We investigate the block basis for modular coinvariants of finite pseudo-reflection groups.We are particularly interested in the case of a subgroup G of the parabolic subgroups of GLn(q)which generalizes the Weyl grou...We investigate the block basis for modular coinvariants of finite pseudo-reflection groups.We are particularly interested in the case of a subgroup G of the parabolic subgroups of GLn(q)which generalizes the Weyl groups of restricted Cartan typeLiealgebra.展开更多
An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the stud...An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic,and some other cases.Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g=Lie(G).It turns out that G has many same properties as reductive groups,such as the Bruhat decomposition.In this note,we obtain an analogue of classical Chevalley restriction theorem for g,which says that the G-invariant ring F[g]~G is a polynomial ring if g satisfies a certain“positivity”condition suited for lots of cases we are interested in.As applications,we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.展开更多
Buckyballs(C_(60)) are linked to one end and two ends of linear poly(ethylene oxide)(PEO) chains through highly efficient click chemistry to obtain giant amphiphilic molecules C_(60)-PEO and C_(60)-PEO-C_(60), respect...Buckyballs(C_(60)) are linked to one end and two ends of linear poly(ethylene oxide)(PEO) chains through highly efficient click chemistry to obtain giant amphiphilic molecules C_(60)-PEO and C_(60)-PEO-C_(60), respectively. C_(60)-PEO and C_(60)-PEO-C_(60) molecules are spread on water surface and then transferred to solid substrates with Langmuir-Blodgett(LB) film deposition approach. C_(60)-PEO and C_(60)-PEO-C_(60) exhibit fractal growth behavior on the solid substrate under certain conditions owing to the crystallization ability of PEO segment. PEO chain length and the end capped mode both affect the fractal growth pattern.展开更多
基金supported by NSFC(No.12101544)Fundamental Research Funds of Yunnan Province(No.202301AT070415).
文摘We investigate the block basis for modular coinvariants of finite pseudo-reflection groups.We are particularly interested in the case of a subgroup G of the parabolic subgroups of GLn(q)which generalizes the Weyl groups of restricted Cartan typeLiealgebra.
基金Supported by NSFC(Grant Nos.12071136,11671138,11771279,12101544)Shanghai Key Laboratory of PMMP(Grant No.13dz2260400)the Fundamental Research Funds of Yunnan Province(Grant No.2020J0375)。
文摘An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical.Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic,and some other cases.Let G be a connected semi-reductive algebraic group over an algebraically closed field F and g=Lie(G).It turns out that G has many same properties as reductive groups,such as the Bruhat decomposition.In this note,we obtain an analogue of classical Chevalley restriction theorem for g,which says that the G-invariant ring F[g]~G is a polynomial ring if g satisfies a certain“positivity”condition suited for lots of cases we are interested in.As applications,we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.
基金financially supported by Fundamental Research Funds for the Central Universitythe National Natural Science Foundation of China (No. 51373032)
文摘Buckyballs(C_(60)) are linked to one end and two ends of linear poly(ethylene oxide)(PEO) chains through highly efficient click chemistry to obtain giant amphiphilic molecules C_(60)-PEO and C_(60)-PEO-C_(60), respectively. C_(60)-PEO and C_(60)-PEO-C_(60) molecules are spread on water surface and then transferred to solid substrates with Langmuir-Blodgett(LB) film deposition approach. C_(60)-PEO and C_(60)-PEO-C_(60) exhibit fractal growth behavior on the solid substrate under certain conditions owing to the crystallization ability of PEO segment. PEO chain length and the end capped mode both affect the fractal growth pattern.
