Unipotent Research Articles

Cayley transform for Toeplitz and dual matrices

Let an n×n complex matrix A be such that I+A is invertible. The Cayley transform of A, denoted by C(A), is defined asC(A)=(I+A)−1(I−A). Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples.

Products of commutators of unipotent matrices of index $2$ in $\mathrm{GL}_n(\mathbb H)$

The aim of this paper is to show that if $\mathbb{H}$ is the real quaternion division ring and $n$ is an integer greater than $1,$ then every matrix in the special linear group $\mathrm{SL}_n(\mathbb{H})$ can be expressed as a product of at most three commutators of unipotent matrices of index $2$.

Products of unipotent elements of index 2 in orthogonal and symplectic groups

An automorphism u of a vector space is called unipotent of index 2 whenever (u−id)2=0. Let b be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space V over a field F of characteristic different from 2.Here, we characterize the elements of the isometry group of b that are the product of two unipotent isometries of index 2. In particular, if b is skewsymmetric and nondegenerate we prove that an element of the symplectic group of b is the product of two unipotent isometries of index 2 if and only if it has no Jordan cell of odd size for the eigenvalue −1. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index 2 (and no fewer in general).For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article [7].

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula II. Non-tube type groups

For every simple Hermitian Lie group G, we consider a certain maximal parabolic subgroup whose unipotent radical N is either abelian (if G is of tube type) or two-step nilpotent (if G is of non-tube type). By the generalized Whittaker Plancherel formula we mean the Plancherel decomposition of L2(G/N,ω), the space of square-integrable sections of the homogeneous vector bundle over G/N associated with an irreducible unitary representation ω of N. Assuming that the central character of ω is contained in a certain cone, we construct embeddings of all holomorphic discrete series representations of G into L2(G/N,ω) and show that the multiplicities are equal to the dimensions of the lowest K-types. The construction is in terms of a kernel function which can be explicitly defined using certain projections inside a complexification of G. This kernel function carries all information about the holomorphic discrete series embedding, the lowest K-type as functions on G/N, as well as the associated Whittaker vectors.

Products of unipotent matrices of index 2 over division rings

Products of unipotent matrices of index 2 over division rings

Symmetric periods for automorphic forms on unipotent groups

Symmetric periods for automorphic forms on unipotent groups

Radiant toric varieties and unipotent group actions

We consider complete toric varieties X such that a maximal unipotent subgroup U of the automorphism group Aut(X) acts on X with an open orbit. It turns out that such varieties can be characterized by several remarkable properties. We study the set of Demazure roots of the corresponding complete fan, describe the structure of a maximal unipotent subgroup U in Aut(X), and find all regular subgroups in U that act on X with an open orbit.

Free groups generated by two unipotent maps

Abstract Let A and B be two unipotent elements of SU ⁢ ( 2 , 1 ) {\mathrm{SU}(2,1)} with distinct fixed points. In [S. B. Kalane and J. R. Parker, Free groups generated by two parabolic maps, Math. Z. 303 2023, 1, Paper No. 9], the authors gave several conditions that guarantee the subgroup 〈 A , B 〉 {\langle A,B\rangle} is discrete and free by using Klein’s combination theorem. We will improve their conditions by using a variant of Klein’s combination theorem. With the same arguments and the additional assumption that AB is unipotent, we also extend Parker and Will’s condition that guarantees the subgroup 〈 A , B 〉 {\langle A,B\rangle} is discrete and free in [J. R. Parker and P. Will, A complex hyperbolic Riley slice, Geom. Topol. 21 2017, 6, 3391–3451].

MV polytopes and reduced double Bruhat cells

When G is a complex reductive algebraic group, MV polytopes are in bijection with the non-negative tropical points of the unipotent group of G . By fixing w from the Weyl group, we can define MV polytopes whose highest vertex is labelled by w . We show that these polytopes are in bijection with the non-negative tropical points of the reduced double Bruhat cell labelled by w^{-1} . To do this, we define a collection of generalized minor functions \Delta_{\gamma}^{\mathrm{new}} which tropicalize on the reduced Bruhat cell to the BZ data of an MV polytope of highest vertex w . We also describe the combinatorial structure of MV polytopes of highest vertex w . We explicitly describe the map from the Weyl group to the subset of elements bounded by w in the Bruhat order which sends u \mapsto v if the vertex labelled by u coincides with the vertex labelled by v for every MV polytope of highest vertex w . As a consequence of this map, we prove that these polytopes have vertices labelled by Weyl group elements less than w in the Bruhat order.

Permawound Unipotent Groups

AbstractWe introduce the class of permawound unipotent groups, and show that they simultaneously satisfy certain “ubiquity” and “rigidity” properties that in combination render them very useful in the study of general wound unipotent groups. As an illustration of their utility, we present two applications: We prove that nonsplit smooth unipotent groups over (infinite) fields finitely generated over $$\textbf{F}_p$$ F p have infinite first cohomology; and we show that every commutative p-torsion wound unipotent group over a field of degree of imperfection 1 is the maximal unipotent quotient of a commutative pseudo-reductive group, thus partially answering a question of Totaro.

Groups, Graphs, and Hypergraphs: Average Sizes of Kernels of Generic Matrices with Support Constraints

We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of F q \mathbf {F}_q -points of the groups under consideration depend polynomially on q q . Our approach combines group theory, graph theory, toric geometry, and p p -adic integration. Our uniformity results are in line with a conjecture of Higman on the numbers of conjugacy classes of unitriangular matrix groups. Our findings are, however, in stark contrast to related results by Belkale and Brosnan on the numbers of generic symmetric matrices of given rank associated with graphs.

The holomorphic discrete series contribution to the generalized Whittaker Plancherel formula

For a Hermitian Lie group G of tube type we find the contribution of the holomorphic discrete series to the Plancherel decomposition of the Whittaker space L2(G/N,ψ), where N is the unipotent radical of the Siegel parabolic subgroup and ψ is a certain non-degenerate unitary character on N. The holomorphic discrete series embeddings are constructed in terms of generalized Whittaker vectors for which we find explicit formulas in the bounded domain realization, the tube domain realization and the L2-model of the holomorphic discrete series. Although L2(G/N,ψ) does not have finite multiplicities in general, the holomorphic discrete series contribution does.Moreover, we obtain an explicit formula for the formal dimensions of the holomorphic discrete series embeddings, and we interpret the holomorphic discrete series contribution to L2(G/N,ψ) as boundary values of holomorphic functions on a domain Ξ in a complexification GC of G forming a Hardy type space H2(Ξ,ψ).

The Wasserstein mean of unipotent matrices

We define the Wasserstein mean of n × n unipotent matrices by solving its corresponding matrix equation, and study its fundamental properties. Under a binomial expansion for the weighted geometric mean of unipotent matrices, we provide the explicit formula of two-variable Wasserstein means. Furthermore, we show that the explicit formula of multi-variable Wasserstein mean for n = 2 , 3.

Arithmetics of homogeneous spaces over p$p$‐adic function fields

AbstractLet be the function field of a smooth projective geometrically integral curve over a finite extension of . Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local–global and weak approximation problems for homogeneous spaces of with geometric stabilizers extension of a group of multiplicative type by a unipotent group. The tools used are arithmetic (local and global) duality theorems in Galois cohomology, in combination with techniques similar to those used by Harari, Szamuely, Colliot‐Thélène, Sansuc, and Skorobogatov. As a consequence, we show that any finite abelian group is a Galois group over , rediscovering the positive answer to the abelian case of the inverse Galois problem over . In the case where the curve is defined over a higher dimensional local field instead of a finite extension of , coarser results are also given.

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let [Formula: see text] be a non-compact irreducible Hermitian symmetric space of rank [Formula: see text] and let [Formula: see text] be an Iwasawa decomposition of [Formula: see text]. The group [Formula: see text] acts on [Formula: see text] by biholomorphisms and the real [Formula: see text]-dimensional submanifold [Formula: see text] intersects every [Formula: see text]-orbit transversally in a single point. Moreover [Formula: see text] is contained in a complex [Formula: see text]-dimensional submanifold of [Formula: see text] biholomorphic to [Formula: see text], the product of [Formula: see text] copies of the upper half-plane in [Formula: see text]. This fact leads to a one-to-one correspondence between [Formula: see text]-invariant domains in [Formula: see text] and tube domains in [Formula: see text]. In this setting we prove an analogue of Bochner’s tube theorem. Namely, an [Formula: see text]-invariant domain [Formula: see text] in [Formula: see text] is Stein if and only if the base [Formula: see text] of the associated tube domain is convex and “cone invariant”. We also prove the univalence of [Formula: see text]-invariant holomorphically separable Riemann domains over [Formula: see text]. This yields a precise description of the envelope of holomorphy of an arbitrary [Formula: see text]-invariant domain in [Formula: see text]. Finally, we obtain a characterization of several classes of [Formula: see text]-invariant plurisubharmonic functions on [Formula: see text] in terms of the corresponding classes of convex functions on [Formula: see text]. As an application we present an explicit Lie group theoretical description of all [Formula: see text]-invariant potentials of the Killing metric on [Formula: see text] and of the associated moment maps.

The Lyndon-Hochschild-Serre spectral sequence for a parabolic subgroup of [formula omitted

Let Γ be a congruence subgroup of level N in GLn(Z). Let P be a maximal Q-parabolic subgroup of GLn/Q, with unipotent radical U, and let Q=(P∩Γ)/(U∩Γ). Let p>dimQ⁡(U(Q))+1 be a prime number that does not divide N. Let M be a (U,p)-admissible Γ-module. Consider the Lyndon-Hochschild-Serre spectral sequence arising from the exact sequence 1→U∩Γ→P∩Γ→Q→1, which abuts to H⁎(P∩Γ,M). We show that if M is a trivial U∩Γ-module, then certain classes in the E2 page survive to E∞. We use this to obtain information about classes in H⁎(P∩Γ,M) even if M is not a trivial U∩Γ-module. This information will be used in future work to prove a Serre-type conjecture for sums of two irreducible Galois representations.

The fundamental group of an extension in a Tannakian category and the unipotent radical of the Mumford–Tate group of an open curve

The fundamental group of an extension in a Tannakian category and the unipotent radical of the Mumford–Tate group of an open curve

Semi-invariant distribution vectors for p-adic unipotent groups

Let G be a unipotent algebraic group defined over a [Formula: see text]-adic field of characteristic zero. We denote by [Formula: see text] the set of rational points of G. It is a [Formula: see text]-adic Lie group with Lie algebra denoted by [Formula: see text]. Let [Formula: see text] be an irreducible unitary representation of [Formula: see text] in a Hilbert space [Formula: see text], [Formula: see text] be a linear form on [Formula: see text] and [Formula: see text] be a polarization at [Formula: see text]. We denote by [Formula: see text] a character of [Formula: see text] related to [Formula: see text]. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors [Formula: see text] of [Formula: see text].

On the Topological Generation of Exceptional Groups by Unipotent Elements

AbstractLet G be a simple algebraic group of exceptional type over an algebraically closed field of characteristic $$p \geqslant 0$$ p ⩾ 0 which is not algebraic over a finite field. Let $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t be non-central conjugacy classes in G. In earlier work with Gerhardt and Guralnick, we proved that if $$t \geqslant 5$$ t ⩾ 5 (or $$t \geqslant 4$$ t ⩾ 4 if $$G = G_2$$ G = G 2 ), then there exist elements $$x_i \in \mathcal {C}_i$$ x i ∈ C i such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , x t ⟩ is Zariski dense in G. Moreover, this bound on t is best possible. Here we establish a more refined version of this result in the special case where $$p>0$$ p > 0 and the $$\mathcal {C}_i$$ C i are unipotent classes containing elements of order p. Indeed, in this setting we completely determine the classes $$\mathcal {C}_1, \ldots , \mathcal {C}_t$$ C 1 , … , C t for $$t \geqslant 2$$ t ⩾ 2 such that $$\langle x_1, \ldots , x_t \rangle $$ ⟨ x 1 , … , x t ⟩ is Zariski dense for some $$x_i \in \mathcal {C}_i$$ x i ∈ C i .

The character table of the finite Chevalley group F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} for q a power of 2

Let q be a prime power and F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} be the Chevalley group of type F4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4$$\\end{document} over a finite field with q elements. Marcelo and Shinoda (Tokyo J Math 18:303–340, 1995) determined the values of the unipotent characters of F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document} on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that q is a power of 2, we explain how to construct the complete character table of F4(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$F_4(q)$$\\end{document}.