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A subalgebra of a semisimple Lie algebra is wide if every simple module of the semisimple Lie algebra remains indecomposable when restricted to the subalgebra. A subalgebra is narrow if the restrictions of all non-trivial simple modules to the subalgebra have proper decompositions. A semisimple Lie algebra is regular extreme if any regular subalgebra of the semisimple Lie algebra is either narrow or wide. We determine necessary and sufficient conditions for a simple module of a semisimple Lie algebra to remain indecomposable when restricted to a regular subalgebra. As a natural consequence, we establish necessary and sufficient conditions for regular subalgebras to be wide, a result which has already been established by Panyushev for essentially all regular solvable subalgebras [10]. Next, we show that establishing whether or not a regular subalgebra of a simple Lie algebra is wide does not require consideration of all simple modules. It is necessary and sufficient to only consider the adjoint representation. Then, we show that all simple Lie algebras are regular extreme. Finally, we show that no non-simple, semisimple Lie algebra is regular extreme. [ABSTRACT FROM AUTHOR]

We propose a refined version of the existing conjectural asymptotic formula for the moments of the family of quadratic Dirichlet L -functions over rational function fields. Our prediction is motivated by two natural conjectures that provide sufficient information to determine the analytic properties (meromorphic continuation, location of poles, and the residue at each pole) of a certain generating function of moments of quadratic L -functions. The number field analogue of our asymptotic formula can be obtained by a similar procedure, the only difference being the contributions coming from the archimedean and even places, which require a separate analysis. To avoid this additional technical issue, we present, for simplicity, the asymptotic formula only in the rational function field setting. This has also the advantage of being much easier to test. [ABSTRACT FROM AUTHOR]

Any symmetric closed subset of a finite crystallographic root system must be a closed subroot system. This is not, in general, true for real affine root systems. In this paper, we determine when this is true and also give a very explicit description of symmetric closed subsets of real affine root systems. At the end, using our results, we study the correspondence between symmetric closed subsets of real affine root systems and the regular subalgebras generated by them. [ABSTRACT FROM AUTHOR]

We study the ring of Weyl invariant E 8 weak Jacobi forms. Wang recently proved that the ring is not a polynomial algebra. We consider a polynomial algebra which contains the ring as a subset and clarify the necessary and sufficient condition for an element of the polynomial algebra to be a Weyl invariant E 8 weak Jacobi form. This serves as a new algorithm for constructing all the Jacobi forms of given weight and index. The algorithm is purely algebraic and does not require Fourier expansion. Using this algorithm we determine the generators of the free module of Weyl invariant E 8 weak Jacobi forms of given index m for m ≤ 20. We also determine the lowest weight generators of the free module of index m for m ≤ 28. Our results support the lower bound conjecture of Sun and Wang and prove explicitly that there exist generators of the ring of Weyl invariant E 8 weak Jacobi forms of weight − 4 m and index m for all 12 ≤ m ≤ 28. [ABSTRACT FROM AUTHOR]

Belowground biomass represents a significant carbon reserve with a crucial role in terms of ecosystem functioning, but very little is known about root systems across tropical dry forests. Here we investigate patterns of root biomass, nutrient concentration/stocks, and the effect of aboveground biomass, chronic anthropogenic disturbance, precipitation, and soil depth as their potential drivers in a Caatinga dry forest, northeastern Brazil. Belowground biomass was obtained by a destructive method from July 2020 to March 2021 across forest stands in three 0.49-m2 and 1-m deep trenches per stand. Root biomass ranged from 3.23 Mg ha−1 to 29.6 Mg ha−1, accounting for less than 1/3 of total forest biomass. Belowground biomass greatly varied across forest stands mainly represented by large roots concentrated in the superficial soil layer. In general, nutrient concentrations and stocks did not vary significantly across forest stands regarding both root size class and soil depth, although fine roots support a higher concentration of P than large-sized roots. Finally, chronic disturbance, aboveground biomass, precipitation, and soil depth were important predictors affecting root biomass and nutrients. Our results suggest that a combination of natural and human-related drivers modulate root biomass and nutrients in Caatinga dry forests immersed in human-modified landscapes. • Belowground biomass is a key component of the Caatinga forest biomass. • Root biomass concentrates in the soil superficial layer and it is highly sensitive to precipitation. • Nutrient stocks are negatively affected by chronic anthropogenic disturbance. • Both natural and human-related drivers modulate root biomass and nutrients. [ABSTRACT FROM AUTHOR]

A Lie algebra L is said to be (Θ n , sl n) -graded if it contains a simple subalgebra g isomorphic to s l n such that the g -module L decomposes into copies of the adjoint module, the trivial module, the natural module V , its symmetric and exterior squares S 2 V and ∧ 2 V and their duals. We describe the multiplicative structures and the coordinate algebras of (Θ n , sl n) -graded Lie algebras for n ≥ 5 , classify these Lie algebras and determine their central extensions. [ABSTRACT FROM AUTHOR]

We describe an algorithm for classifying the closed subsets of a root system, up to conjugation by the associated Weyl group. Our algorithm is implemented in the language of the computer algebra system GAP 4. We discuss the implementation and give runtimes on some sample inputs. The classification of the closed subsets of an irreducible root system is closely related to the classification of the regular subalgebras, up to inner automorphism, of the corresponding simple Lie algebra. We show how to obtain the regular subalgebras corresponding to a given closed set for the root systems of rank 3. The version of this paper on the arxiv has tables describing the subalgebras that were obtained. [ABSTRACT FROM AUTHOR]

In previous joint work with J.-S. Koskivirta, we introduced the notion of "quasi-constant" character (of a maximal torus of a connected reductive group over a field); we showed that over an algebraically closed field it naturally unifies the notions "minuscule" and "co-minuscule". In this note we characterize quasi-constant fundamental weights in terms of the Weyl group of the corresponding maximal Levi subgroup. Equivalently, purely in the language of root systems, the result characterizes special and co-special vertices of Dynkin diagrams in terms of the Weyl group of the corresponding maximal sub-root system. [ABSTRACT FROM AUTHOR]

Abstract In this note we classify the Coxeter systems for which the Brink-Howlett automaton is minimal. We show that this automaton is minimal if and only if each elementary root is supported on a standard spherical subsystem, thereby resolving a conjecture of Hohlweg, Nadeau and Williams. [ABSTRACT FROM AUTHOR]

We generalize the definition and properties of root systems to complex reflection groups — roots become rank one projective modules over the ring of integers of a number field k . In the irreducible case, we provide a classification of root systems over the field of definition k of the reflection representation. In the case of spetsial reflection groups, we generalize as well the definition and properties of bad primes. [ABSTRACT FROM AUTHOR]

We study a combinatorial object, which we call a GRRS (generalized reflection root system); the classical root systems and GRSs introduced by V. Serganova are examples of finite GRRSs. A GRRS is finite if it contains a finite number of vectors and is called affine if it is infinite and has a finite minimal quotient. We prove that an irreducible GRRS containing an isotropic root is either finite or affine; we describe all finite and affine GRRSs and classify them in most of the cases. [ABSTRACT FROM AUTHOR]

We consider twisted standard filtrations of Soergel bimodules associated to arbitrary Coxeter groups and show that the graded multiplicities in these filtrations can be interpreted as structure constants in the Hecke algebra. This corresponds to the positivity of the polynomials occurring when expressing an element of the canonical basis in a generalized standard basis twisted by a biclosed set of roots in the sense of Dyer, and comes as a corollary of Soergel's conjecture. We then show the positivity of the corresponding inverse polynomials in the case where the biclosed set is an inversion set of an element or its complement by generalizing a result of Elias and Williamson on the linearity of the Rouquier complexes associated to lifts of these basis elements in the Artin–Tits group. These lifts turn out to be generalizations of Mikado braids as introduced in a joint work with Digne. This second positivity property generalizes a result of Dyer and Lehrer from finite to arbitrary Coxeter groups. [ABSTRACT FROM AUTHOR]

An idempotent in a Jordan algebra induces a Peirce decomposition of the algebra into subspaces whose pairwise multiplication satisfies a certain fusion rule Φ ( 1 2 ) . On the other hand, 3-transposition groups ( G , D ) can be algebraically characterised as Matsuo algebras M α ( G , D ) with idempotents satisfying the fusion rule Φ ( α ) for some α . We classify the Jordan algebras J which are isomorphic to a Matsuo algebra M 1 2 ( G , D ) , in which case ( G , D ) is a subgroup of the (algebraic) automorphism group of J ; the only possibilities are G = Sym ( n ) and G = 3 2 : 2 . Along the way, we also obtain results about Jordan algebras associated to root systems. [ABSTRACT FROM AUTHOR]

Dunkl operators are differential-difference operators parametrized by a finite reflection group and a weight function. The commutative algebra generated by these operators generalizes the algebra of standard differential operators and intertwines with this latter by the so-called intertwining operator. In this paper, we give an integral representation for the operator V k ∘ e Δ / 2 for an arbitrary Weyl group and a large class of regular weights k containing those of nonnegative real parts. Our representing measures are absolutely continuous with respect the Lebesgue measure in R d , which allows us to derive out new results about the intertwining operator V k and the Dunkl kernel E k . We show in particular that the operator V k ∘ e Δ / 2 extends uniquely as a bounded operator to a large class of functions which are not necessarily differentiables. In the case of nonnegative weights, this operator is shown to be positivity-preserving. [ABSTRACT FROM AUTHOR]

In this article we provide a new finite class of elements in any Coxeter system ( W , S ) called low elements . They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of ( W , S ) . Our first main result is to show that they form a Garside shadow in ( W , S ) , i.e., they contain S and are closed under join (for the right weak order) and by taking suffixes. These low elements are the key to prove that all finitely generated Artin–Tits groups have a finite Garside family. This result was announced in a note with P. Dehornoy in Comptes rendus mathématiques [9] in which the present article was referred to under the following working title: Monotonicity of dominance-depth on root systems and applications . The proof is based on a fundamental property enjoyed by small roots and which is our second main result; the set of small root is bipodal . For a natural number n , we define similarly n -low elements from n -small roots and conjecture that the set of n -small roots is bipodal, implying the set of n -low elements is a Garside shadow; we prove this conjecture for affine Coxeter groups and Coxeter groups whose graph is labelled by 3 and ∞. To prove the latter, we extend the root poset on positive roots to a weak order on the root system and define a Bruhat order on the root system , and study the paths in those orders in order to establish a criterion to prove bipodality involving only finite dihedral reflection subgroups. [ABSTRACT FROM AUTHOR]

We study the cluster automorphism group Aut ( A ) of a coefficient free cluster algebra A of finite type. A cluster automorphism of A is a permutation of the cluster variable set X that is compatible with cluster mutations. We show that, on the one hand, by the well-known correspondence between X and the almost positive root system Φ ≥ − 1 of the corresponding Dynkin type, the piecewise-linear transformations τ + and τ − on Φ ≥ − 1 induce cluster automorphisms f + and f − of A respectively; on the other hand, excepting type D 2 n ( n ⩾ 2 ), all the cluster automorphisms of A are compositions of f + and f − . For a cluster algebra of type D 2 n ( n ⩾ 2 ), there exists an exceptional cluster automorphism induced by a permutation of negative simple roots in Φ ≥ − 1 , which is not a composition of τ + and τ − . By using these results and folding a simply laced cluster algebra, we compute the cluster automorphism group for a non-simply laced finite type cluster algebra. As an application, we show that Aut ( A ) is isomorphic to the cluster automorphism group of the FZ -universal cluster algebra of A . [ABSTRACT FROM AUTHOR]

We present an explicit difference equation for the Heckman–Opdam hypergeometric function associated with root systems. Via a confluent hypergeometric limit, an analogous difference equation is obtained for the class-one Whittaker function diagonalizing the open quantum Toda chain. [ABSTRACT FROM AUTHOR]

In this study, we investigate a new class of Lie algebras, i.e., tame locally extended affine Lie algebras of nullity 1, which are an infinite-rank analog of affine Lie algebras. This type of algebra is called a locally affine Lie algebra. A certain ideal of a locally affine Lie algebra, called a core, is a universal extension of a local version of a loop algebra, which is called a locally loop algebra. We classify locally loop algebras and locally affine Lie algebras. [ABSTRACT FROM AUTHOR]

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t → ∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q -Whittaker function. These coefficients are pure q -powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebras for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G 2 , and also establish the relations of our extremal degrees to minimal q -degrees of the extremal terms of the Kostant q -partition function; they coincide with the latter only for some root systems. [ABSTRACT FROM AUTHOR]

Our study addressed biomass dynamics in traditional and newly introduced Jatropha curcas production systems in Burkina Faso, West Africa. Five prevailing J. curcas systems included interplanting with annual crops, intensely managed plantations, afforestation of abandoned land, plantings along contour stone walls, and traditional living fences. Measurements of stem diameter, tree height (n = 670) and above- and below-ground biomass (n = 157) enabled the development of generic allometric models relating shoot and root biomass with stem diameter. The relations showed very good fits (R 2 > 0.9) for all studied systems, except afforestation sites which largely perished. Considering system-specific height-diameter (HD) relationship improved the model performance for living fences where trees allocated more biomass in the height rather than diameter growth. Self-propagation in living fences results in variable stand density, tree age, and consequently shifting HD relations. Therefore these models would benefit from local calibrations should they be applied elsewhere. We argue that for the other systems the developed generic equations are applicable subject to accounting for the tree ontogenetic stages deduced from HD relations. In this respect, the allometric models for juvenile trees are most robust whereas overall validity range of the equations can be improved with more observations of large-size trees. [ABSTRACT FROM AUTHOR]

Abstract: A minimal family of curves on an embedded surface is defined as a 1-dimensional family of rational curves of minimal degree, which cover the surface. We classify such minimal families using constructive methods. This allows us to compute the minimal families of a given surface. The classification of minimal families of curves can be reduced to the classification of minimal families which cover weak Del Pezzo surfaces. We classify the minimal families of weak Del Pezzo surfaces and present a table with the number of minimal families of each weak Del Pezzo surface up to Weyl equivalence. As an application of this classification we generalize some results of Schicho. We classify algebraic surfaces that carry a family of conics. We determine the minimal lexicographic degree for the parametrization of a surface that carries at least 2 minimal families. [Copyright &y& Elsevier]

Abstract: A series expansion for Heckman–Opdam hypergeometric functions is obtained for all . As a consequence, estimates for away from the walls of a Weyl chamber are established. We also characterize the bounded hypergeometric functions and thus prove an analogue of the celebrated theorem of Helgason and Johnson on the bounded spherical functions on a Riemannian symmetric space of the noncompact type. The -theory for the hypergeometric Fourier transform is developed for . In particular, an inversion formula is proved when . [Copyright &y& Elsevier]

Abstract: We introduce a representation of the double affine Hecke algebra at the critical level in terms of difference-reflection operators and use it to construct an explicit integrable discrete Laplacian on the Weyl alcove corresponding to an element in the center. The Laplacian in question is to be viewed as an integrable discretization of the conventional Laplace operator on Euclidean space perturbed by a delta-potential supported on the reflection hyperplanes of the affine Weyl group. The Bethe Ansatz method is employed to show that our discrete Laplacian and its commuting integrals are diagonalized by a finite-dimensional basis of periodic Macdonald spherical functions. [Copyright &y& Elsevier]

Abstract: We establish orthogonality relations for the Baker–Akhiezer (BA) eigenfunctions of the Macdonald difference operators. We also obtain a version of Cherednik–Macdonald–Mehta integral for these functions. As a corollary, we give a simple derivation of the norm identity and Cherednik–Macdonald–Mehta integral for Macdonald polynomials. In the appendix written by the first author, we prove a summation formula for BA functions. We also consider more general identities of Cherednik type, which we use to introduce and construct more general, twisted BA functions. This leads to a construction of new quantum integrable models of Macdonald–Ruijsenaars type. [Copyright &y& Elsevier]

Abstract: Let be a semisimple Lie algebra of finite dimension. The nullcone of is the set of in such that x and y are nilpotents and are in the same Borel subalgebra. The main result of this paper is that is a closed and irreducible subvariety of and its normalization morphism is bijective. [Copyright &y& Elsevier]

Abstract: If x and y are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group W, we say that x dominates y if for all , wy is a negative root whenever wx is a negative root. We call a positive root elementary if it does not dominate any positive root other than itself. The set of all elementary roots is denoted by . It has been proved by B. Brink and R.B. Howlett [B. Brink, R.B. Howlett, A finiteness property and an automatic structure of Coxeter groups, Math. Ann. 296 (1993) 179–190] that is finite if (and only if) W is a finite-rank Coxeter group. Amongst other things, this finiteness property enabled Brink and Howlett to establish the automaticity of all finite-rank Coxeter groups. Later Brink has also given a complete description of the set for arbitrary finite-rank Coxeter groups [B. Brink, The set of dominance-minimal roots, J. Algebra 206 (1998) 371–412]. However the set of non-elementary positive roots has received little attention in the literature. In this paper we answer a collection of questions concerning the dominance behavior between such non-elementary positive roots. In particular, we show that for any finite-rank Coxeter group and for any non-negative integer n, the set of roots each dominating precisely n other positive roots is finite. We give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation. [Copyright &y& Elsevier]

Abstract: For certain Lie algebras , we can use a grading and define a quartic form and a skew-symmetric bilinear form on , thereby constructing a Freudenthal triple system. The structure of the Freudenthal triple system is examined using root system methods available in the Lie algebra context. In the cases (where is the minuscule representation of ) and , we determine the groups stabilizing the quartic form and both the quartic and bilinear forms. [Copyright &y& Elsevier]

Abstract: Let be a finite-dimensional Lie algebra and M be a -module. The Fernando–Kac subalgebra of associated to M is the subset of all elements which act locally finitely on M. A subalgebra for which there exists an irreducible module M with is called a Fernando–Kac subalgebra of . A Fernando–Kac subalgebra of is of finite type if in addition M can be chosen to have finite Jordan–Hölder -multiplicities. Under the assumption that is simple, I. Penkov has conjectured an explicit combinatorial criterion describing all Fernando–Kac subalgebras of finite type which contain a Cartan subalgebra. In the present paper we prove this conjecture for . [Copyright &y& Elsevier]

Abstract: In this paper we show that for a simply-laced root system a choice of a Coxeter element C gives rise to a natural construction of the Dynkin diagram, in which vertices of the diagram correspond to C-orbits in R; moreover, it gives an identification of R with a certain subset of , where h is the Coxeter number. The set has a natural quiver structure; we call it the periodic Auslander–Reiten quiver. This gives a combinatorial construction of the root system associated with the Dynkin diagram I: roots are vertices of , and the root lattice and the inner product admit an explicit description in terms of . Finally, we relate this construction to the theory of quiver representations. [Copyright &y& Elsevier]

Abstract: Let G be the group or . We compute the integral of on the Hurwitz locus of curves admitting a degree 4 cover of having monodromy group G. We compute the generating functions for these integrals and write them as a trigonometric expression summed over the positive roots of the and root systems, respectively. As an application, we prove the Crepant Resolution Conjecture for the orbifolds and . [Copyright &y& Elsevier]

Abstract: The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite-dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood–Richardson coefficients. [Copyright &y& Elsevier]

Abstract: We show that lattice polytopes cut out by root systems of classical type are normal and Koszul, generalizing a well-known result of Bruns, Gubeladze, and Trung in type A. We prove similar results for Cayley sums of collections of polytopes whose Minkowski sums are cut out by root systems. The proofs are based on a combinatorial characterization of diagonally split toric varieties. [Copyright &y& Elsevier]

Abstract: This paper studies a class of embeddings of “permutation” root systems, with properties abstracting those of examples of embeddings of linear root systems over commutative rings, as considered in the first paper in this series. The root system of one Coxeter group W (regarded as a permutation representation) is realized with its roots as sets of roots of another Coxeter group , with action induced by an embedding of W in (a completion of) . [Copyright &y& Elsevier]

Abstract: There is a well-known presentation for finite and affine Weyl groups called the presentation by conjugation. Recently, it has been proved that this presentation holds for certain sub-classes of extended affine Weyl groups, the Weyl groups of extended affine root systems. In particular, it is shown that if nullity is ⩽2, an -type extended affine Weyl group has the presentation by conjugation. We set up a general framework for the study of simply laced extended affine Weyl groups. As a result, we obtain certain necessary and sufficient conditions for an -type extended affine Weyl group of arbitrary nullity to have the presentation by conjugation. This gives an affirmative answer to a conjecture that there are extended affine Weyl groups which are not presented by “presentation by conjugation.” [Copyright &y& Elsevier]

Abstract: We complete the classification of the irreducible 2F-modules for quasi-simple groups by dealing with the two open cases for sporadic groups from [R.M. Guralnick, G. Malle, Classification of 2F-modules, I, J. Algebra 257 (2002) 348–372]. The techniques developed involve abelian sets of roots within root systems, and enable us to determine the 2-ranks of the Monster, the Baby Monster and the double cover of the latter. [Copyright &y& Elsevier]

Abstract: Let W be a Weyl group whose type is a simply laced Dynkin diagram. On several W-orbits of sets of mutually commuting reflections, a poset is described which plays a role in linear representations of the corresponding Artin group A. The poset generalizes many properties of the usual order on positive roots of W given by height. In this paper, a linear representation of the positive monoid of A is defined by use of the poset. [Copyright &y& Elsevier]

In this article we introduce the notion of a regular partition of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi's classical result that each component of the Shi arrangement of an affine Coxeter group has a unique minimal length element. We further develop the theory of cone types in Coxeter groups by identifying the minimal set of roots required to express a cone type as an intersection of half-spaces. This set of boundary roots is closely related to the elementary inversion sets of Brink and Howlett, and also to the notion of the base of an inversion set introduced by Dyer. [ABSTRACT FROM AUTHOR]

The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type E 8. This approach is also applicable to E 8. Even if the algebra of E 8 Jacobi forms is known to be non-free, we still derive a new structure result. [ABSTRACT FROM AUTHOR]

Coastal areas are at increasing risk from flooding and erosion due to coastal development and climate change. Seagrass meadows, like other coastal ecosystems, can stabilise sediments, indirectly reducing coastal erosion. There is evidence that seagrass can reduce both the erosion of seabeds and lateral erosion of seagrass cliffs. However, these previous studies have only considered seagrass species whose below-ground biomass form dense root mats. This study aimed to investigate whether Enhalus acoroides , a species which does not form root mats, reduces lateral erosion rates of seagrass cliffs. To do this, samples of E. acoroides collected in sandy sediments within Xincun Bay China were transplanted into a wave flume. Lateral erosion rates were measured under exposure to waves for five cliff heights over 1 h. There was an interaction between cliff height and time on the rate of erosion, with higher erosion rates earlier in the experiment for larger cliff heights. The cores with the E. acoroides persisted longer than unvegetated cores, regardless of cliff height, but this result was not assessed statistically due to constraints in the analyses. The below-ground biomass of E. acoroides samples in this study was higher than values previously reported in the literature. Overall, these results suggest that non-dense root-mat forming seagrass species may not provide equivalent sediment stabilisation as dense root-mat-forming species, likely due to the reduced binding structure of the roots in species such as Enhalus. [Display omitted] • Erosion rates of submerged Enhalus acoroides cliffs increased with cliff height. • Presence of the seagrass Enhalus acoroides did not affect rates of erosion. • Unlike all seagrass species used in previous measurements of erosion, Enhalus acoroides does not form dense root mats. • There is a significant interaction between cliff height and exposure time on the volume of sediment lost from the cores. • The effect of sediment grain size on rates of erosion within seagrass cliffs is unknown. [ABSTRACT FROM AUTHOR]

For 25 orthogonal groups of signature (2 , n) related to the root lattices A 1 , 2 A 1 , 3 A 1 , 4 A 1 , A 2 , A 3 , A 4 , A 5 , A 6 , A 7 , D 4 , D 5 , D 6 , D 7 , D 8 , E 6 , E 7 , we prove that the algebras of modular forms on symmetric domains of type IV are freely generated by the additive lifts of some special Jacobi forms. The proof is universal and elementary. [ABSTRACT FROM AUTHOR]

We study the polyhedral geometry of the hyperplanes orthogonal to the weights of the first and the second fundamental representations of sl n inside the dual fundamental Weyl chamber. We obtain generating functions that enumerate the flats and the faces of a fixed dimension. In addition, we describe the extreme rays of the incidence geometry and classify simplicial faces. From the perspective of supersymmetric gauge theories with 8 supercharges in five-dimensional spacetime, the poset of flats is isomorphic to the network of mixed Coulomb-Higgs branches. On the other hand, the poset of faces is conjectured to be isomorphic to the network of crepant partial resolutions of an elliptic fibration with gauge algebra sl n and "matter representation" given by the sum of the first two fundamental representations. [ABSTRACT FROM AUTHOR]