B-Orbits of Square Zero in Nilradical of the Symplectic Algebra

Barnea, Nurit; Melnikov, Anna

doi:10.1007/s00031-016-9401-x

Cited by 7 publications

(29 citation statements)

References 17 publications

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“…These elementary column operations result in multiplying A from the right by a matrix B ′ (1) ∈ B n . Denote 1) . Then for each i ∈ [n − 1], add a multiple (−a iσ(n) /a nσ(n) ) of the nth row of A ′ 1 to the ith row of A ′ 1 .…”

Section: Preliminarymentioning

confidence: 99%

“…Then for each i ∈ [n − 1], add a multiple (−a iσ(n) /a nσ(n) ) of the nth row of A ′ 1 to the ith row of A ′ 1 . These elementary row operations result in multiplying A ′ 1 from the left by a matrix B (1) 1) . Then a (1) nσ(n) = a nσ(n) is the only nonzero entry of its row and column in A 1 .…”

Section: Preliminarymentioning

confidence: 99%

“…(2) (Normalizing the first entry) Let (A (0) , B (0) ) := (A, B n ). Find A (1) in the B (0) -similarity orbit of A (0) = [a ij ] such that the (n − 1, n) entry of A (1) is either 0 or 1. For example,…”

Section: And Only If a Meets The Following Conditionsmentioning

confidence: 99%

“…Lemma 3.2. A ∈ M n is in QU n for a subpermutation Q ∈ N n if and only if the weighted graph of A satisfies the following conditions: (1) shows that i is not a chain head of G Q , and Lemma 2.4 (2) and the assumption Q ∈ N n show that i < σ(i) = i + < j.…”

Section: Graph Representations and Graph Operationsmentioning

confidence: 99%

“…M. Boos and M. Reineke described the B n -orbits and their closure relations of all 2-nilpotent matrices [4]. N. Barnea and A. Melnikov described the Borel orbits of 2-nilpotent elements in nilradicals for the symplectic algebra in 2017 [1]. M. Boos et al described the parabolic orbits of 2-nilpotent elements for classical groups [2,3].…”

Section: Introductionmentioning

confidence: 99%

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On triangular similarity of nilpotent triangular matrices

Tsai

Bogale

Huang

2020

Linear Algebra and its Applications

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Let Bn (resp. Un, Nn) be the set of n × n nonsingular (resp. unit, nilpotent) upper triangular matrices. We use a novel approach to explore the Bn-similarity orbits in Nn. The Belitskiȋ's canonical form of A ∈ Nn under Bn-similarity is in QUn where Q is the subpermutation such that A ∈ BnQBn. Using graph representations and Un-similarity actions stabilzing QUn, we obtain new properties of the Belitskiȋ's canonical forms and present an efficient algorithm to find the Belitskiȋ's canonical forms in Nn. As consequences, we construct new Belitskiȋ's canonical forms in all Nn's, list all Belitskiȋ's canonical forms for n = 7, 8, and show examples of 3-nilpotent Belitskiȋ's canonical forms in Nn with arbitrary numbers of parameters up to O(n 2 ).Mathematics Subject Classification 2020: Primary 15A21, Secondary 15A23, 14L30.

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Section: Preliminarymentioning

confidence: 99%

Section: Preliminarymentioning

confidence: 99%

Section: And Only If a Meets The Following Conditionsmentioning

confidence: 99%

Section: Graph Representations and Graph Operationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On triangular similarity of nilpotent triangular matrices

Tsai

Bogale

Huang

2020

Linear Algebra and its Applications

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B–Orbits in Abelian Nilradicals of Types B, C and D: Towards a Conjecture of Panyushev

Barnea

Melnikov

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. Let B be a Borel subgroup of a semisimple algebraic group G and let m be an abelian nilradical in b = Lie(B). Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to m, D. Panyushev [5] gives in particular classification of B−orbits in m and m * and states general conjectures on the closure and dimensions of the B−orbits in both m and m * in terms of involutions of the Weyl group. Using Pyasetskii correspondence between B−orbits in m and m * he shows the equivalence of these two conjectures. In this Note we prove his conjecture in types B n , C n and D n for adjoint case.1. Abelian nilradicals and Panyushev's conjecture 1.1. Minimal nilradicals. Let G be a semisimple linear algebraic group over C and let g be its Lie algebra. Let B be its Borel subgroup and b = Lie(B). Let g = n ⊕ h ⊕ n − be its corresponding triangular decomposition, where b = n ⊕ h. B acts adjointly on n. For x ∈ n let B.x denote its orbit.Since the description of B−orbits in n immediately reduces to simple Lie algebras in what follows we assume that g is simple.Let R be the root system of g and W its Weyl group. For α ∈ R let s α be the corresponding reflection in W .Let R + (resp. R − ) denote the subset of positive (resp. negative) roots. For α ∈ R let X α denote the standard root vector in g so that n =+ be a set of simple roots. Let θ be the maximal root in R + . Recall that any standard parabolic subgroup P of G is of the form P = L ⋉ M where L is a standard Levy subgroup and M is the unipotent radical of P. If R L is the root system of l = Lie(L) then ∆ L = ∆ ∩ R L . Let W P denote Weyl group of l. Let w be the longest element of W P .P is maximal if and only if ∆ L = ∆ \ {α i }. We will write

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