scholarly journals Generators and relations for the unitary group of a skew hermitian form over a local ring

2018 ◽  
Vol 552 ◽  
pp. 1-28 ◽  
Author(s):  
J. Cruickshank ◽  
F. Szechtman
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Hermitian Form ◽  
Generators And Relations

Normal structure of the unitary group over a local ring

1984 ◽  
Vol 27 (4) ◽  
pp. 2944-2954 ◽  
Author(s):  
S. L. Krupetskii
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Normal Structure

The Weil Character of the Unitary Group Associated to a Finite Local Ring

2002 ◽  
Vol 54 (6) ◽  
pp. 1229-1253 ◽  
Author(s):  
Roderick Gow ◽  
Fernando Szechtman
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Weil Representation ◽  
Finite Local Ring

AbstractLetR/R be a quadratic extension of finite, commutative, local and principal rings of odd characteristic. Denote byUn(R) the unitary group of ranknassociated toR/R. The Weil representation ofUn(R) is defined and its character is explicitly computed.

scholarly journals UNITARY GROUPS OVER LOCAL RINGS

2013 ◽  
Vol 13 (02) ◽  
pp. 1350093 ◽  
Author(s):  
J. CRUICKSHANK ◽  
A. HERMAN ◽  
R. QUINLAN ◽  
F. SZECHTMAN
Keyword(s):  
Local Ring ◽  
Unitary Group ◽  
Quadratic Extension ◽  
Unitary Groups ◽  
Weil Representation ◽  
Local Rings ◽  
Hermitian Forms ◽  
Finite Local Ring

We study hermitian forms and unitary groups defined over a local ring, not necessarily commutative, equipped with an involution. When the ring is finite we obtain formulae for the order of the unitary groups as well as their point stabilizers, and use these to compute the degrees of the irreducible constituents of the Weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring.

scholarly journals The build-up construction of quasi self-dual codes over a non-unital ring

2021 ◽  
pp. 2250143
Author(s):  
Adel Alahmadi ◽  
Alaa Altassan ◽  
Hatoon Shoaib ◽  
Amani Alkathiry ◽  
Alexis Bonnecaze ◽  
...  
Keyword(s):  
Local Ring ◽  
Recursive Construction ◽  
Dual Codes ◽  
Generators And Relations ◽  
Unital Ring ◽  
Type Iv ◽  
Orthogonal Codes ◽  
Construction Of Self

There is a local ring [Formula: see text] of order [Formula: see text] without identity for the multiplication, defined by generators and relations as [Formula: see text] We study a recursive construction of self-orthogonal codes over [Formula: see text] We classify, up to permutation equivalence, self-orthogonal codes of length [Formula: see text] and size [Formula: see text] (called here quasi self-dual codes or QSD) up to the length [Formula: see text]. In particular, we classify Type IV codes (QSD codes with even weights) up to [Formula: see text].

scholarly journals The subnormal structure of classical-like groups over commutative rings

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Raimund Preusser
Keyword(s):  
Unitary Group ◽  
Hermitian Form ◽  
Commutative Rings ◽  
Unitary Groups ◽  
Elementary Subgroup ◽  
Sandwich Theorem ◽  
Subnormal Structure ◽  
Form Ring ◽  
K 12 ◽  
Subnormal Subgroups

AbstractLet 𝑛 be an integer greater than or equal to 3, and let (R,\Delta) be a Hermitian form ring, where 𝑅 is commutative. We prove that if 𝐻 is a subgroup of the odd-dimensional unitary group \operatorname{U}_{2n+1}(R,\Delta) normalised by a relative elementary subgroup \operatorname{EU}_{2n+1}((R,\Delta),(I,\Omega)), then there is an odd form ideal (J,\Sigma) such that\operatorname{EU}_{2n+1}((R,\Delta),(JI^{k},\Omega_{\mathrm{min}}^{JI^{k}}\dotplus\Sigma\circ I^{k}))\leq H\leq\operatorname{CU}_{2n+1}((R,\Delta),(J,\Sigma)),where k=12 if n=3 respectively k=10 if n\geq 4. As a consequence of this result, we obtain a sandwich theorem for subnormal subgroups of odd-dimensional unitary groups.

scholarly journals A Herman–Avila–Bochi formula for higher-dimensional pseudo-unitary and Hermitian-symplectic cocycles

2014 ◽  
Vol 35 (5) ◽  
pp. 1582-1591 ◽  
Author(s):  
CHRISTIAN SADEL
Keyword(s):  
Lyapunov Exponents ◽  
Unitary Group ◽  
Symplectic Group ◽  
Hermitian Form ◽  
Schrödinger Operators ◽  
Parameter Family ◽  
Schrodinger Operators ◽  
Type Formula ◽  
Higher Dimensional

A Herman–Avila–Bochi type formula is obtained for the average sum of the top$d$Lyapunov exponents over a one-parameter family of$\mathbb{G}$-cocycles, where$\mathbb{G}$is the group that leaves a certain, non-degenerate Hermitian form of signature$(c,d)$invariant. The generic example of such a group is the pseudo-unitary group$\text{U}(c,d)$or, in the case$c=d$, the Hermitian-symplectic group$\text{HSp}(2d)$which naturally appears for cocycles related to Schrödinger operators. In the case$d=1$, the formula for$\text{HSp}(2d)$cocycles reduces to the Herman–Avila–Bochi formula for$\text{SL}(2,\mathbb{R})$cocycles.

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