scholarly journals Structure of Hyperbolic Unitary Groups II: Classification of E-Normal Subgroups

2017 ◽  
Vol 24 (02) ◽  
pp. 195-232 ◽  
Author(s):  
Raimund Preusser
Keyword(s):  
Unitary Group ◽  
Direct Limit ◽  
Finite Ring ◽  
Unitary Groups ◽  
Normal Subgroups ◽  
Finite Rings ◽  
Elementary Subgroup ◽  
Rings With Involution ◽  
Hyperbolic Unitary Group

This paper proves the sandwich classification conjecture for subgroups of an even dimensional hyperbolic unitary group [Formula: see text] which are normalized by the elementary subgroup [Formula: see text], under the condition that R is a quasi-finite ring with involution, i.e., a direct limit of module finite rings with involution, and [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.

ON -DISTINGUISHED REPRESENTATIONS OF THE QUASI-SPLIT UNITARY GROUPS

2019 ◽  
pp. 1-52 ◽  
Author(s):  
A. Mitra ◽  
Omer Offen
Keyword(s):  
Unitary Group ◽  
Discrete Series ◽  
Large Family ◽  
Quadratic Extension ◽  
Base Change ◽  
Unitary Groups ◽  
Distinguished Representations ◽  
Tempered Representation

We study $\text{Sp}_{2n}(F)$ -distinction for representations of the quasi-split unitary group $U_{2n}(E/F)$ in $2n$ variables with respect to a quadratic extension $E/F$ of $p$ -adic fields. A conjecture of Dijols and Prasad predicts that no tempered representation is distinguished. We verify this for a large family of representations in terms of the Mœglin–Tadić classification of the discrete series. We further study distinction for some families of non-tempered representations. In particular, we exhibit $L$ -packets with no distinguished members that transfer under base change to $\text{Sp}_{2n}(E)$ -distinguished representations of $\text{GL}_{2n}(E)$ .

A classification of subgroups of odd unitary groups

2018 ◽  
Vol 46 (9) ◽  
pp. 3795-3805 ◽  
Author(s):  
Weibo Yu ◽  
Yaya Li ◽  
Hang Liu
Keyword(s):  
Unitary Groups

Rings With Involution Whose Symmetric Units Commute

1976 ◽  
Vol 28 (5) ◽  
pp. 915-928 ◽  
Author(s):  
Charles Lanski
Keyword(s):  
Unitary Group ◽  
Simple Ring ◽  
Algebraic Properties ◽  
Rings With Involution ◽  
Additional Assumptions

In the last few years many results have appeared which deal with questions of how various algebraic properties of the symmetric elements of a ring with involution, or the subring they generate, affect the structure of the whole ring. If the ring has an identity, similar questions may be posed by making assumptions about the symmetric units or subgroup they generate. Little seems to be known about the special units which exist in rings with involution, although several questions of importance have existed for some time. For example, given a simple ring with appropriate additional assumptions, is the unitary group essentially simple? Also, what can be said about the structure of subspaces invariant under conjugation by all unitary or symmetric units (see [7])?

scholarly journals Fast Constructive Recognition of Black-Box Unitary Groups

2003 ◽  
Vol 6 ◽  
pp. 162-197 ◽  
Author(s):  
Peter A. Brooksbank
Keyword(s):  
Homomorphic Image ◽  
Polynomial Time ◽  
Unitary Group ◽  
Black Box ◽  
Unitary Groups ◽  
Discrete Logarithms ◽  
Cyclic Groups

AbstractIn this paper, the author presents a new algorithm to recognise, constructively, when a given black-box group is a homomorphic image of the unitary group SU(d, q) for known d and q. The algorithm runs in polynomial time, assuming the existence of oracles for handling SL(2, q) subgroups, and for computing discrete logarithms in cyclic groups of order q ± 1.

scholarly journals THE COMPLEXITY OF THE EQUIVALENCE PROBLEM OVER FINITE RINGS

2011 ◽  
Vol 54 (1) ◽  
pp. 193-199 ◽  
Author(s):  
GÁBOR HORVÁTH
Keyword(s):  
Polynomial Time ◽  
Jacobson Radical ◽  
Equivalence Problem ◽  
Finite Ring ◽  
Polynomial Equations ◽  
Finite Rings

AbstractWe investigate the complexity of the equivalence problem over a finite ring when the input polynomials are written as sum of monomials. We prove that for a finite ring if the factor by the Jacobson radical can be lifted in the centre, then this problem can be solved in polynomial time. This result provides a step in proving a dichotomy conjecture of Lawrence and Willard (J. Lawrence and R. Willard, The complexity of solving polynomial equations over finite rings (manuscript, 1997)).

Normality in Elementary Subgroups of Chevalley Groups Over Rings

1976 ◽  
Vol 28 (2) ◽  
pp. 420-428 ◽  
Author(s):  
James F. Hurley
Keyword(s):  
Normal Subgroup ◽  
Commutative Ring ◽  
Lie Algebra ◽  
Chevalley Group ◽  
Complex Field ◽  
Normal Subgroups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Elementary Subgroup ◽  
Finite Dimensional

In [6] we have constructed certain normal subgroups G7 of the elementary subgroup GR of the Chevalley group G(L, R) over R corresponding to a finite dimensional simple Lie algebra L over the complex field, where R is a commutative ring with identity. The method employed was to augment somewhat the generators of the elementary subgroup EI of G corresponding to an ideal I of the underlying Chevalley algebra LR;EI is thus the group generated by all xr(t) in G having the property that ter ⊂ I. In [6, § 5] we noted that in general EI actually had to be enlarged for a normal subgroup of GR to be obtained.

scholarly journals Commuting elements in central products of special unitary groups

2012 ◽  
Vol 56 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Alejandro Adem ◽  
F. R. Cohen ◽  
José Manuel Gómez
Keyword(s):  
Prime Number ◽  
Moduli Space ◽  
Unitary Group ◽  
Unitary Groups ◽  
Connected Components ◽  
Flat Connections ◽  
Isomorphism Classes ◽  
Central Product ◽  
Central Products ◽  
Path Connected

AbstractWe study the space of commuting elements in the central product Gm,p of m copies of the special unitary group SU(p), where p is a prime number. In particular, a computation for the number of path-connected components of these spaces is given and the geometry of the moduli space Rep(ℤn, Gm,p) of isomorphism classes of flat connections on principal Gm,p-bundles over the n-torus is completely described for all values of n, m and p.

scholarly journals LOCAL BORCHERDS PRODUCTS FOR UNITARY GROUPS

2017 ◽  
Vol 234 ◽  
pp. 139-169
Author(s):  
ERIC HOFMANN
Keyword(s):  
Boundary Component ◽  
Unitary Group ◽  
Picard Group ◽  
Unitary Groups ◽  
Weil Representation ◽  
Cusp Forms ◽  
Modular Variety ◽  
Arithmetic Subgroup ◽  
Torsion Element ◽  
Vector Valued

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

On strongly right bounded finite rings

1991 ◽  
Vol 44 (3) ◽  
pp. 353-355 ◽  
Author(s):  
Weimin Xue
Keyword(s):  
Left Ideal ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Finite Ring ◽  
Finite Rings

An associative ring is called strongly right (left) bounded if every nonzero right (left) ideal contains a nonzero ideal. We prove that if R is a strongly right bounded finite ring with unity and the order |R| of R has no factors of the form p5, then R is strongly left bounded. This answers a question of Birkenmeier and Tucci.

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