In this paper, first we investigate the invariant rings of the finite groups G ≤ GL（n, F;） generated by i-transvections and i-reflections with given invariant subspaces H over a finite field F;in the modular case. ...In this paper, first we investigate the invariant rings of the finite groups G ≤ GL（n, F;） generated by i-transvections and i-reflections with given invariant subspaces H over a finite field F;in the modular case. Then we are concerned with general groups G;（ω） and G;（ω）;named generalized transvection groups where ωis a k-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay,Gorenstein, complete intersection, polynomial and Poincare series of these rings.展开更多
The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R3n Jordan blocks....The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R3n Jordan blocks. We use an algorithm based on the notion of transvectants from classical invariant theory known as boosting to equivariants in determining the normal form when the Stanley decomposition for the ring of invariants is known.展开更多
We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpo...We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions.Let G=U_(n)(2).It is well known that G is a 3-transposition group with the set D,the conjugacy class consisting of its transvections,as the set of 3-transpositions.Let L be a set of basic transpositions in D.We give general descriptions of L and 1-(ν,κ,λ)designs D=(P,B),with P=D and B={L^(9)|g∈G}.The parameters k=|L|,λ and further properties of D are determined.We also,as examples,apply the method to the unitary simple groups U_(4)(2),U_(5)(2),U_(6)(2),U_(7)(2),U_(8)(2)and U_(9)(2).展开更多
文摘In this paper, first we investigate the invariant rings of the finite groups G ≤ GL（n, F;） generated by i-transvections and i-reflections with given invariant subspaces H over a finite field F;in the modular case. Then we are concerned with general groups G;（ω） and G;（ω）;named generalized transvection groups where ωis a k-th root of unity. By constructing quotient group and tensor, we calculate their invariant rings. In the end, we determine the properties of Cohen-Macaulay,Gorenstein, complete intersection, polynomial and Poincare series of these rings.
文摘The concept of normal form is used to study the dynamics of non-linear systems. In this work we describe the normal form for vector fields on 3 × 3 with linear nilpotent part made up of coupled R3n Jordan blocks. We use an algorithm based on the notion of transvectants from classical invariant theory known as boosting to equivariants in determining the normal form when the Stanley decomposition for the ring of invariants is known.
文摘We aim to study maximal pairwise commuting sets of 3-transpositions(transvections)of the simple unitary group U_(n)(2)over GF(4),and to construct designs from these sets.Any maximal set of pairwise commuting 3-transpositions is called a basic set of transpositions.Let G=U_(n)(2).It is well known that G is a 3-transposition group with the set D,the conjugacy class consisting of its transvections,as the set of 3-transpositions.Let L be a set of basic transpositions in D.We give general descriptions of L and 1-(ν,κ,λ)designs D=(P,B),with P=D and B={L^(9)|g∈G}.The parameters k=|L|,λ and further properties of D are determined.We also,as examples,apply the method to the unitary simple groups U_(4)(2),U_(5)(2),U_(6)(2),U_(7)(2),U_(8)(2)and U_(9)(2).
