Search

Let $G$ be a subgroup of ${\rm PGL}_2({\mathbb F}_q)$, where $q$ is any prime power, and let $Q \in {\mathbb F}_q[x]$ such that ${\mathbb F}_q(x)/{\mathbb F}_q(Q(x))$ is a Galois extension with group $G$. By explicitly computing the Artin map on unramified degree-1 primes in ${\mathbb F}_q(Q)$ for various groups $G$, interesting new results emerge about finite fields, additive polynomials, and conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. For example, by taking $G$ to be a unipotent group, one obtains a new characterization for when an additive polynomial splits completely over ${\mathbb F}_q$. When $G = {\rm PGL}_2({\mathbb F}_q)$, one obtains information about conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. When $G$ is the group of order 3 generated by $x \mapsto 1 - 1/x$, one obtains a natural tripartite symbol on ${\mathbb F}_q$ with values in ${\mathbb Z}/3{\mathbb Z}$. Some of these results generalize to ${\rm PGL}_2(K)$ for arbitrary fields $K$. Apart from the introduction, this article is written from first principles, with the aim to be accessible to graduate students or advanced undergraduates. An earlier draft of this article was published on the Math arXiv in June 2019 under the title {\it More structure theorems for finite fields}., Comment: Version 4 contains minor corrections and updates to the bibliograpy. Version 3 is a major revision, including a change in the title from "More structure theorems for finite fields" to "Explicit Artin maps into PGL2". The author thanks Xander Faber for insightful comments that led to the change in the title

We discuss some of the contributions of T.A. Springer (1926–2011) to the theory of algebraic groups, with emphasis on his work on unipotent classes and representations of Weyl groups.

We examine the cohomology and representation theory of a family of finite supergroup schemes of the form $(\mathbb G_a^-\times \mathbb G_a^-)\rtimes (\mathbb G_{a(r)}\times (\mathbb Z/p)^s)$. In particular, we show that a certain relation holds in the cohomology ring, and deduce that for finite supergroup schemes having this as a quotient, both cohomology mod nilpotents and projectivity of modules is detected on proper sub-super\-group schemes. This special case feeds into the proof of a more general detection theorem for unipotent finite supergroup schemes, in a separate work of the authors joint with Iyengar and Krause. We also completely determine the cohomology ring in the smallest cases, namely $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb G_{a(1)}$ and $(\mathbb G_a^- \times \mathbb G_a^-) \rtimes \mathbb Z/p$. The computation uses the local cohomology spectral sequence for group cohomology, which we describe in the context of finite supergroup schemes., 19 pages

We give a short combinatorial proof of Asai's decomposition formula for Lusztig induction of unipotent characters in groups of classical type, relying solely on the Mackey formula.

We study the local zeta integrals attached to a pair of generic representations ( π , τ ) of GL n × GL m , n > m , over a p-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of π and τ. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin–Selberg (local) L-function.

Given a Speh representation π of GL n ( F ) for a non-archimedean local field F, we obtain necessary and sufficient conditions on standard parabolic subgroups for the Jacquet module of π with respect to the parabolic subgroup to be generic. Using this we describe the precise set of characters Θ of the maximal unipotent radical U n of GL n ( F ) such that Hom U n ( π , Θ ) is non-zero. We then describe the behavior of this set under the base change map (with respect to a finite cyclic extension of F of prime degree).

We compute the K-automorphism group of A K = K ⊗ F A f of a weak crossed product algebra A f for a weak 2-cocycle f over a Galois extension K / F with Galois group G. The K-automorphism group of A K decomposes into its unipotent part and the reductive part H ˆ . Automorphisms in H ˆ are computed via their restriction to semi-simple part of A K . There is a strong relationship between H ˆ and the lower-subtractive relation (≤) induced by f on G. We introduce a subgroup of H ˆ , namely Λ, which contains interesting combinatorial information of ≤. We also present a duality on lower subtractive relations which simplifies the computation of the automorphism group. For the Weak Bruhat order of a Coxeter system, which is an important example of a lower-subtractive relation, it is shown that the automorphism group of the corresponding idempotent algebra is related to the diagram automorphisms of the associated graph.

We construct two supercharacter theories (in the sense of P. Diaconis and I.M. Isaacs) for the parabolic subgroups in orthogonal and symplectic groups. For each supercharacter theory, we obtain a supercharacter analog of the A.A. Kirillov formula for irreducible characters of finite unipotent groups.

We establish an effective version of the classical Lie–Kolchin Theorem. Namely, let A , B ∈ GL m ( C ) be quasi-unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k ⩾ 0 the matrix A B k is also quasi-unipotent. Then A and B have a common eigenvector. In particular, 〈 A , B 〉 GL m ( C ) is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces.

Let $G$ be a finite Chevalley group. We are concerned with computing the values of the unipotent characters of $G$ by making use of Lusztig's theory of character sheaves. In this framework, one has to find the transformation between several bases for the class functions on $G$. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. We shall determine these scalars in the specific case where $G$ is the (twisted or non-twisted) group of type $E_6$ over the finite field with $q$ elements, for $q$ a power of the bad prime $p=3$, by exploiting known facts about the representation theory of the Hecke algebra associated with $G$., Comment: 12 pages

This paper has two main parts. Firstly, we give a classification of the $\ell$-blocks of finite special linear and unitary groups $SL_n(\epsilon q)$ in the non-defining characteristic $\ell\ge 3$. Secondly, we describe how the $\ell$-weights of $SL_n(\epsilon q)$ can be obtained from the $\ell$-weights of $GL_n(\epsilon q)$ when $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$, and verify the Alperin weight conjecture for $SL_n(\epsilon q)$ under the condition $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$. As a step to establish the Alperin weight conjecture for all finite groups, we prove the inductive blockwise Alperin weight condition for any unipotent $\ell$-block of $SL_n(\epsilon q)$ if $\ell\nmid\mathrm{gcd}(n,q-\epsilon)$., Comment: revised version, to appear in Journal of Algebra

We prove the vanishing of z -eigen distributions on a quasi-split real reductive group which change according to a non-degenerate character under the left action of the unipotent radical of the Borel subgroup, and are equivariant under the right action of a spherical subgroup.

We give a combinatorial formula to calculate the Euler characteristic of an analogue of a Deligne–Lusztig variety, denoted Y w , g , which is attached to an element w in the Weyl group of G L n and g ∈ G L n . The main theorem of this paper states that the Euler characteristic of Y w , g only depends on the unipotent part of the Jordan decomposition of g and the conjugacy class of w. It generalizes the formula of the Euler characteristic of Springer fibers for type A.

We study the decomposition of certain reducible characters of classical groups as the sum of irreducible ones. Let G be an algebraic group of classical type with defining characteristic p > 0 , μ a dominant weight and W the Weyl group of G. Let G = G ( q ) be a finite classical group, where q is a p-power. For a weight μ of G the sum s μ of distinct weights w ( μ ) with w ∈ W viewed as a function on the semisimple elements of G is known to be a generalized Brauer character of G called an orbit character of G. We compute, for certain orbit characters and every maximal torus T of G, the multiplicity of the trivial character 1 T of T in s μ . The main case is where μ = ( q − 1 ) ω and ω is a fundamental weight of G. Let St denote the Steinberg character of G. Then we determine the unipotent characters occurring as constituents of s μ ⋅ S t defined to be 0 at the p-singular elements of G. Let β μ denote the Brauer character of a representation of S L n ( q ) arising from an irreducible representation of G with highest weight μ. Then we determine the unipotent constituents of the characters β μ ⋅ S t for μ = ( q − 1 ) ω , and also for some other μ (called strongly q-restricted). In addition, for strongly restricted weights μ, we compute the multiplicity of 1 T in the restriction β μ | T for every maximal torus T of G.

We construct a monoidal category C w , v which categorifies the doubly-invariant algebra C N ′ ( w ) [ N ] N ( v ) associated with Weyl group elements w and v. It gives, after a localization, the coordinate algebra C [ R w , v ] of the open Richardson variety associated with w and v. The category C w , v is realized as a subcategory of the graded module category of a quiver Hecke algebra R. When v = id , C w , v is the same as the monoidal category which provides a monoidal categorification of the quantum unipotent coordinate algebra A q ( n ( w ) ) Z [ q , q − 1 ] given by Kang–Kashiwara–Kim–Oh. We show that the category C w , v contains special determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) for k = 1 , … , l ( w ) , which commute with each other. When the quiver Hecke algebra R is symmetric, we find a formula of the degree of R-matrices between the determinantial modules M ( w ≤ k Λ , v ≤ k Λ ) . When it is of finite ADE type, we further prove that there is an equivalence of categories between C w , v and C u for w , u , v ∈ W with w = v u and l ( w ) = l ( v ) + l ( u ) .

Let G be a linear algebraic group over an algebraically closed field of characteristic p ≥ 0 . We show that if H 1 and H 2 are connected subgroups of G such that H 1 and H 2 have a common maximal unipotent subgroup and H 1 / R u ( H 1 ) and H 2 / R u ( H 2 ) are semisimple, then H 1 and H 2 are G-conjugate. Moreover, we show that if H is a semisimple linear algebraic group with maximal unipotent subgroup U then for any algebraic group homomorphism σ : U → G , there are only finitely many G-conjugacy classes of algebraic group homomorphisms ρ : H → G such that ρ | U is G-conjugate to σ. This answers an analogue for connected algebraic groups of a question of B. Kulshammer. In Kulshammer's original question, H is replaced by a finite group and U by a Sylow p-subgroup of H; the answer is then known to be no in general. We obtain some results in the general case when H is non-connected and has positive dimension. Along the way, we prove existence and conjugacy results for maximal unipotent subgroups of non-connected linear algebraic groups. When G is reductive, we formulate Kulshammer's question and related conjugacy problems in terms of the nonabelian 1-cohomology of unipotent radicals of parabolic subgroups of G, and we give some applications of this cohomological approach. In particular, we analyse the case when G is a semisimple group of rank 2.

In this article, we consider the actions of subgroups of the general linear group G L r ( Z n ) on Z n r , including groups of upper triangular matrices in G L r ( Z n ) , unipotent groups, Heisenberg groups and extended Heisenberg groups. By applying the Cauchy–Frobenius–Burnside lemma, we obtain several generalizations of the well known Menon's identity.

In this paper, bifurcation of limit cycles is considered for planar cubic-order systems with an isolated nilpotent critical point. Normal form theory is applied to compute the generalized Lyapunov constants and to prove the existence of at least 9 small-amplitude limit cycles in the neighborhood of the nilpotent critical point. In addition, the method of double bifurcation of nilpotent focus is used to show that such systems can have 10 small-amplitude limit cycles near the nilpotent critical point. These are new lower bounds on the number of limit cycles in planar cubic-order systems near an isolated nilpotent critical point. Moreover, a set of center conditions is obtained for such cubic systems.

Let p and l be distinct primes, K a finite extension of Q p , and ρ ‾ : Gal ( K ‾ / K ) → GL n ( F ‾ l ) a mod l Galois representation. In this paper, we show that the generic fiber of universal lifting space of ρ ‾ is equidimensional of dimension n 2 . We also characterize the irreducible components of the generic fiber of the universal lifting space which represent the liftings ρ's of ρ ‾ with unipotent ρ | I K 's, when ρ ‾ | I K is trivial or n ≤ 4 and the square of the order of the residue field of K is not equal to 1 mod l.

For x an element of a field other than 0 or 1, we compute the order n Massey products 〈 ( 1 − x ) − 1 , x − 1 , … , x − 1 , ( 1 − x ) − 1 〉 of n − 2 factors of x − 1 and two factors of ( 1 − x ) − 1 by embedding P 1 − { 0 , 1 , ∞ } into its Picard variety and constructing Gal ( k s / k ) equivariant maps from π 1 et applied to this embedding to unipotent matrix groups. This method produces obstructions to π 1 -sections of P 1 − { 0 , 1 , ∞ } , partial computations of obstructions of Jordan Ellenberg, and also computes the Massey products 〈 x − 1 , ( − x ) − 1 , … , ( − x ) − 1 , x − 1 〉 .

Given a regular subgroup R of AGL n ( F ) , one can ask if R contains nontrivial translations. A negative answer to this question was given by Liebeck, Praeger and Saxl for AGL 2 ( p ) (p a prime), AGL 3 ( p ) (p odd) and for AGL 4 ( 2 ) . A positive answer was given by Hegedűs for AGL n ( p ) when n ≥ 4 if p is odd and for n = 3 or n ≥ 5 if p = 2 . A first generalization to finite fields of Hegedűs' construction was recently obtained by Catino, Colazzo and Stefanelli. In this paper we give examples of such subgroups in AGL n ( F ) for any n ≥ 5 and any field F . For n 5 we provide necessary and sufficient conditions for their existence, assuming R to be unipotent if char F = 0 .

Given an arbitrary field F, we describe all Galois extensions L / F whose Galois groups are isomorphic to the group of upper triangular unipotent 4-by-4 matrices with entries in the field of two elements.

We investigate support schemes for infinitesimal unipotent supergroups and their representations. Our main results provide a non-cohomological description of these schemes which generalizes the classical work of Suslin, Friedlander, and Bendel. As a consequence, support schemes in this setting have the desired features of such a theory, including naturality with respect to group homomorphisms, the tensor product property, and realizability. As an application of the theory developed here, we investigate support varieties for certain finite-dimensional Hopf subalgebras of the Steenrod algebra., Comment: 45 pages. Comments welcome

Let $G$ be a reductive group over an algebraically closed field and let $W$ be its Weyl group. In a series of papers, Lusztig introduced a map from the set $[W]$ of conjugacy classes of $W$ to the set $[G_u]$ of unipotent classes of $G$. This map, when restricted to the set of elliptic conjugacy classes $[W_e]$ of $W$, is injective. In this paper, we show that Lusztig's map $[W_e] \to [G_u]$ is order-reversing, with respect to the natural partial order on $[W_e]$ arising from combinatorics and the natural partial order on $[G_u]$ arising from geometry., Comment: 25 pages

For all primes p and for all fields, we find a sufficient and necessary condition of the existence of a unipotent Galois extension of degree p 6 . The main goal of this paper is to describe an explicit construction of such a Galois extension over fields admitting such a Galois extension. This construction is surprising in its simplicity and generality. The problem of finding such a construction has been left open since 2003. Recently a possible solution of this problem gained urgency because of an effort to extend new advances in Galois theory and its relations with Massey products in Galois cohomology.

In order to find a tractable theory to substitute for the wild character theory of the group of n × n unipotent upper-triangular matrices over a finite field F q , Andre and Yan introduced the notion of supercharacter theory. In this paper, we construct a supercharacter theory from an arbitrary set S of normal subgroups of G . We call such supercharacter theory the normal supercharacter theory generated by S . It is shown that normal supercharacter theories are integral, and a recursive formula for supercharacters of the normal supercharacter theory is provided. Also, we indicate that the superclasses of the normal supercharacter theory generated by all normal subgroups of G are given by certain values on the primitive central idempotents. We study the connection between the finest normal supercharacter theory and faithful irreducible characters. Moreover, an algorithm is presented to construct the supercharacter table of the finest normal supercharacter theory from the character table. Finally, we argue that normal supercharacter theories cannot be obtained by previously known supercharacter theory constructions.

We study the simultaneously nilpotent index of a simultaneously nilpotent set of matrices over an antinegative commutative semiring S . We find an upper bound for this index and give some characterizations of the simultaneously nilpotent sets when this upper bound is met. In the special case of antinegative semirings with all zero divisors nilpotent, we also find a bound on the simultaneously nilpotent index for all nonmaximal simultaneously nilpotent sets of matrices and establish their cardinalities in case of a finite S .

Let k be a nonperfect field of characteristic 2. Let G be a k-split simple algebraic group of type E 6 (or G 2 ) defined over k. In this paper, we present the first examples of nonabelian non-G-completely reducible k-subgroups of G which are G-completely reducible over k. Our construction is based on that of subgroups of G acting non-separably on the unipotent radical of a proper parabolic subgroup of G in our previous work. We also present examples with the same property for a non-connected reductive group G. Along the way, several general results concerning complete reducibility over nonperfect fields are proved using the recently proved Tits center conjecture for spherical buildings. In particular, we show that under mild conditions a connected k-subgroup of G is pseudo-reductive if it is G-completely reducible over k.

We provide a general formula and give an explicit expression of the Lefschetz zeta function for any quasi-unipotent map on the n -dimensional torus. The Lefschetz zeta function is used to characterize the minimal set of Lefschetz periods for Morse–Smale diffeomorphisms on the n -dimensional torus; we completely describe this set, for different families containing infinitely many Morse–Smale diffeomorphisms. Moreover we show that for any given odd integer, there are Morse–Smale diffeomorphisms such that the corresponding minimal set of Lefschetz periods consists of all square free divisors of the given number. The results of the present article generalize the previous results of Berrizbeitia–Sirvent [7] . The methods used are based on the arithmetical properties of the number n .

Let k be an algebraically closed field and L = k ( ( ϵ ) ) the field of Laurent series over k. Let G be a connected reductive group over k such that the characteristic of k does not divide the order of the Weyl group of G. For q ∈ k × , the automorphism of L, defined by ∑ a i ϵ i ↦ ∑ a i ( q ϵ ) i , induces an automorphism σ q of the loop group G ( L ) . We show that, when q is not a root of unity, the classification of σ q -conjugacy classes of G ( L ) can be reduced to the classification of unipotent classes of (non-connected) reductive groups. As an application, we recover the classification (of σ q -conjugacy classes) for G = G L n .

In this paper we start a classification of certain global integrals. First, we use the language of unipotent orbits to write down a family of global integrals. We then classify all those integrals which satisfy the dimension equation we set. After doing so, we check which of these integrals are global unipotent integrals. We do all this for groups of type A n , and using all this we derive a certain interesting conjecture about the length of these integrals.

In this note, we prove that the category of unipotent torsion Breuil modules is an abelian category, under the condition e r = p − 1 , r p − 1 .

Let F be an algebraically closed field of positive characteristic p > 3 , and A the divided power algebra in one indeterminate, which, as a vector space, coincides with the truncated polynomial ring of F [ T ] by T p n . Let g be the special derivation algebra over A which is a simple Lie algebra, and additionally non-restricted as long as n > 1 . Let N be the nilpotent cone of g , and G = Aut ( g ) , the automorphism group of g . In contrast with only finitely many nilpotent orbits in a classical simple Lie algebra, there are infinitely many nilpotent orbits in g . In this paper, we parameterize all nilpotent orbits, and obtain their dimensions. Furthermore, the nilpotent cone N is proven to be reducible and not normal. There are two irreducible components in N . The dimension of N is determined.

Gordon James investigated the Specht modules of the symmetric groups: For each partition of n there is a Specht module defined in terms of the intersection of the kernels of certain homomorphisms. Following the philosophy that GLn(q) is a q analog to Sn, Gordon James defined the unipotent Specht modules over a field F for GLn(q). If the characteristic of F is coprime to q then the dimension of the unipotent Specht module is independent of F. Indeed one expects the representation theory for GLn(q) to translate into that for Sn by setting q equal 1. In the sense of the conjecture by Richard Dipper and Gordon James, we have the analogous concept of standard basis. This conjecture was proved by Marco Brandt, Richard Dipper, Gordon James and Sinead Lyle for the case of a two part partition. But the proof is rather combinational hence it seems the method there will not work for an arbitrary partition. Our new strategy is to investigate the FU module structure of the unipotent Specht module. The advantage of restriction to FU module is that FU is semisimple, since by general assumption the characteristic of the field is coprime to q. Gordon James definiert Spechtmoduln fur symmetrische Gruppen: Fur jede Partition von n gibt es einen Spechtmodul von Sn, der als Schnitt von Kernen gewisser Homomorphismen definiert ist. In der Tat gibt es eine Basis von den Spechtmoduln, die durch standard Tableaux indiziert ist. Diese Basis heist Standardbasis. In der Philosophie, dass GLn(q) ein q Analog zu Sn ist, definiert Gordon James den unipotenten Spechtmoduln uber einem Korper F fur GLn(q). Die Dimension von den unipotenten Spechtmoduln ist unabhangig von F, falls die Charakteristik von F koprim zu q ist. Wir hoffen, die Darstellungstheorie von GLn(q) zu der von Sn ubersetzen zu konnen. Im Sinne der Vermutung von Richard Dipper und Gordon James haben wir den analogen Begriff der Standardbasis. Marco Brandt, Richard Dipper, Gordon James und Sinead Lyle zeigen diese Vermutung fur den Fall, dass Partition eine 2 Teile Partition ist. Da der Beweis sehr kombinatorisch ist, scheint es unwahrscheinlich, dass diese Methoden fur beliebige partition funktionieren. Dies motiviert uns neue Methoden zu finden. Unsere neue Strategie ist: Untersuchung der FU Modul Struktur von den unipotenten Spechtmoduln. Der Vorteil der Einschrankung zu FU Moduln ist, dass FU halbeinfach ist, wenn die Charakteristik von F koprim zu q ist.

It is well known that if the ground field K has characteristic zero and G is a connected algebraic group, defined over K, then the Lie algebra Lie(G′) of the commutant G′ of G coincides with the commutant Lie(G)′ of Lie(G). We show that this result is no longer true in the category of algebraic supergroups. We also construct a reductive supergroup H=X⋊G, where X and G are connected, reduced and abelian supergroups, such that Xu≠1 and (Hev)u is non-trivial connected (super)group. Quasi-reductive supergroups have been introduced in [10]. We prove that a supergroup H is quasi-reductive if and only if the largest even (super)subgroup of the solvable radical R(H) is a torus, H˜=H/R(H) contains a normal supersubgroup U, which is quasi-isomorphic to a direct product of normal supersubgroups Ui, and H˜/U is a triangulizable supergroup with odd unipotent radical. Moreover, for every i, Lie(Ui)=Ui⊗Sym(ni) are such that either ni=0 and Ui is a classical simple Lie superalgebra, or ni=1 and Ui is a simple Lie algebra.

The maximal subgroup of unipotent upper-triangular matrices of the finite general linear groups is a fundamental family of p-groups. Their representation theory is well known to be wild, but there is a standard supercharacter theory, replacing irreducible representations by super-representations, that gives us some control over its representation theory. While this theory has a beautiful underlying combinatorics built on set partitions, the structure constants of restricted super-representations remain mysterious. This paper proposes a new approach to solving the restriction problem by constructing natural intermediate modules that help “factor” the computation of the structure constants. We illustrate the technique by solving the problem completely in the case of rainbow supercharacters (and some generalizations). Along the way we introduce a new q-analogue of the binomial coefficients that depend on an underlying poset.

We determine the decomposition matrices of unipotent l-blocks of defect Φ 4 2 for exceptional groups of Lie type up to a few unknowns. For this we employ the new cohomological methods of the first author, together with properties of generalised Gelfand–Graev characters which were recently shown to hold whenever the underlying characteristic is good.

We study Galois descents for categories of mixed Tate motives over O N [ 1 / N ] , for N ∈ { 2 , 3 , 4 , 8 } or O N for N = 6 , with O N the ring of integers of the Nth cyclotomic field, and construct families of motivic iterated integrals with prescribed properties. In particular this gives a basis of multiple zeta values via multiple zeta values at roots of unity μ N . It also gives a new proof, via Goncharov's coproduct, of Deligne's results [9] : the category of mixed Tate motives over O k N [ 1 / N ] , for N ∈ { 2 , 3 , 4 , 8 } is spanned by the motivic fundamental groupoid of P 1 ∖ { 0 , μ N , ∞ } with an explicit basis. By applying the period map, we obtain a generating family for multiple zeta values relative to μ N .

We reconfigure the Milnor invariant of links in terms of central group extensions and unipotent Magnus embeddings. We also develop a diagrammatic computation of the invariant and compute the first non-vanishing invariants of the Milnor link and of several other links. Moreover, we refine the original Milnor invariants of higher degree., Comment: 15 pages.We revised minors mistakes, and added a comparison with the original Milnor invariant

We introduce the notion of a lowering–raising (or LR) triple of linear transformations on a nonzero finite-dimensional vector space. We show how to normalize an LR triple, and classify up to isomorphism the normalized LR triples. We describe the LR triples using various maps, such as the reflectors, the inverters, the unipotent maps, and the rotators. We relate the LR triples to the equitable presentation of the quantum algebra U q ( sl 2 ) and Lie algebra sl 2 .

For G a connected, reductive group over an algebraically closed field k of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of g : = Lie ( G ) and the unipotent variety of G to study the projective variety of elementary subalgebras of g of rank r, denoted E ( r , g ) . In the case that G is defined over F p , we define the category of F q -expressible subalgebras of g for q = p d , and prove that this category is isomorphic to a subcategory of Quillen's category of elementary abelian subgroups of the finite Chevalley group G ( F q ) . This isomorphism of categories leads to a correspondence between G-orbits of E ( r , g ) defined over F q and G-conjugacy classes of certain elementary abelian subgroups of rank rd in G ( F q ) which satisfy a closure property characterized by the Springer isomorphism. We use Magma to compute examples for G = GL n , n ≤ 5 .

Let H be a quadratic homogeneous polynomial map of dimension n over an infinite field in which 2 is invertible such that its Jacobian JH is nilpotent. Meisters and Olech have shown that JH is strongly nilpotent if n ≤ 4 . They also proved that it is not true when n = 5 . We show that if rank J H ≤ 2 and n arbitrary, then JH is strongly nilpotent. We also give examples to show that this is no longer true for any rank and dimension as long as the rank is greater than 2.

Let G = GL n ( C ) and 1 ≠ ψ : C → C × be an additive character. Let U be the subgroup of upper triangular unipotent matrices in G. Denote by θ the character θ : U → C given by θ ( u ) : = ψ ( u 1 , 2 + u 2 , 3 + … + u n − 1 , n ) . Let P be the mirabolic subgroup of G consisting of all matrices in G with the last row equal to ( 0 , 0 , … , 0 , 1 ) . We prove that if π is an irreducible generic representation of GL n ( C ) and W ( π , ψ ) is its Whittaker model, then the space { f | P : P → C : f ∈ W ( π , ψ ) } contains the space of infinitely differentiable functions f : P → C that satisfy f ( u p ) = ψ ( u ) f ( p ) for all u ∈ U and p ∈ P and that have a compact support modulo U. A similar result was proven for GL n ( F ) , where F is a p-adic field by Gelfand and Kazhdan (1975) [1] and for GL n ( R ) by Jacquet (2010) [2] .

We investigate when a skew polynomial extension T=R[x;σ,δ] of a Hopf algebra R admits a Hopf algebra structure, substantially generalising a theorem of Panov. When this construction is applied iteratively in characteristic 0 one obtains a large family of connected noetherian Hopf algebras of finite Gelfand–Kirillov dimension, including for example all enveloping algebras of finite dimensional solvable Lie algebras and all coordinate rings of unipotent groups. The properties of these Hopf algebras are investigated.

We consider monodromy groups of the generalized hypergeometric equation \begin{equation*} \big[z(\theta+\alpha_{1})\cdots (\theta+\alpha_{n})-(\theta+\beta_{1}-1)\cdots (\theta+\beta_{n}-1)\big]f(z) = 0\text{, where }\theta = z d/dz, \end{equation*} in a suitable basis, closely related to the Frobenius basis. We pay particular attention to the maximally unipotent case, where $\beta_{1}=\ldots=\beta_{n}=1$, and present a theorem that enables us to determine the form of the corresponding monodromy matrices in the case where $(X-e^{-2\pi i\alpha_{1}})\cdots (X-e^{-2\pi i\alpha_{n}})$ is a product of cyclotomic polynomials., Comment: 20 pages

This article addresses questions about the double centralizer C G ( C G ( u ) ) = Z ( C G ( u ) ) of unipotent elements u in simple algebraic groups G of type E 7 and E 8 defined over algebraically closed fields of bad characteristic. We use the method developed in [14] to determine Z ( C G ( u ) ) ∘ , deduce its dimension and recognize if it is an overgroup for u. The method used requires explicit representatives of the component group of C G ( u ) which we produce in all cases. This article extends the results of [14] to all exceptional type groups.

Let G be a simple algebraic group defined over an algebraically closed field K and let G = G ( K ) . We consider here the problem when an element g ∈ G can be presented in the form g = e n ( g 1 , g 2 ) : = [ g 1 , [ g 1 , ⋯ [ g 1 ︸ n -times , g 2 ] ⋯ ] ] for some g 1 , g 2 ∈ G . This can always be done if g is a semisimple or a unipotent element. Also, it is known that this holds for G = SL 2 ( K ) . Here we prove that g = e n ( g 1 , g 2 ) for every n if G is a group of type B 2 or G 2 and we prove g = e 2 ( g 1 , g 2 ) = [ g 1 , [ g 1 , g 2 ] ] if G = PGL 3 ( K ) . We also show g = e 2 ( g 1 , g 2 ) = [ g 1 , [ g 1 , g 2 ] ] if g is a regular element of G and G is a group of rank 3. For any simple group G we give a criterion for some “general” regular elements of G to be presented in the form [ g 1 , [ g 1 , g 2 ] ] .

We consider positive entropy G-systems for certain countable, discrete, infinite left-orderable amenable groups G. We will use local analysis to study the connection between the existence of asymptotic pairs, chaotic sets and stable sets. Examples are given for the case of integer lattice groups, the Heisenberg group, and the groups of integral unipotent upper triangular matrices.

Fourier coefficients of automorphic representations $\pi$ of $\Sp_{2n}(\BA)$ are attached to unipotent adjoint orbits in $\Sp_{2n}(F)$, where $F$ is a number field and $\BA$ is the ring of adeles of $F$. We prove that for a given $\pi$, all maximal unipotent orbits, which gives nonzero Fourier coefficients of $\pi$ are special, and prove, under a well acceptable assumption, that if $\pi$ is cuspidal, then the stabilizer attached to each of those maximal unipotent orbits is $F$-anisotropic as algebraic group over $F$. These results strengthen, refine and extend the earlier work of Ginzburg, Rallis and Soudry on the subject. As a consequence, we obtain constraints on those maximal unipotent orbits if $F$ is totally imaginary, further applications of which to the discrete spectrum with the Arthur classification will be considered in our future work., Comment: 48 pages. Accepted by Journal of Number Theory (Mar. 2014)

In this paper we find noncommutative analogues of the coordinate rings of the unipotent radicals of the standard Borel subgroups in SL n + 1 . Two subalgebras of the quantized coordinate ring of the standard Borel in SL n + 1 are defined, both of which can be considered quantizations of the unipotent radical. Presentations are given for these algebras and they are proven to be isomorphic. It is the shown that these algebras also arise as coinvariants of a natural comodule algebra action using the Hopf Algebra structure of O q ( SL n + 1 ) . Finally, using a dual paring, it is shown that these algebras are isomorphic U q ± ( sl n + 1 ) .