Sub-root Systems Research Articles

Subroot systems and total positivity in finite reflection groups

Given a root system R R and the corresponding finite reflection group W W let Hom ⁡ ( W , Z ^ 2 ) \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2) be the group of homomorphisms from W W into Z ^ 2 \widehat {\mathbb Z}_2 , where Z ^ 2 = { 1 , − 1 } \widehat {\mathbb Z}_2=\{1,-1\} with multiplication. We propose a procedure of constructing subroot systems of R R by using homomorphisms η ∈ Hom ⁡ ( W , Z ^ 2 ) \eta \in \operatorname {Hom}(W,\,\widehat {\mathbb Z}_2) . This construction is next used for establishing a relation between concepts of total positivity and η \eta -total positivity.

Borel–de Siebenthal Theory for Affine Reflection Systems

We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.

Changes in root hydraulic conductance in relation to the overall growth response of maize seedlings to partial root-zone nitrogen application

The influence mechanism of heterogeneous nutrient distribution on compensatory growth of maize seedlings is a key topic in the interaction between crops and soil environment. We evaluated the significance of root hydraulic conductance (Lp) and related parameter changes in water uptake and growth regulation of plants under different nitrogen (N) conditions. Maize seedlings were grown in split-root containers containing N solutions. Three N treatments were applied: (1) full-strength N (control: C), in which both sub-root systems received 4.0 mM N (each sub-root: C44); (2) partial N deficit (D), in which each sub-root system received 2.0 mM N (D42) or full-strength N (D44); and (3) partial N resupply (S), in which both sub-root systems received 2.0 mM N for six days, followed by 2.0 mM N (S22) or full-strength N (S24). The shoot dry mass in D increased gradually with early development, and was greater than that in C and S within 15–21 day, suggesting that the superiority of partial N deficit (D) in term of maize seedling growth was apparent and caused compensatory growth. Moreover, the slope and intercept in D44 and D42 between ΨL vs. Lp and shoot N content vs. Lp were obviously greater than that in C44, S24 and S22, indicating that higher plant dry mass in D might be attributable to the maintenance of a similar ΨL and improved shoot N content. In addition, the slopes and intercepts in D44 and D42 between Lp vs. root surface area and shoot N content vs. root surface area were higher than that in S24 and S22, suggesting that compared with partial N resupply, partial N deficit was more advantageous to root water uptake and N accumulation at the same level of root surface area, thus resulted in higher dry mass of maize seedlings. However, partial N resupply significantly increased root cortex thickness/diameter ratio and reduced root vessel diameter, which resulted in lower Lp, ΨL and shoot N content during 12–21 day. Unexpectedly, at 21 day, the shoot dry mass in S could recover to the level of control. The regulation mechanisms in partial N resupply will be the focus of future studies. Thus, when adopting the method of partial N application, it is necessary to consider the soil N condition before partial application.

PARABOLIC SUBROOT SYSTEMS AND THEIR APPLICATIONS

AbstractIn this note we consider parabolic subroot systems of a complex simple Lie Algebra. We describe root theoretic data of the subroot systems in terms of that of the root system and we give a selection of applications of our results to the study of generalized flag manifolds.

Root systems and quotients of deformations of simple singularities

In this article we study quotients of deformations of simple singularities, and attempt to characterise them using subsystems of simple root systems. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type Br (r≥2), Cr (r≥3), F4 or G2 by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type X=Ds, E6 or E7, but not semiuniversal anymore. Therefore not all subdiagrams of X appear as singular configurations of the fibres of the deformation. We propose a conjecture for the types of singular configurations in terms of sub-root systems of a root system of type X and prove it for types B2,B3,C3,F4 and G2.

MAXIMAL CLOSED SUBROOT SYSTEMS OF REAL AFFINE ROOT SYSTEMS

We completely classify and give explicit descriptions of all maximal closed subroot systems of real affine root systems. As an application, we describe a procedure to get the classification of all regular subalgebras of affine Kac-Moody algebras in terms of their root systems.

The root nitrogen uptake response to partial nitrogen stress is related to previous nutritional status

We investigated the dynamics and factors influencing the N uptake capacity in two sub-root systems of maize seedlings under partial N stress. Maize seedlings were grown in split-root containers containing nutrient solutions. Two N application conditions prior to partial N stress (full-strength N (4.0 mM): CP; N stress: SP) were considered. Thus, two experiments were conducted: (A) four treatments: CPNc, CPN2, CPN1 and CPN0, in which half of the root system was exposed to 4.0 mM, 2 mM, 1 mM and 0 mM N (CPNc–c, CPN2–2, CPN1–1 and CPN0–0), respectively, and the other half received full-strength N (CPNc–c, CPN2–c, CPN1–c and CPN0–c); (B) four treatments: SPNc, SPN2, SPN1 and SPN0, in which both sub-root systems received 4.0 mM, 2 mM, 1 mM and 0 mM N for 6 days, respectively, after which half of the root system was maintained at original N level (SPNc–c, SPN2–2, SPN1–1 and SPN0–0) and the other half received full-strength N (SPNc–c, SPN2–c, SPN1–c and SPN0–c). At 0.25 days after treatment (DAT), CPN2–c and CPN1–c enhanced N inflow rates (Iroot), whereas CPN0–c decreased it by 27.3% compared with CPNc–c, Iroot in CPN2–c, CPN1–c and CPN0–c was uniformly enhanced at 1 DAT, but were only enhanced in CPN0–c at 5 DAT compared with CPNc–c. In contrast, SPN1–c had a significantly increased Iroot by 10.52% compared with SPNc–c, although other treatments showed a negative effect on Iroot at 0.25 DAT. At 1 and 5 DAT, Iroot in SPN2–c, SPN2–2, SPN1–c, SPN1–1 and SPN0–c were significantly lower than that in SPNc–c. Furthermore, CP significantly enhanced Iroot in non-stressed sub-roots compared with SP. Additionally, CPN2 improved shoot dry mass and N use efficiency even under SPN2. Thus, N uptake capacity in each sub-root zone varied not only depending on the severity and duration of the N stress, but was also related to the N status prior to partial N stress. Moreover, the occurrence and disappearance of the root compensatory effect were delayed with increasing N stress severity. A higher compensatory effect developed following CP, even under CPN0.

THE -SINGULAR DICHOTOMY FOR EXCEPTIONAL LIE GROUPS AND ALGEBRAS

AbstractWe show that every orbital measure, ${\mu }_{x} $, on a compact exceptional Lie group or algebra has the property that for every positive integer either ${ \mu }_{x}^{k} \in {L}^{2} $ and the support of ${ \mu }_{x}^{k} $ has non-empty interior, or ${ \mu }_{x}^{k} $ is singular to Haar measure and the support of ${ \mu }_{x}^{k} $ has Haar measure zero. We also determine the index $k$ where the change occurs; it depends on properties of the set of annihilating roots of $x$. This result was previously established for the classical Lie groups and algebras. To prove this dichotomy result we combinatorially characterize the subroot systems that are kernels of certain homomorphisms.

Reflection subgroups and sub-root systems of the imprimitive complex reflection groups

Reflection subgroups and sub-root systems of the imprimitive complex reflection groups

Reflection subgroups and sub-root systems of the imprimitive complex reflection groups

Based on a graph-theoretic analysis, we determine all the irreducible reflection subgroups of the imprimitive complex reflection groups G(m, p, n), and describe the irreducible subsystems of all possible types in the root system R(m, p, n) of G(m, p, n).

Universal Lax Pair for Generalised CalogerosMoser Models

In this talk we introduce generalised Calogero–Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems ofthe finite reflection groups, H3, H4, and the dihedral group I2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero–Moser models and for any choices ofthe potentials are linear combinations ofthe reflection operators. The equivalence ofthe Lax pair with the equations ofmotion is proved by decomposing the root system into a sum oftwo-dimensional sub-root systems, A2, B2, G2 ,a ndI2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators.

Universal Lax Pair for Generalised Calogero–Moser Models

In this talk we introduce generalised Calogero–Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H 3, H 4, and the dihedral group I 2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero–Moser models and for any choices of the potentials are linear combinations of the reflection operators. The equivalence of the Lax pair with the equations of motion is proved by decomposing the root system into a sum of two-dimensional sub-root systems, A 2, B 2, G 2, and I 2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators.

Generalised Calogero-Moser Models and Universal Lax Pair Operators

Calogero-Moser models can be generalised for all of the finite reflection groups. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H_3, H_4, and the dihedral group I_2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero-Moser models and for any choices of the potentials are constructed as linear combinations of the reflection operators. The consistency conditions are reduced to functional equations for the coefficient functions of the reflection operators in the Lax pair. There are only four types of such functional equations corresponding to the two-dimensional sub-root systems, A_2, B_2, G_2, and I_2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators, a powerful tool for solving quantum Calogero-Moser models.

On infinite root systems

We define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a Kac-Moody Lie algebra. We study these objects from a purely formal and geometrical point of view as well as in relation to their associated Lie algebras. This leads to a coherent theory of root systems, bases, subroot systems, Lie algebras defined by root data, and subalgebras.