scholarly journals Finite free convolutions via Weingarten calculus

2020 ◽  
pp. 2150038
Author(s):  
Jacob Campbell ◽  
Zhi Yin
Keyword(s):  
Unitary Groups ◽  
Orthogonal Groups ◽  
Signed Permutation ◽  
Permutation Matrices ◽  
Explicit Formulae ◽  
Free Convolutions

We consider the three finite free convolutions for polynomials studied in a recent paper by Marcus, Spielman and Srivastava. Each can be described either by direct explicit formulae or in terms of operations on randomly rotated matrices. We present an alternate approach to the equivalence between these descriptions, based on combinatorial Weingarten methods for integration over the unitary and orthogonal groups. A key aspect of our approach is to identify a certain quadrature property, which is satisfied by some important series of subgroups of the unitary groups (including the groups of unitary, orthogonal, and signed permutation matrices), and which yields the desired convolution formulae.

scholarly journals Sur l'homologie des groupes unitaires à coefficients polynomiaux

2012 ◽  
Vol 10 (1) ◽  
pp. 87-139 ◽  
Author(s):  
Aurélien Djament
Keyword(s):  
Unitary Groups ◽  
Symmetric Power ◽  
Orthogonal Groups ◽  
Tensor Power ◽  
Full Generality ◽  
Standard Methods ◽  
Homology Groups ◽  
Special Cases ◽  
Functor Homology ◽  
Kunneth Formula

AbstractLet A be a ring with anti-involution and F a nice functor (tensor or symmetric power, for example) from finitely-generated projective A-modules to abelian groups. We show that the homology of the hyperbolic unitary groups Un,n(A) with coefficients in F(A2n) can be expressed stably (i.e. after taking the colimit over n) by the homology of these groups with untwisted coefficients and functor homology groups that we can compute in suitable cases (for example, when A is a field of characteristic 0 or a ring without ℤ-torsion and F a tensor power). This extends the result where A is a finite field, which was dealt with previously by C. Vespa and the author (Ann. Sci. ENS, 2010).The proof begins by relating, without any assumption on F, our homology groups to the homology of a category of hermitian spaces with coefficients twisted by F. Then, when F is polynomial, we establish — following a method due to Scorichenko — an isomorphism between this homology and the homology of another category of (possibly degenerate) hermitian spaces, which is computable (in good cases) by standard methods of homological algebra in functor categories (using adjunctions, Künneth formula…). We give some examples.Finally, we deal with the analogous problem for non-hyperbolic unitary groups in some special cases, for example euclidean orthogonal groups On (A) (the ring A being here commutative). The isomorphism between functor homology and group homology with twisted coefficients does not hold in full generality; nevertheless we succeed to get it when A is a field or, for example, a subring of ℚ containing ℤ[1/2]. The method, which is similar to that in the previous case, uses a general result of symmetrisation in functor homology proved at the beginning of the article.

Sandwich classification for O2n+1(R) and U2n+1(R,Δ) revisited

2018 ◽  
Vol 21 (4) ◽  
pp. 539-571 ◽  
Author(s):  
Raimund Preusser
Keyword(s):  
Natural Number ◽  
Commutative Ring ◽  
Unitary Groups ◽  
Orthogonal Groups

AbstractIn a recent paper, the author proved that if {n\geq 3} is a natural number, R a commutative ring and {\sigma\in GL_{n}(R)}, then {t_{kl}(\sigma_{ij})} where {i\neq j} and {k\neq l} can be expressed as a product of 8 matrices of the form {{}^{\varepsilon}\sigma^{\pm 1}} where {\varepsilon\in E_{n}(R)}. In this article we prove similar results for the odd-dimensional orthogonal groups {O_{2n+1}(R)} and the odd-dimensional unitary groups {U_{2n+1}(R,\Delta)} under the assumption that R is commutative and {n\geq 3}. This yields new, short proofs of the Sandwich Classification Theorems for the groups {O_{2n+1}(R)} and {U_{2n+1}(R,\Delta)}.

scholarly journals Odd-dimensional orthogonal groups as amalgams of unitary groups.

2007 ◽  
Vol 316 (2) ◽  
pp. 591-607 ◽  
Author(s):  
R. Gramlich ◽  
M. Horn ◽  
W. Nickel
Keyword(s):  
Unitary Groups ◽  
Orthogonal Groups

scholarly journals GENERAL MOMENTS OF MATRIX ELEMENTS FROM CIRCULAR ORTHOGONAL ENSEMBLES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250005 ◽  
Author(s):  
SHO MATSUMOTO
Keyword(s):  
Matrix Element ◽  
Unitary Groups ◽  
Matrix Elements ◽  
Orthogonal Groups ◽  
Systematic Method ◽  
Single Matrix ◽  
Orthogonal Ensembles ◽  
Explicit Expressions

The aim of this paper is to present a systematic method for computing moments of matrix elements taken from circular orthogonal ensembles (COE). The formula is given as a sum of Weingarten functions for orthogonal groups but the technique for its proof involves Weingarten calculus for unitary groups. As an application, explicit expressions for the moments of a single matrix element of a COE matrix are given.

scholarly journals Even-dimensional orthogonal groups as amalgams of unitary groups

2005 ◽  
Vol 284 (1) ◽  
pp. 141-173 ◽  
Author(s):  
R. Gramlich ◽  
C. Hoffman ◽  
W. Nickel ◽  
S. Shpectorov
Keyword(s):  
Unitary Groups ◽  
Orthogonal Groups

scholarly journals Quantum symmetries on noncommutative complex spheres with partial commutation relations

2018 ◽  
Vol 21 (04) ◽  
pp. 1850028
Author(s):  
Simeng Wang
Keyword(s):  
Quantum Groups ◽  
Unitary Groups ◽  
Symmetry Groups ◽  
Orthogonal Groups ◽  
Commutation Relations ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Free Independence

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

scholarly journals Odd-dimensional orthogonal groups as amalgams of unitary groups. Part 1: General simple connectedness

2007 ◽  
Vol 312 (1) ◽  
pp. 426-444 ◽  
Author(s):  
C. Bennett ◽  
R. Gramlich ◽  
C. Hoffman ◽  
S. Shpectorov
Keyword(s):  
Unitary Groups ◽  
Orthogonal Groups ◽  
Simple Connectedness

scholarly journals Existence of finite groups with classical commutator subgroup

1978 ◽  
Vol 25 (1) ◽  
pp. 41-44 ◽  
Author(s):  
Michael D. Miller
Keyword(s):  
Abelian Group ◽  
Finite Fields ◽  
Finite Groups ◽  
Commutator Subgroup ◽  
Unitary Groups ◽  
The Other ◽  
Linear Groups ◽  
Normal Subgroups ◽  
Orthogonal Groups ◽  
General Linear Groups

AbstractGiven a group G, we may ask whether it is the commutator subgroup of some group G. For example, every abelian group G is the commutator subgroup of a semi-direct product of G x G by a cyclic group of order 2. On the other hand, no symmetric group Sn(n>2) is the commutator subgroup of any group G. In this paper we examine the classical linear groups over finite fields K of characteristic not equal to 2, and determine which can be commutator subgroups of other groups. In particular, we settle the question for all normal subgroups of the general linear groups GLn(K), the unitary groups Un(K) (n≠4), and the orthogonal groups On(K) (n≧7).

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