of characterizing topology vary, this paper presents an exploration of several different methods for assigning order within branched root systems on the basis of (a) developmental (centrifugal) vs. functional (centripetal) ordering sequences and (b) whether orders are assigned to individual links or groups of adjacent links (segments). For each ordering system, patterns of scaling in relation to various aspects of link and segment size are explored using regression analyses. Segment-based ordering systems resulted in better fits for simple scaling relationships with size, but these patterns varied between developmental vs. functional ordering as well as the different size metrics examined. The functional (centripetal), link-based ordering system showed complex, non-linear scaling in relation to numbers of links per order. Using a simple simulation model of root growth, it is demonstrated that this method of characterizing root topology in relation to root size might be a more powerful tool for characterizing root system architecture than in the use of simple, single-index characterizations of topology.