Finite Unitary Groups Research Articles

Powers in finite unitary groups

Let U(n,Fq2) denote the subgroup of unitary matrices of the general linear group GL(n,Fq2) which fix a Hermitian form and let M≥2 be an integer. In earlier work, elements of the groups GL(n,Fq), Sp(2n,Fq), O±(2n,Fq) and O(2n+1,Fq) with an M-th root have been described. Here we will describe the M-th powers in unitary groups for the regular semisimple, semisimple and cyclic elements, under the assumption that M and q are coprime. We will further describe the generating functions in the corresponding cases.

Symbols of mixed unitary quantum channels generated by finite unitary groups

We study the symbols of linear operators generated by a finite unitary group. Under some assumptions, it is shown that the operator can be restored from its symbol. We also introduce symbols of mixed unitary quantum channels generated by finite unitary groups. An example illustrating the technique is given.

Wavefront sets and descent method for finite unitary groups

Wavefront sets and descent method for finite unitary groups

Trace formulae for actions of finite unitary groups on cohomology of Artin–Schreier varieties

Associated to a certain additive polynomial, we introduce an Artin–Schreier variety admitting an action of a finite unitary group. We calculate the character of the cohomology as a representation of a finite unitary group. One of our main ingredients is explicit character formulae for Weil representations of unitary groups due to Gérardin. We give another trace formula for a projective hypersurface admitting an action of a finite unitary group.

Geometric construction of Heisenberg–Weil representations for finite unitary groups and Howe correspondences

We give a geometric construction of the Heisenberg–Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for (textrm{Sp}_{2n},textrm{O}_2^-) over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.

О базисных инвариантах некоторых конечных подгрупп в SL3(C)

The work is devoted to the study of algebras of invariants of finite unitary groups G' = G∩SL3(C), where G is a finite unitary irreducible primitive group generated by reflections in the unitary space U3. It is known that the system of invariants of the group G' that form an algebra is obtained from the system of invariants of the group G that form the algebra by adding all semi-invariants of the group G of a special form. In the paper, generators of the algebras of invariants of all the indicated groups G' are constructed.

Nonabelian Ramified Coverings and $$L^p$$-boundedness of Bergman Projections in $${\mathbb {C}}^2$$

In this work we explore the theme of $L^p$-boundedness of Bergman projections of domains that can be covered, in the sense of ramified coverings, by "nice" domains (e.g. strictly pseudoconvex domains with real analytic boundary). In particular, we focus on two-dimensional normal ramified coverings whose covering group is a finite unitary reflection group. In an infinite family of examples we are able to prove $L^p$-boundedness of the Bergman projection for every $p\in(1,\infty)$.

COHOMOLOGY OF THE BRUHAT–TITS STRATA IN THE UNRAMIFIED UNITARY RAPOPORT–ZINK SPACE OF SIGNATURE

AbstractIn their renowned paper (2011, Inventiones Mathematicae 184, 591–627), I. Vollaard and T. Wedhorn defined a stratification on the special fiber of the unitary unramified PEL Rapoport–Zink space with signature $(1,n-1)$ . They constructed an isomorphism between the closure of a stratum, called a closed Bruhat–Tits stratum, and a Deligne–Lusztig variety which is not of classical type. In this paper, we describe the $\ell $ -adic cohomology groups over $\overline {{\mathbb Q}_{\ell }}$ of these Deligne–Lusztig varieties, where $\ell \not = p$ . The computations involve the spectral sequence associated with the Ekedahl–Oort stratification of a closed Bruhat–Tits stratum, which translates into a stratification by Coxeter varieties whose cohomology is known. Eventually, we find out that the irreducible representations of the finite unitary group which appear inside the cohomology contribute to only two different unipotent Harish-Chandra series, one of them belonging to the principal series.

Constructing group-invariant CR mappings

We construct CR mappings between spheres that are invariant under actions of finite unitary groups. In particular, we combine a tensoring procedure with D’Angelo’s construction of a canonical group-invariant CR mapping to obtain new invariant mappings. We also explore possible gap phenomena in this setting.

A question of Katz and Tiep on representations of finite general unitary groups

Katz and Tiep proved a characterization of the total Weil character of finite special unitary groups for any integer and any prime power q other than in terms of the character values and irreducible constituents. They asked whether a similar characterization can be done for finite general unitary groups In this article, we prove that such characterization for can be done up to endomorphisms and tensoring by real linear characters.

Infinite-dimensional groups over finite fields and Hall-Littlewood symmetric functions

The groups mentioned in the title are certain matrix groups of infinite size with elements in a finite field Fq. They are built from finite classical groups and at the same time they are similar to reductive p-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. Of special interest for us are ergodic invariant measures, which are a substitute of orbital measures.We examine first the group GLB, a topological completion of the inductive limit group lim→GL(n,Fq). The traceable factor representations of GLB were studied by Gorin, Kerov, and Vershik (2014) [12]. We show that there exists a parallel theory for ergodic coadjoint-invariant measures, which is linked with harmonic functions on the “HL-deformed Young graph” YHL(t). Here the deformation means that the edges of the Young graph Y are endowed with certain formal multiplicities depending on the Hall-Littlewood (HL) parameter t specialized to q−1.This fact serves as a prelude to our main results, which concern topological completions of two inductive limit groups built from finite unitary groups, of even and odd dimension. We show that in this case, coadjoint-invariant measures are linked to some new branching graphs. They are still related to the HL symmetric functions, but in a nonstandard way, and the HL parameter t takes now the negative value −q−1. As an application, we find several families of unitarily invariant measures, including analogues of the Plancherel measure.

Traces of powers of matrices over finite fields

Let $M$ be a random matrix chosen according to Haar measure from the unitary group $\mathrm{U}(n,\mathbb{C})$. Diaconis and Shahshahani proved that the traces of $M,M^2,\ldots,M^k$ converge in distribution to independent normal variables as $n \to \infty$, and Johansson proved that the rate of convergence is superexponential in $n$. We prove a finite field analogue of these results. Fixing a prime power $q = p^r$, we choose a matrix $M$ uniformly from the finite unitary group $\mathrm{U}(n,q)\subseteq \mathrm{GL}(n,q^2)$ and show that the traces of $\{ M^i \}_{1 \le i \le k,\, p \nmid i}$ converge to independent uniform variables in $\mathbb{F}_{q^2}$ as $n \to \infty$. Moreover we show the rate of convergence is exponential in $n^2$. We also consider the closely related problem of the rate at which characteristic polynomial of $M$ equidistributes in `short intervals' of $\mathbb{F}_{q^2}[T]$. Analogous results are also proved for the general linear, special linear, symplectic and orthogonal groups over a finite field. In the two latter families we restrict to odd characteristic. The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial.

Hypergeometric sheaves and finite symplectic and unitary groups

Hypergeometric sheaves and finite symplectic and unitary groups

On modular Harish-Chandra series of finite unitary groups

In the modular representation theory of finite unitary groups when the characteristic ℓ \ell of the ground field is a unitary prime, the s l ^ e \widehat {\mathfrak {sl}}_e -crystal on level 2 2 Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for ℓ \ell sufficiently large, the s l ∞ \mathfrak {sl}_\infty -crystal on the same level 2 2 Fock spaces provides the remaining piece of the puzzle for the full Harish-Chandra branching rule.

On the Gan–Gross–Prasad problem for finite unitary groups

In this paper we study the Gan–Gross–Prasad problem for unitary groups over finite fields. Our results provide complete answers for unipotent representations, and we obtain the explicit branching of these representations.

Complete set of translation invariant measurements with Lipschitz bounds

In image and audio signal classification, a major problem is to build stable representations that are invariant under rigid motions and, more generally, to small diffeomorphisms. Translation invariant representations of signals in C n are of particular importance. The existence of such representations is intimately related to classical invariant theory, inverse problems in compressed sensing and deep learning. Despite an impressive body of literature on the subject, most representations available are either not stable due to the presence of high frequencies or non discriminative. In the present paper, we construct low dimensional representations of signals in C n that are invariant under finite unitary group actions, as a special case we establish the existence of low-dimensional and complete Z m -invariant representations for any m ∈ N . Our construction yields a stable, discriminative transform with semi-explicit Lipschitz bounds on the dimension; this is particularly relevant for applications. Using some tools from Algebraic Geometry, we define a high dimensional homogeneous function that is injective. We then exploit the projective character of this embedding and see that the target space can be reduced significantly by using a generic linear transformation. Finally, we introduce the notion of non-parallel map, which is enjoyed by our function and employ this to construct a Lipschitz modification of it.

Normalizers of Sylow subgroups in finite linear and unitary groups

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.

Classification of Sylow classes of parabolic and reflection subgroups in unitary reflection groups

Let be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an -Sylow subgroup. The classification assists in describing the -Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an -Sylow subgroup is G itself, the reflection subgroup within the parabolic minimally containing an -Sylow subgroup is the whole parabolic subgroup.

Descents of unipotent representations of finite unitary groups

Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent representations of unitary groups over finite fields. We give the first descents of unipotent representations explicitly, which are unipotent as well. Our results include both the Bessel case and Fourier-Jacobi case, which are related via theta correspondence.

The (2,3)-generation of the finite unitary groups

In this paper we prove that the unitary groups SUn(q2) 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 SUn(q2) and PSUn(q2) are (2,3)-generated, except when (n,q)∈{(3,2),(3,3),(3,5),(4,2),(4,3),(5,2)}.