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Suppose that is some nonempty set of finite groups. A group is saturated with groups from if each finite subgroup of lies in a subgroup isomorphic to an element of . We prove that a periodic group with locally finite centralizers of involutions which is saturated with simple unitary groups of degree 4 over finite fields of fixed odd characteristic is isomorphic to a simple unitary group of degree 4 over some locally finite field of characteristic . [ABSTRACT FROM AUTHOR]

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics. [ABSTRACT FROM AUTHOR]

Frames with special structures play a crucial role in various industrial applications, such as medical imaging, quantum communication, recognition and identification software, and so on. In this paper, we will discuss a more general setting, i.e., a g-frame induced by the projective unitary representation. We show some new results on the dilation property for g-frame generator sets for unitary groups and projective unitary representations. In particular, by using complete wandering operators, several properties of g-frame generators for projective unitary representations have been obtained. Moreover, we explore some characterizations of the g-frame generator dual pairs.

Let ϕ : P c (M 1) → P c (M 2) be a surjective Lp-isometry between Grassmann spaces of projections with the trace value c in semifinite factors M 1 and M 2 . Based on the characterization of surjective Lp-isometries of unitary groups in finite factors, we show that φ or I — φ can be extended to a *-isomorphism or a *-anti-isomorphism. In particular, φ is given by a *-(anti-)isomorphism unless M 1 and M 2 are finite and c = 1 2 . [ABSTRACT FROM AUTHOR]

We study the homotopy type of the space of the unitary group U 1 (C u * (| ℤ n |)) of the uniform Roe algebra C u * (| ℤ n |) of ℤ n . We show that the stabilizing map U 1 (C u * (| ℤ n |)) → U ∞ (C u * (| ℤ n |)) is a homotopy equivalence. Moreover, when n = 1 , 2 , we determine the homotopy type of U 1 (C u * (| ℤ n |)) , which is the product of the unitary group U 1 (C * (| ℤ n |)) (having the homotopy type of U ∞ (ℂ) or ℤ × B U ∞ (ℂ) depending on the parity of n) of the Roe algebra C * (| ℤ n |) and rational Eilenberg–MacLane spaces. [ABSTRACT FROM AUTHOR]

Suppose given a Hamiltonian and holomorphic action of G = U (2) on a compact Kähler manifold M, with nowhere vanishing moment map. Given an integral coadjoint orbit O for G, under transversality assumptions we shall consider two naturally associated 'conic' reductions. One, which will be denoted M ¯ O G , is taken with respect to the action of G and the cone over O ; another, which will be denoted M ¯ ν T , is taken with respect to the action of the standard maximal torus T ⩽ G and the ray R + ı ν along which the cone over O intersects the positive Weyl chamber. These two reductions share a common 'divisor', which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some indications in this direction. Furthermore, we give explicit transversality criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatorial data associated to the action and the orbit. [ABSTRACT FROM AUTHOR]

We describe a method to compute mod ℓ Galois representations contained in the H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in GL 3 (F 9) attached to an eigenform over a congruence subgroup of SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group PSU 3 (F 9) and ramified at 2 and 3 only. [ABSTRACT FROM AUTHOR]

We characterize those modular group algebras FG whose group of unitary units is locally nilpotent or nilpotent under the classical involution of FG. The main results of [19, 20] are trivial consequences of this note. [ABSTRACT FROM AUTHOR]

Some rank metric codes are constructed based on the actions of cyclic unitary groups and Abelian noncyclic unitary groups. All the rank metric codes studied in this paper consist of proper rectangular matrices. Most of these codes are maximum linear rank metric codes, and one is nonlinear maximum rank metric codes. [ABSTRACT FROM AUTHOR]

In this paper, we study the isospectrality problem for a free quantum particle confined in a ring with a junction, analyzing all the self-adjoint realizations of the corresponding Hamiltonian in terms of a boundary condition at the junction. In particular, by characterizing the energy spectrum in terms of a spectral function, we classify the self-adjoint realizations in two classes, identifying all the families of isospectral Hamiltonians. These two classes turn out to be discerned by the action of parity (i.e. space reflection), which plays a central role in our discussion. [ABSTRACT FROM AUTHOR]

We study the nonrigid dynamics induced by the standard birational actions of the split unitary groups , and on the compact classical Lie groups , and , respectively. More precisely, we study the geometry of endowed with the kinetic energy metric associated with the action of on assuming that carries its canonical bi-invariant Riemannian metric and has initially a homogeneous distribution of mass. By the least action principle, force-free motions (thought of as curves in ) correspond to geodesics of . The geodesic equation may be understood as an inviscid Burgers equation with Möbius constraints. We prove that the kinetic energy metric on is not complete and in particular not invariant, find symmetries and totally geodesic submanifolds of and address the question under which conditions geodesics of rigid motions are geodesics of . Besides, we study equivalences with the dynamics of conformal and projective motions of the sphere in low dimensions. [ABSTRACT FROM AUTHOR]

Orbit codes, as special constant dimension codes, have attracted much attention due to their applications for error correction in random network coding. This paper is devoted to constructing large orbit codes by making full use of unitary space. Firstly, we construct a cyclic unitary group of order $q^{2n}-1$ by means of the companion matrix of a primitive polynomial over finite fields $\mathbb {F}_{q^{2}}$ , and so the corresponding code is unitary cyclic orbit code. As a special application, a new quaternary orbit code $(6,63,4,3)$ is given. Secondly, we obtain orbit codes with large size using the external direct product of unitary groups acting on the direct sum of subspaces. Finally, a table is given for illustrating our codes improve upon those constructed by Trautmann et al. and Poroch et al.

We consider unitals of order q with two points which are centers of translation groups of order q. The group G generated by these translations induces a Moufang set on the block joining the two points. We show that G is either SL(2,Fq) (as in all classical unitals and also in some nonclassical examples), or PSL(2,Fq), or a Suzuki, or a Ree group. Moreover, G is semiregular outside the special block. [ABSTRACT FROM AUTHOR]

We give the complete description of bijective real-linear transformations on B (H) , the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space H , where every unitary U ∈ B (H) is mapped to a unitary ϕ (U). In turn, bijective real-linear maps on B (H) preserving equivalence by unitaries will be determined. [ABSTRACT FROM AUTHOR]

This article is part of a series with the aim of classifying all non-hyperoctahedral categories of two-colored partitions. Those constitute by some Tannaka-Krein type result the representation categories of a specific class of quantum groups. In Part I we introduced a class of parameters which gave rise to many new non-hyperoctahedral categories of partitions. In the present article we show that this class actually contains all possible parameter values of all non-hyperoctahedral categories of partitions. This is an important step towards the classification of all non-hyperoctahedral categories. [ABSTRACT FROM AUTHOR]

In 1895 Wiman introduced the Riemann surface 𝒲 of genus 6 over the complex field ℂ defined by the equation X6+Y6+ℨ6+(X2+Y2+ℨ2)(X4+Y4+ℨ4)−12X2Y2ℨ2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field 𝕂 of characteristic p ≥ 7. For p = 2, 3 the above polynomial is reducible over 𝕂, and for p = 5 the curve 𝒲 is rational and Aut(𝒲) ≅ PGL(2,𝕂). We also show that Wiman's 𝔽192-maximal sextic 𝒲 is not Galois covered by the Hermitian curve H19 over the finite field 𝔽192. [ABSTRACT FROM AUTHOR]

Compact quantum groups can be studied by investigating their representation categories in analogy to the Schur–Weyl/Tannaka–Krein approach. For the special class of (unitary) "easy" quantum groups, these categories arise from a combinatorial structure: rows of two-colored points form the objects, partitions of two such rows the morphisms. Vertical/horizontal concatenation and reflection give composition, monoidal product and involution. Of the four possible classes O , B , S and H of such categories (inspired, respectively, by the classical orthogonal, bistochastic, symmetric and hyperoctahedral groups), we treat the first three—the non-hyperoctahedral ones. We introduce many new examples of such categories. They are defined in terms of subtle combinations of block size, coloring and non-crossing conditions. This article is part of an effort to classify all non-hyperoctahedral categories of two-colored partitions. It is purely combinatorial in nature. The quantum group aspects are left out. [ABSTRACT FROM AUTHOR]

In this note we point out an error in a recent paper on flag-transitive block designs with gcd (r , λ) = 1 , and we prove a result correcting the error. [ABSTRACT FROM AUTHOR]

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The aim of this note is to compare work of Formanek [7] on a certain construction of central polynomials with that of Collins [3] on integration on unitary groups. These two quite disjoint topics share the construction of the same function on the symmetric group, which the second author calls Weingarten function. By joining these two approaches we succeed in giving a simplified and very natural presentation of both Formanek and Collins's Theory. [ABSTRACT FROM AUTHOR]

We discuss two fragments of a large problem that extends the author's recently completed similar studies in the space to the next dimension. One of the fragments is related to the local description of nonspherical holomorphically homogeneous strictly pseudoconvex hypersurfaces in with stabilizers of submaximal dimension. Using the Moser normal form technique and the properties of subgroups of the unitary group , we show that up to holomorphic equivalence there exist only two such surfaces. Both of them are natural generalizations of known homogeneous hypersurfaces in the space . In the second part of the paper, we consider a technique of holomorphic realization in of abstract seven-dimensional Lie algebras that correspond, in particular, to homogeneous hypersurfaces with trivial stabilizer. Some sufficient conditions for the Lie algebras are obtained under which the orbits of all realizations of such algebras are Levi degenerate. The schemes of studying holomorphically homogeneous hypersurfaces that were used in the two-dimensional (É. Cartan) and three-dimensional (Doubrov, Medvedev, and The; Fels and Kaup; Beloshapka and Kossovskiy; Loboda) situations and resulted in full descriptions of such hypersurfaces turn out to be quite efficient in the case of greater dimension of the ambient space as well. [ABSTRACT FROM AUTHOR]

In this article, we determine all pairs (D , G) , where D is a 2-design with gcd (r , λ) = 1 and G is a flag-transitive almost simple automorphism group of D whose socle is PSU (n , q) with (n , q) ≠ (3 , 2) . We prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt–Bose–Shrikhande spaces, or it is isomorphic to a design with parameters (6, 3, 2), (7, 3, 1), (8, 4, 3), (10, 6, 5), (11, 5, 2) or (28, 7, 2). In particular, if D is symmetric, then it is either the Fano plane with parameters (7, 3, 1), or the unique Hadamard design with parameters (11, 5, 2). [ABSTRACT FROM AUTHOR]

We propose a gradient-based Jacobi algorithm for a class of maximization problems on the unitary group, with a focus on approximate diagonalization of complex matrices and tensors by unitary transformations. We provide weak convergence results, and prove local linear convergence of this algorithm. The convergence results also apply to the case of real-valued tensors. [ABSTRACT FROM AUTHOR]

We classify certain categories of partitions of finite sets subject to specific rules on the coloring of points and the sizes of blocks. More precisely, we consider pair partitions such that each block contains exactly one white and one black point when rotated to one line; however, crossings are allowed. There are two families of such categories, the first of which is indexed by cyclic groups and is covered in the present article; the second family will be the content of a follow-up article. Via a Tannaka–Krein result, the categories in the two families correspond to easy quantum groups interpolating the classical unitary group U n and Wang's free unitary quantum group U n + . In fact, they are all half-liberated in some sense and our results imply that there are many more half-liberation procedures than previously expected. However, we focus on a purely combinatorial approach leaving quantum group aspects aside. [ABSTRACT FROM AUTHOR]

This paper concerns the enumeration of simultaneous conjugacy classes of tuples of commuting unitary matrices and of commuting symplectic matrices over a finite field F q of odd size. For any given conjugacy class, the orbits for the action of its centralizer group on itself by conjugation (that is, the conjugacy classes within the centralizer group) are called branches. We determine the branching rules for the unitary groups U 2 (F q) , U 3 (F q) , and for the symplectic groups S p 2 (F q) , S p 4 (F q). [ABSTRACT FROM AUTHOR]

The purpose of this article is to give a new, explicit and elementary construction of the F i 22 in 2 E 6 (2) , using the notion of M-sets introduced by the second author, and properties of the generalized quadrangle (P , L) of type O 6 − (2). This construction is elementary and explicit as the transpositions generating F i 22 in 2 E 6 (2) have been explicitly constructed and implemented in GAP, which might be very helpful and well suited to people who do computations with this group. It is remarkable to mention that this work supports the work of Cuypers et al., where the existence of F i 22 has been assumed then the embedding of the sporadic simple group F i 22 in the group 2 E 6 (2) has been given. In fact our approach for the construction is completely different from Fischer's construction. [ABSTRACT FROM AUTHOR]

In this paper we prove that the unitary groups SU n (q 2) are (2 , 3) -generated for any prime power q and any integer n ≥ 8. By previous results this implies that, if n ≥ 3 , the groups SU n (q 2) and PSU n (q 2) are (2 , 3) -generated, except when (n , q) ∈ { (3 , 2) , (3 , 3) , (3 , 5) , (4 , 2) , (4 , 3) , (5 , 2) }. [ABSTRACT FROM AUTHOR]

Let B be a ring, not necessarily commutative, having an involution ⁎ and let U 2 m (B) be the unitary group of rank 2 m associated to a hermitian or skew hermitian form relative to ⁎. When B is finite, we construct a Weil representation of U 2 m (B) via Heisenberg groups and find its explicit matrix form on the Bruhat elements. As a consequence, we derive information on generalized Gauss sums. On the other hand, there is an axiomatic method to define a Weil representation of U 2 m (B) , and we compare the two Weil representations thus obtained under fairly general hypotheses. When B is local, not necessarily finite, we compute the index of the subgroup of U 2 m (B) generated by its Bruhat elements. Besides the independent interest, this subgroup and index are involved in the foregoing comparison of Weil representations. [ABSTRACT FROM AUTHOR]

We specify normalizers of Sylow r-subgroups in finite simple linear and unitary groups for the case where r is an odd prime distinct from the characteristic of a definition field of a group. [ABSTRACT FROM AUTHOR]

We study the equivariant homotopy type of the poset Lpk of orthogonal decompositions of Cpk. The fixed point space of the p-radical subgroup Γk⊂U(pk) acting on Lpk is shown to be homeomorphic to a symplectic Tits building, a wedge of (k-1)-dimensional spheres. Our second result concerns Δk=(Z/p)k⊂U(pk) acting on Cpk by the regular representation. We identify a retract of the fixed point space of Δk acting on Lpk. This retract has the homotopy type of the unreduced suspension of the Tits building for GLk(Fp), also a wedge of (k-1)-dimensional spheres. As a consequence of these results, we find that the fixed point space of any coisotropic subgroup of Γk contains, as a retract, a wedge of (k-1)-dimensional spheres. We make a conjecture about the full homotopy type of the fixed point space of Δk acting on Lpk, based on a more general branching conjecture, and we show that the conjecture is consistent with our results. [ABSTRACT FROM AUTHOR]

We study the geometry of Lie groups G with a continuous Finsler metric, in presence of a subgroup K such that the metric is right-invariant for the action of K. We present a systematic study of the metric and geodesic structure of homogeneous spaces M obtained by the quotient M ≃ G / K {M\simeq G/K}. Of particular interest are left-invariant metrics of G which are then bi-invariant for the action of K. We then focus on the geodesic structure of groups K that admit bi-invariant metrics, proving that one-parameter groups are short paths for those metrics, and characterizing all other short paths. We provide applications of the results obtained, in two settings: manifolds of Banach space linear operators, and groups of maps from compact manifolds. [ABSTRACT FROM AUTHOR]

We compute the characters of simple supercuspidal representations of a twisted GLN and an unramified quasi-split unitary group UN over a p-adic field. Comparing them by the endoscopic character relation, we determine the endoscopic lifts of simple supercuspidal representations of UN to GLN, under the assumption that p is not equal to 2. [ABSTRACT FROM AUTHOR]

We compute the characters of simple supercuspidal representations of a twisted GLN and an unramified quasi-split unitary group UN over a p-adic field. Comparing them by the endoscopic character relation, we determine the endoscopic lifts of simple supercuspidal representations of UN to GLN, under the assumption that p is not equal to 2. [ABSTRACT FROM AUTHOR]

We call a function of k variables reducing for a multiplicative semigroup S ⊆ M n (C) , if the semigroup is reducible whenever f sends S to zero. For continuous reducing functions on groups of unitary matrices we prove that such a group is reducible whenever f is uniformly small enough on the group. Extensions of this result for certain types of semigroups are also given. Note that our results contain and extend many known results of the kind. In particular, it is shown that a group of complex unitary matrices is reducible if, for fixed indices i and j , its (i , j) entries are uniformly small. [ABSTRACT FROM AUTHOR]