674 results on '"Unipotent"'
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2. Revolutionizing medicine practice using stem cells in healthcare: review article
- Author
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Prajnashree Acharya and Sanatkumar B Nyamagoud
- Subjects
differentiation ,embryonic stem cells (esc) ,induced pluripotent stem cells (ipsc) ,pluripotent ,regenerative medicine ,self-renew ,stem cells ,tissue bank ,totipotent ,unipotent ,Medicine - Abstract
This review delves into the transformative potential of stem cells in healthcare, particularly within regenerative medicine. With their unique ability to self-renew and differentiate into various cell types, stem cells offer groundbreaking possibilities for treating various medical conditions. The review begins by thoroughly exploring different types of stem cells, from totipotent to pluripotent, highlighting their specific capabilities. This foundational understanding sets the stage for examining the therapeutic potential of stem cells. A key focus is the practical application of stem cell-based therapies, particularly in treating conditions like epidermolysis bullosa and macular degeneration. These examples showcase how stem cell research translates into real-world treatments, helping individuals with debilitating illnesses regain functionality and improve their quality of life. The review further emphasizes advancements in clinical trials, particularly in neurodegenerative diseases and spinal cord injuries, demonstrating significant progress in these fields. Additionally, the importance of stem cell banking is underscored as an essential resource for future regenerative medicine, offering a readily available source of cells for personalized treatments. Integrating stem cell research into therapeutic applications represents a revolutionary leap in modern medicine, potentially disrupting traditional treatment paradigms and providing new hope for previously incurable diseases.
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- 2024
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3. Symmetric and reversible properties of bi-amalgamated rings.
- Author
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Aruldoss, Antonysamy and Selvaraj, Chelliah
- Abstract
Let f: A → B and g: A → C be two ring homomorphisms and let K and K′ be two ideals of B and C, respectively, such that f
−1 (K) = g−1 (K′). We investigate unipotent, symmetric and reversible properties of the bi-amalgamation ring A ⋈f,g (K, K′) of A with (B, C) along (K, K′) with respect to (f, g). [ABSTRACT FROM AUTHOR]- Published
- 2024
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4. 弱左型 B 半群的半格分解.
- Author
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李春华, 方洁莹, 孟令香, and 徐保根
- Subjects
- *
GENERALIZATION - Abstract
Weakly type B semi-groups are generalized inverse semigroups on semi-abundant semigroups. This paper studied the semi group by the method of idempotent. As a generalization, the notion of a weakly type B unary semi-group was introduced by an idempotent method. Some basic properties of such unary semi-group were given. Moreover, some equivalent conditions for an arbitrary unary semigroup to be a weakly left type B semi-group were obtained. Finally, a semi-lattice decomposition of a weakly left B semi-group was given, and some results were obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2022
5. Enumeration of Latin squares with conjugate symmetry.
- Author
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McKay, Brendan D. and Wanless, Ian M.
- Subjects
- *
MAGIC squares , *SYMMETRY , *IDEMPOTENTS - Abstract
A Latin square has six conjugate Latin squares obtained by uniformly permuting its (row, column, symbol) triples. We say that a Latin square has conjugate symmetry if at least two of its six conjugates are equal. We enumerate Latin squares with conjugate symmetry and classify them according to several common notions of equivalence. We also do similar enumerations under additional hypotheses, such as assuming the Latin square is reduced, diagonal, idempotent or unipotent. Our data corrected an error in earlier literature and suggested several patterns that we then found proofs for, including (1) the number of isomorphism classes of semisymmetric idempotent Latin squares of order n equals the number of isomorphism classes of semisymmetric unipotent Latin squares of order n+1, and (2) suppose A and B are totally symmetric Latin squares of order n≢0 mod3. If A and B are paratopic then A and B are isomorphic. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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6. Explicit Artin maps into PGL2
- Author
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Antonia W. Bluher
- Subjects
Mathematics - Number Theory ,Group (mathematics) ,General Mathematics ,Order (ring theory) ,Unipotent ,Characterization (mathematics) ,Additive polynomial ,Combinatorics ,11R58, 11T30 ,Conjugacy class ,FOS: Mathematics ,Number Theory (math.NT) ,Galois extension ,Prime power ,Mathematics - Abstract
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
- Published
- 2022
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7. Unipotent representations attached to the principal nilpotent orbit
- Author
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Lucas Mason-Brown
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Pure mathematics ,Mathematics (miscellaneous) ,Principal (computer security) ,FOS: Mathematics ,Nilpotent orbit ,Representation Theory (math.RT) ,Unipotent ,Reductive group ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper, we construct and classify the special unipotent representations of a real reductive group attached to the principal nilpotent orbit. We give formulas for the $\mathbf{K}$-types, associated varieties, and Langlands parameters of all such representations., revised exposition
- Published
- 2021
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8. Overgroups of regular unipotent elements in simple algebraic groups
- Author
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Gunter Malle and Donna Testerman
- Subjects
Connected component ,Mathematics::Group Theory ,Pure mathematics ,Simple (abstract algebra) ,010102 general mathematics ,Torus ,010103 numerical & computational mathematics ,General Medicine ,0101 mathematics ,Algebraic number ,Unipotent ,01 natural sciences ,Mathematics - Abstract
We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.
- Published
- 2021
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9. Springer’s work on unipotent classes and Weyl group representations
- Author
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George Lusztig
- Subjects
Algebra ,Weyl group ,symbols.namesake ,Work (electrical) ,General Mathematics ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,symbols ,Unipotent ,Algebraic number ,Mathematics::Representation Theory ,Mathematics - Abstract
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.
- Published
- 2021
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10. On Chief Factors of Parabolic Maximal Subgroups of the Group $${}^{2}F_{4}(2^{2n+1})$$
- Author
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V. V. Korableva
- Subjects
Combinatorics ,Physics ,Maximal subgroup ,Mathematics (miscellaneous) ,Group (mathematics) ,Chief series ,Classification of finite simple groups ,Unipotent ,Type (model theory) - Abstract
This study continues the author’s previous papers where a refined description of the chief factors of a parabolic maximal subgroup involved in its unipotent radical was obtained for all (normal and twisted) finite simple groups of Lie type except for the groups $${}^{2}F_{4}(2^{2n+1})$$ and $$B_{l}(2^{n})$$ . In present paper, such a description is given for the group $${}^{2}F_{4}(2^{2n+1})$$ . We prove a theorem in which, for every parabolic maximal subgroup of $${}^{2}F_{4}(2^{2n+1})$$ , a fragment of the chief series involved in the unipotent radical of this subgroup is given. Generators of the corresponding chief factors are presented in a table.
- Published
- 2021
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11. Equivalence of categories between coefficient systems and systems of idempotents
- Author
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Thomas Lanard
- Subjects
Subcategory ,Pure mathematics ,Equivalence of categories ,Group (mathematics) ,Block (permutation group theory) ,Zero (complex analysis) ,Reductive group ,Unipotent ,Mathematics (miscellaneous) ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Equivalence (measure theory) ,Mathematics - Representation Theory ,Mathematics - Abstract
The consistent systems of idempotents of Meyer and Solleveld allow to construct Serre subcategories of $Rep_R(G)$, the category of smooth representations of a $p$-adic group $G$ with coefficients in $R$. In particular, they were used to construct level 0 decompositions when $R=\overline{\mathbb{Z}}_{\ell}$, $\ell \neq p$, by Dat for $GL_n$ and the author for a more general group. Wang proved in the case of $GL_n$ that the subcategory associated with a system of idempotents is equivalent to a category of coefficient systems on the Bruhat-Tits building. This result was used by Dat to prove an equivalence between an arbitrary level zero block of $GL_n$ and a unipotent block of another group. In this paper, we generalize Wang's equivalence of category to a connected reductive group on a non-archimedean local field., 17 pages, in English
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- 2021
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12. Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)
- Author
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Dmitry Chelkak and Mikhail Basok
- Subjects
Loop (topology) ,Combinatorics ,Applied Mathematics ,General Mathematics ,Entire function ,Simply connected space ,Domain (ring theory) ,Disjoint sets ,Unipotent ,Invariant (mathematics) ,Lambda ,Mathematics - Abstract
Building upon recent results of Dub\'edat on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations $\Omega^\delta$ to a simply connected domain $\Omega\subset\mathbb C$ we prove the convergence of probabilities of cylindrical events for the \emph{double-dimer loop ensembles} on $\Omega^\delta$ as $\delta\to 0$. More precisely, let $\lambda_1,\dots,\lambda_n\in\Omega$ and $L$ be a macroscopic lamination on $\Omega\setminus\{\lambda_1,\dots,\lambda_n\}$, i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities $P_L^\delta$ that one obtains $L$ after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on $\Omega^\delta$ converge to a conformally invariant limit $P_L$ as $\delta \to 0$, for each $L$. Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety $\mathrm{Hom}(\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C))$ and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do \emph{not} use any RSW-type arguments for double-dimers. The limits $P_L$ of the probabilities $P_L^\delta$ are defined as coefficients of the isomonodormic tau-function studied by Dub\'edat with respect to the Fock--Goncharov lamination basis on the representation variety. The fact that $P_L$ coincides with the probability to obtain $L$ from a sample of the nested CLE(4) in $\Omega$ requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.
- Published
- 2021
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13. A combinatorial approach to Donkin-Koppinen filtrations of general linear supergroups
- Author
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František Marko
- Subjects
Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,Unipotent ,01 natural sciences ,Rings and Algebras (math.RA) ,Natural transformation ,FOS: Mathematics ,High Energy Physics::Experiment ,Representation Theory (math.RT) ,0101 mathematics ,Nuclear Experiment ,Mathematics::Representation Theory ,Supergroup ,Mathematics - Representation Theory ,Mathematics - Abstract
For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$. We compute the action of odd superderivations on the images $\phi^*(x_{ij})$ of the generators of $K[G]$. We describe a specific ordering of the dominant weights $X(T)^+$ of $GL(m|n)$ for which there exists a Donkin-Koppinen filtration of the coordinate algebra $K[G]$. Let $\Gamma$ be a finitely generated ideal $\Gamma$ of $X(T)^+$ and $O_{\Gamma}(K[G])$ be the largest $\Gamma$-subsupermodule of $K[G]$ having simple composition factors of highest weights $\lambda\in \Gamma$. We apply combinatorial techniques, using generalized bideterminants, to determine a basis of $G$-superbimodules appearing in Donkin-Koppinen filtration of $O_{\Gamma}(K[G])$.
- Published
- 2021
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14. The Deligne–Illusie Theorem and exceptional Enriques surfaces
- Author
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Stefan Schröer
- Subjects
Ring (mathematics) ,Pure mathematics ,Mathematics::Algebraic Geometry ,Mathematics::K-Theory and Homology ,Group scheme ,General Mathematics ,Pushforward (differential) ,Homological algebra ,Vector bundle ,Unipotent ,Witt vector ,Cohomology ,Mathematics - Abstract
Building on the results of Deligne and Illusie on liftings to truncated Witt vectors, we give a criterion for non-liftability that involves only the dimension of certain cohomology groups of vector bundles arising from the Frobenius pushforward of the de Rham complex. Using vector bundle methods, we apply this to show that exceptional Enriques surfaces, a class introduced by Ekedahl and Shepherd-Barron, do not lift to truncated Witt vectors, yet the base of the miniversal formal deformation over the Witt vectors is regular. Using the classification of Bombieri and Mumford, we also show that bielliptic surfaces arising from a quotient by a unipotent group scheme of order p do not lift to the ring of Witt vectors. These results hinge on some observations in homological algebra that relates splittings in derived categories to Yoneda extensions and certain diagram completions.
- Published
- 2021
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15. Free by cyclic groups and linear groups with restricted unipotent elements.
- Author
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Button, Jack O.
- Subjects
- *
CYCLIC groups , *LINEAR statistical models , *CURVATURE , *AUTOMORPHISMS , *MATHEMATICAL mappings - Abstract
We introduce the class of linear groups that do not contain unipotent elements of infinite order, which includes all linear groups in positive characteristic.We show that groups in this class have good closure properties, in addition to having properties akin to non-positive curvature, which were proved in [6].We give examples of abstract groups lying in this class, but also show that Gersten's free by cyclic group does not. This implies that it has no faithful linear representation of any dimension over any field of positive characteristic, nor can it be embedded in any complex unitary group. [ABSTRACT FROM AUTHOR]
- Published
- 2017
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16. A unipotent circle action on 𝑝-adic modular forms
- Author
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Sean Howe
- Subjects
Pure mathematics ,Action (philosophy) ,010102 general mathematics ,0103 physical sciences ,Modular form ,010307 mathematical physics ,General Medicine ,0101 mathematics ,Unipotent ,01 natural sciences ,Mathematics - Abstract
Following a suggestion of Peter Scholze, we construct an action of G m ^ \widehat {\mathbb {G}_m} on the Katz moduli problem, a profinite-étale cover of the ordinary locus of the p p -adic modular curve whose ring of functions is Serre’s space of p p -adic modular functions. This action is a local, p p -adic analog of a global, archimedean action of the circle group S 1 S^1 on the lattice-unstable locus of the modular curve over C \mathbb {C} . To construct the G m ^ \widehat {\mathbb {G}_m} -action, we descend a moduli-theoretic action of a larger group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the action explicitly on local expansions and find it is given by a simple multiplication of the cuspidal and Serre-Tate coordinates q q ; along the way we also prove a natural generalization of Dwork’s equation τ = log q \tau =\log q for extensions of Q p / Z p \mathbb {Q}_p/\mathbb {Z}_p by μ p ∞ \mu _{p^\infty } valid over a non-Artinian base. Finally, we give a direct argument (without appealing to local expansions) to show that the action of G m ^ \widehat {\mathbb {G}_m} integrates the differential operator θ \theta coming from the Gauss-Manin connection and unit root splitting, and explain an application to Eisenstein measures and p p -adic L L -functions.
- Published
- 2020
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17. Cohomological equation and cocycle rigidity of discrete parabolic actions in some higher-rank Lie groups
- Author
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James Tanis and Zhenqi Jenny Wang
- Subjects
Pure mathematics ,General Mathematics ,Simple Lie group ,010102 general mathematics ,Lie group ,Unipotent ,Type (model theory) ,01 natural sciences ,Representation theory ,Cohomology ,010101 applied mathematics ,Sobolev space ,0101 mathematics ,Abelian group ,Analysis ,Mathematics - Abstract
Let $$\mathbb{G}$$ denote a higher-rank ℝ-split simple Lie group of the following type: SL(n, ℝ), SOo(m, m), E6(6), E7(7) and E8(8), where m ≥ 4 and n ≥ 3. We study the cohomological equation for discrete, abelian parabolic actions on $$\mathbb{G}$$ via representation theory. Specifically, we characterize the obstructions to solving the cohomological equation and construct smooth solutions with Sobolev estimates. We prove that global estimates of the solution are generally not tame, and our non-tame estimates in the case $$\mathbb{G}$$ = SL(n, ℝ) are sharp up to finite loss of regularity. Moreover, we prove that for general $$\mathbb{G}$$ the estimates are tame in all but one direction, and as an application, we obtain tame estimates for the common solution of the cocycle equations. We also give a sufficient condition for which the first cohomology with coefficients in smooth vector fields is trivial. In the case that $$\mathbb{G}$$ = SL(n, ℝ), we show this condition is also necessary. A new method is developed to prove tame directions involving computations within maximal unipotent subgroups of the unitary duals of SL(2, ℝ) ⋉ ℝ2 and SL(2, ℝ) ⋉ ℝ4. A new technique is also developed to prove non-tameness for solutions of the cohomological equation.
- Published
- 2020
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18. Representations and cohomology of a family of finite supergroup schemes
- Author
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Julia Pevtsova and Dave Benson
- Subjects
Pure mathematics ,Algebra and Number Theory ,Group cohomology ,010102 general mathematics ,Mathematics - Rings and Algebras ,Unipotent ,Local cohomology ,01 natural sciences ,Cohomology ,Cohomology ring ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,0103 physical sciences ,Spectral sequence ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Supergroup ,Mathematics - Representation Theory ,Quotient ,Mathematics - Abstract
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
- Published
- 2020
- Full Text
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19. Properties of the commutators of some elements of linear groups over divisions rings
- Author
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Yu. V. Petechuk and V. M. Petechuk
- Subjects
Combinatorics ,Physics ,Kernel (algebra) ,law ,Group (mathematics) ,General Mathematics ,Image (category theory) ,Division ring ,Commutator (electric) ,Unipotent ,Element (category theory) ,Commutative property ,law.invention - Abstract
Inclusions resulting from the commutativity of elements and their commutators with trans\-vections in the language of residual and fixed submodules are found. The residual and fixed submodules of an element $\sigma $ of the complete linear group are defined as the image and the kernel of the element $\sigma -1$ and are denoted by $R(\sigma )$ and $P(\sigma )$, respectively. It is shown that for an arbitrary element $g$ of a complete linear group over a division ring whose characteristic is different from 2 and the transvection $\tau $ from the commutativity of the commutator $\left[g,\tau \right]$ with $g$ is followed by the inclusion of $R(\left[g,\tau \right])\subseteq P(\tau )\cap P(g)$. It is proved that the same inclusions occur over an arbitrary division ring if $g$ is a unipotent element, $\mathrm{dim}\mathrm{}(R\left(\tau \right)+R\left(g\right))\le 2$ and the commutator $\left[g,\tau \right]$ commutes with $\tau $ or if $g$ is a unipotent commutator of some element of the complete linear group and transvection $\ \tau $.
- Published
- 2020
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20. The decomposition of Lusztig induction in classical groups
- Author
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Gunter Malle
- Subjects
Classical group ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Combinatorial proof ,Unipotent ,01 natural sciences ,Classical type ,0103 physical sciences ,Decomposition (computer science) ,010307 mathematical physics ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
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.
- Published
- 2020
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21. Brauer trees of unipotent blocks
- Author
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Olivier Dudas, David A. Craven, and Raphaël Rouquier
- Subjects
Pure mathematics ,Brauer tree ,Group of Lie type ,Applied Mathematics ,General Mathematics ,Unipotent ,Mathematics - Published
- 2020
- Full Text
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22. The Grothendieck group of unipotent representations: A new basis
- Author
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George Lusztig
- Subjects
Pure mathematics ,Group (mathematics) ,Mathematics::Number Theory ,Basis (universal algebra) ,Unipotent ,Mathematics (miscellaneous) ,Finite field ,Simple (abstract algebra) ,Algebraic group ,FOS: Mathematics ,Grothendieck group ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Let G(F_q) be the group of rational points of a simple algebraic group defined and split over a finite field F_q. In this paper we define a new basis for the Grothendieck group of unipotent representations of G(F_q)., 36 pages. arXiv admin note: text overlap with arXiv:1805.03770
- Published
- 2020
- Full Text
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23. Word Maps of Chevalley Groups Over Infinite Fields
- Author
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E. A. Egorchenkova
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Image (category theory) ,010102 general mathematics ,Cohomological dimension ,Type (model theory) ,Unipotent ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,Group of Lie type ,Algebraic group ,0103 physical sciences ,Perfect field ,0101 mathematics ,Word (group theory) ,Mathematics - Abstract
Let G be a simply connected Chevalley group over an infinite field K, and let $$ \tilde{w} $$ : Gn → G be a word map that corresponds to a nontrivial word w. In 2015, it has been proved that if w = w1w2w3w4 is the product of four words in independent variables, then every noncentral element of G is contained in the image of $$ \tilde{w} $$. A similar result for a word w = w1w2w3, which is the product of three independent words, was obtained in 2019 under the condition that the group G is not of type B2 or G2. In the present paper, it is proved that for a group of type B2 or G2, all elements of the large Bruhat cell B nw0B are contained in the image of the word map $$ \tilde{w} $$, where w = w1w2w3 is the product of three independent words. For a group G of type Ar, Cr, or G2 (respectively, for a group of type Ar) or a group over a perfect field K (respectively, over a perfect field K the characteristic of which is not a bad prime for G) with dim K ≤ 1 (here, dim K is the cohomological dimension of K), it is proved that all split regular semisimple elements (respectively, all regular unipotent elements) of G are contained in the image of $$ \tilde{w} $$, where w = w1w2 is the product of two independent words. Also, for any isotropic (but not necessary split) simple algebraic group G over a field K of characteristic zero, it is shown that for a word map $$ \tilde{w} $$ : G(K)n → G(K), where w = w1w2 is a product of two independent words, all unipotent elements are contained in Im $$ \tilde{w} $$.
- Published
- 2020
- Full Text
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24. Test vectors for Rankin–Selberg L-functions
- Author
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M. Krishnamurthy, Andrew R. Booker, and Min Lee
- Subjects
Pure mathematics ,Automorphic representations ,Algebra and Number Theory ,Test vectors ,Test vector ,Mathematics::Number Theory ,Automorphic form ,Field (mathematics) ,Unipotent ,Mathematics::Representation Theory ,Mathematics - Abstract
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.
- Published
- 2020
- Full Text
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25. A note on degenerate Whittaker models for general linear groups
- Author
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Arnab Mitra
- Subjects
Base change ,Pure mathematics ,Algebra and Number Theory ,Degenerate energy levels ,Zero (complex analysis) ,Extension (predicate logic) ,Unipotent ,Representation (mathematics) ,Local field ,Mathematics ,Jacquet module - Abstract
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).
- Published
- 2020
- Full Text
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26. The Block Structure of the Images of Regular Unipotent Elements from Subsystem Symplectic Subgroups of Rank 2 in Irreducible Representations of Symplectic Groups. I
- Author
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I. D. Suprunenko and T. S. Busel
- Subjects
Pure mathematics ,Rank (linear algebra) ,Simple (abstract algebra) ,General Mathematics ,Irreducible representation ,Unipotent ,Type (model theory) ,Algebraic number ,Representation (mathematics) ,Symplectic geometry ,Mathematics - Abstract
The dimensions of the Jordan blocks in the images of regular unipotent elements from subsystem subgroups of type C2 in p-restricted irreducible representations of groups of type Cn in characteristic p ≥ 11 with locally small highest weights are found. These results can be applied for investigating the behavior of unipotent elements in modular representations of simple algebraic groups and recognizing representations and linear groups. The article consists of 3 parts. In the first one, preliminary lemmas that are necessary for proving the principal results, are contained and the case where all weights of the restriction of a representation considered to a subgroup of type A1 containing a relevant unipotent element are less than p, is investigated.
- Published
- 2020
- Full Text
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27. K-automorphisms of a weak-crossed product F-algebra over a Galois extension K/F
- Author
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Jayampathy Ratnayake
- Subjects
Mathematics::Group Theory ,Pure mathematics ,Algebra and Number Theory ,F-algebra ,Crossed product ,Coxeter group ,Galois group ,Galois extension ,Unipotent ,Automorphism ,Bruhat order ,Mathematics - Abstract
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.
- Published
- 2019
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28. Two supercharacter theories for the parabolic subgroups in orthogonal and symplectic groups
- Author
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Alexander Nikolaevich Panov
- Subjects
Mathematics::Group Theory ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Unipotent ,Mathematics::Representation Theory ,01 natural sciences ,Mathematics ,Symplectic geometry - Abstract
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.
- Published
- 2019
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- View/download PDF
29. An effective Lie–Kolchin Theorem for quasi-unipotent matrices
- Author
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Thomas Koberda, Feng Luo, and Hongbin Sun
- Subjects
Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Block (permutation group theory) ,010103 numerical & computational mathematics ,Unipotent ,01 natural sciences ,Representation theory ,Lie–Kolchin theorem ,Matrix (mathematics) ,Solvable group ,Discrete Mathematics and Combinatorics ,Canonical form ,Geometry and Topology ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
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.
- Published
- 2019
- Full Text
- View/download PDF
30. On the values of unipotent characters of finite Chevalley groups of type E6 in characteristic 3
- Author
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Jonas Hetz
- Subjects
Pure mathematics ,Hecke algebra ,Class (set theory) ,Algebra and Number Theory ,010102 general mathematics ,Type (model theory) ,Unipotent ,20C33, 20G40, 20G41 ,01 natural sciences ,Representation theory ,Prime (order theory) ,Character (mathematics) ,Group of Lie type ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
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
- Published
- 2019
- Full Text
- View/download PDF
31. Bands of E-Inversive Unipotent Semigroups
- Author
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Roman S. Gigoń
- Subjects
Pure mathematics ,Class (set theory) ,General Mathematics ,010102 general mathematics ,Inversive ,010103 numerical & computational mathematics ,Unipotent ,01 natural sciences ,Subclass ,Set (abstract data type) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Retract ,0101 mathematics ,Mathematics - Abstract
We study some special types of bands of E-inversive unipotent semigroups. It has been proved that in any R-semigroup S, which is a band of E-inversive unipotent semigroups, the set of its regular elements is a retract of S. Also, some characterizations of E-inversive rectangular bands of unipotent semigroups are given. This theorem extends nearly 40-old results from the theory of epigroups. In fact, a more general result is valid in some special subclass of the class of E-inversive semigroups; this result seems to be (partially) new for all epigroups.
- Published
- 2019
- Full Text
- View/download PDF
32. The blocks and weights of finite special linear and unitary groups
- Author
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Zhicheng Feng
- Subjects
Algebra and Number Theory ,Conjecture ,Mathematics::Number Theory ,010102 general mathematics ,Group Theory (math.GR) ,Unipotent ,01 natural sciences ,Unitary state ,20C20, 20C33 ,Combinatorics ,Mathematics::Group Theory ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
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
- Published
- 2019
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33. Parabolic Perturbations of Unipotent Flows on Compact Quotients of SL $${(3,\mathbb{R})}$$ ( 3 , R )
- Author
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Davide Ravotti
- Subjects
Physics ,Pure mathematics ,Homogeneous ,010102 general mathematics ,0103 physical sciences ,Statistical and Nonlinear Physics ,Vector field ,010307 mathematical physics ,0101 mathematics ,Unipotent ,01 natural sciences ,Mathematical Physics ,Quotient - Abstract
We consider a family of smooth perturbations of unipotent flows on compact quotients of SL $${(3,\mathbb{R})}$$ which are not time-changes. More precisely, given a unipotent vector field, we perturb it by adding a non-constant component in a commuting direction. We prove that, if the resulting flow preserves a measure equivalent to Haar, then it is parabolic and mixing. The proof is based on a geometric shearing mechanism together with a non-homogeneous version of Mautner Phenomenon for homogeneous flows. Moreover, we characterize smoothly trivial perturbations and we relate the existence of non-trivial perturbations to the failure of cocycle rigidity of parabolic actions in SL $${(3,\mathbb{R})}$$ .
- Published
- 2019
- Full Text
- View/download PDF
34. Malcev products of weakly cancellative monoids and varieties of bands.
- Author
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Petrich, Mario
- Subjects
- *
MONOIDS , *SEMIGROUPS (Algebra) , *GROUP theory , *MATHEMATICAL mappings , *MATHEMATICAL analysis - Abstract
We consider unipotent, weakly cancellative, left cancellative, right cancellative, cancellative monoids and groups with the unary operation of mapping onto the identity element and bands with the identity mapping as unary operations. Then form Malcev products of each of the former with varieties of rectangular bands, semilattices, normal bands and all bands. Regarding the resulting semigroups with the induced unary operations, we characterize these classes, some of them we provide with a construction, and determine a basis for their implications, for they all turn out to be quasivarieties. We also determine their relationship which we clarify by an inclusion diagram. [ABSTRACT FROM AUTHOR]
- Published
- 2015
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35. Realm of matrices.
- Author
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Biswas, Debapriya
- Subjects
TRANSMISSION line matrix methods ,QUANTITATIVE research ,PHYSICAL sciences research ,TRANSCENDENTAL functions ,MATRICES (Mathematics) - Abstract
In this article, we discuss the exponential and the logarithmic functions in the realm of matrices. These notions are very useful in the mathematical and the physical sciences [1,2]. We discuss some important results including the connections established between skew-symmetric and orthogonal matrices, etc., through the exponentialmap. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
36. ON WEAKLY AMPLE SEMIGROUPS.
- Author
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PETRICH, MARIO
- Subjects
- *
SEMIGROUPS (Algebra) , *SEMIGROUP algebras , *GROUP theory , *IDEMPOTENTS , *LINEAR algebra - Abstract
Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a semigroup. Elements $a,b$ of $S$ are $\widetilde{\mathscr{R}}$-related if they have the same idempotent left identities. Then $S$ is weakly left ample if (1) idempotents of $S$ commute, (2) $\widetilde{\mathscr{R}}$ is a left congruence, (3) for any $a \in S$, $a$ is $\widetilde{\mathscr{R}}$-related to a (unique) idempotent, say $a^+$, and (4) for any element $a$ and idempotent $e$ of $S$, $ae=(ae)^+a$. Elements $a,b$ of $S$ are $\mathscr{R}^*$-related if, for any $x,y \in S^1$, $xa=ya$ if and only if $xb=yb$. Then $S$ is left ample if it satisfies (1), (3) and (4) relative to $\mathscr{R}^*$ instead of $\widetilde{\mathscr{R}}$. Further, $S$ is (weakly) ample if it is both (weakly) left and right ample. We establish several characterizations of these classes of semigroups. For weakly left ample ones we provide a construction of all such semigroups with zero all of whose nonzero idempotents are primitive. Among characterizations of weakly ample semigroups figure (strong) semilattices of unipotent monoids, and among those for ample semigroups, (strong) semilattices of cancellative monoids. This describes the structure of these two classes of semigroups in an optimal way, while, for the ‘one-sided’ case, the problem of structure remains open. [ABSTRACT FROM PUBLISHER]
- Published
- 2014
- Full Text
- View/download PDF
37. Lattice points counting and bounds on periods of Maass forms
- Author
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Feng Su and Andre Reznikov
- Subjects
Pointwise ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Automorphic form ,Lattice (group) ,Unipotent ,Type (model theory) ,01 natural sciences ,Matrix (mathematics) ,Homogeneous ,0101 mathematics ,Mathematics::Representation Theory ,Fourier series ,Mathematics - Abstract
We provide a “soft” proof for nontrivial bounds on spherical, hyperbolic, and unipotent Fourier coefficients of a fixed Maass form for a general cofinite lattice Γ \Gamma in PGL 2 ( R ) {\operatorname {PGL}_2(\mathbb {R})} . We use the amplification method based on the Airy type phenomenon for corresponding matrix coefficients and an effective Selberg type pointwise asymptotic for the lattice points counting in various homogeneous spaces for the group PGL 2 ( R ) {\operatorname {PGL}_2(\mathbb {R})} . This requires only L 2 L^2 -theory. We also show how to use the uniform bound for the L 4 L^4 -norm of K K -types in a fixed automorphic representation of PGL 2 ( R ) {\operatorname {PGL}_2(\mathbb {R})} in order to slightly improve these bounds.
- Published
- 2019
- Full Text
- View/download PDF
38. Permutative k-exponential epigroups
- Author
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Roman S. Gigoń
- Subjects
Pure mathematics ,Algebra and Number Theory ,Normal band ,010102 general mathematics ,Epigroup ,Congruence (manifolds) ,010103 numerical & computational mathematics ,0101 mathematics ,Unipotent ,01 natural sciences ,Mathematics ,Exponential function - Abstract
We prove that any permutative k-exponential epigroup is a normal band of unipotent epigroups. Moreover, we describe the least regular congruence on such semigroups. Finally, we give an example of a...
- Published
- 2019
- Full Text
- View/download PDF
39. On the values of unipotent characters in bad characteristic
- Author
-
Meinolf Geck
- Subjects
Pure mathematics ,Class (set theory) ,Algebra and Number Theory ,010102 general mathematics ,Type (model theory) ,Unipotent ,Symbolic computation ,01 natural sciences ,Prime (order theory) ,Character (mathematics) ,Finite field ,Group of Lie type ,20C33, 20G40 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematical Physics ,Analysis ,Mathematics - Abstract
Let $G(q)$ be a Chevalley group over a finite field $F_q$. By Lusztig's and Shoji's work, the problem of computing the values of the unipotent characters of $G(q)$ is solved, in principle, by the theory of character sheaves; one issue in this solution is the determination of certain scalars relating two types of class functions on $G(q)$. We show that this issue can be reduced to the case where $q$ is a prime, which opens the way to use computer algebra methods. Here, and in a sequel to this article, we use this approach to solve a number of cases in groups of exceptional type which seemed hitherto out of reach., 20 pages; minor corrections and additions
- Published
- 2018
- Full Text
- View/download PDF
40. Representations associated to small nilpotent orbits for complex Spin groups
- Author
-
Wan-Yu Tsai and Dan Barbasch
- Subjects
Group (mathematics) ,Complexification (Lie group) ,010102 general mathematics ,Unipotent ,Type (model theory) ,01 natural sciences ,Combinatorics ,Identity (mathematics) ,Nilpotent ,Mathematics (miscellaneous) ,0101 mathematics ,Maximal compact subgroup ,Mathematics ,Spin-½ - Abstract
This paper provides a comparison between the K K -structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type D D . Precisely, let G 0 = Spin ( 2 n , C ) G_0 =\operatorname {Spin}(2n,\mathbb {C}) be the Spin complex group as a real group, and let K ≅ G 0 K\cong G_0 be the complexification of the maximal compact subgroup of G 0 G_0 . We compute K K -spectra of the regular functions on some small nilpotent orbits O \mathcal {O} transforming according to characters ψ \psi of C K ( O ) C_{ K}(\mathcal {O}) trivial on the connected component of the identity C K ( O ) 0 C_{ K}(\mathcal {O})^0 . We then match them with the K {K} -types of the genuine (i.e., representations which do not factor to SO ( 2 n , C ) \operatorname {SO}(2n,\mathbb {C}) ) unipotent representations attached to O \mathcal {O} .
- Published
- 2018
- Full Text
- View/download PDF
41. Supercharacters of Unipotent and Solvable Groups
- Author
-
Alexander Nikolaevich Panov
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,0102 computer and information sciences ,Unipotent ,Type (model theory) ,Hopf algebra ,01 natural sciences ,Algebra ,Finite field ,General theory ,010201 computation theory & mathematics ,Solvable group ,0101 mathematics ,Algebra over a field ,Mathematics::Representation Theory ,Mathematics - Abstract
The notion of the supercharacter theory was introduced by P. Diaconis and I. M. Isaaks in 2008. In this paper, we present a review of the main notions and facts of the general theory and discuss the construction of the supercharacter theory for algebra groups and the theory of basic characters for unitriangular groups over a finite field. Based on his earlier papers, the author constructs the supercharacter theory for finite groups of triangular type. The structure of the Hopf algebra of supercharacters for triangular groups over finite fields is also characterized.
- Published
- 2018
- Full Text
- View/download PDF
42. Möbius disjointness for models of an ergodic system and beyond
- Author
-
Joanna Kułaga-Przymus, El Houcein El Abdalaoui, Mariusz Lemańczyk, Thierry de la Rue, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Marseille (I2M), Centre National de la Recherche Scientifique (CNRS)-École Centrale de Marseille (ECM)-Aix Marseille Université (AMU), Faculty of Mathematics and Computer Science, Nicolaus Copernicus University [Toruń], Research supported by the special program in the framework of the Jean Morlet semester'Ergodic Theory and Dynamical Systems in their Interactions with Arithmetic and Combina-torics', and by Narodowe Centrum Nauki grant UMO-2014/15/B/ST1/03736., and Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Mathematics - Number Theory ,General Mathematics ,Liouville function ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,010102 general mathematics ,Topological entropy ,Disjoint sets ,Unipotent ,Physics::Classical Physics ,Dynamical system ,Stationary ergodic process ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,010101 applied mathematics ,Combinatorics ,Ergodic theory ,Mathematics - Dynamical Systems ,0101 mathematics ,Entropy (arrow of time) ,Mathematics - Abstract
Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of $\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It is proved that, given an ergodic measure-preserving system $(Z,\mathcal{D},\kappa,R)$, the strong MOMO property (relatively to $\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all other uniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are M\"obius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to $\boldsymbol{\mu}$ in all zero entropy systems, in particular, it makes $\boldsymbol{\mu}$-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero., Comment: 35 pages
- Published
- 2018
- Full Text
- View/download PDF
43. On a minimal counterexample to Brauer’s k(B)-conjecture
- Author
-
Gunter Malle
- Subjects
Classical group ,Conjecture ,General Mathematics ,010102 general mathematics ,Unipotent ,01 natural sciences ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Algebra over a field ,Abelian group ,Minimal counterexample ,Counterexample ,Mathematics - Abstract
We study Brauer’s long-standing k(B)-conjecture on the number of characters in p-blocks for finite quasi-simple groups and show that their blocks do not occur as a minimal counterexample for p ≥ 5 nor in the case of abelian defect. For p = 3 we obtain that the principal 3-blocks do not provide minimal counterexamples. We also determine the precise number of irreducible characters in unipotent blocks of classical groups for odd primes.
- Published
- 2018
- Full Text
- View/download PDF
44. The finite unipotent groups consisting of bireflections
- Author
-
Katherine Horan and Peter Fleischmann
- Subjects
Ring (mathematics) ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Field (mathematics) ,Codimension ,Unipotent ,Space (mathematics) ,01 natural sciences ,010101 applied mathematics ,Converse ,0101 mathematics ,Element (category theory) ,QA ,Subspace topology ,Mathematics - Abstract
Let k be a field of characteristic p and V a finite-dimensional k-vector space. An element g ∈ GL ( V ) {g\in{\rm GL}(V)} is called a bireflection if it centralizes a subspace of codimension less than or equal to 2. It is known by a result of Kemper that if for a finite p-group G ≤ GL ( V ) {G\leq{\rm GL}(V)} the ring of invariants Sym ( V * ) G {{\rm Sym}(V^{*})^{G}} is Cohen–Macaulay, G is generated by bireflections. Although the converse is false in general, it holds in special cases e.g. for particular families of groups consisting of bireflections. In this paper we give, for p > 2 {p>2} , a classification of all finite unipotent subgroups of GL ( V ) {{\rm GL}(V)} consisting of bireflections. Our description of the groups is given explicitly in terms useful for exploring the corresponding rings of invariants. This further analysis will be the topic of a forthcoming paper.
- Published
- 2018
- Full Text
- View/download PDF
45. Parabolic induction in characteristic p
- Author
-
Rachel Ollivier and Marie-France Vignéras
- Subjects
Hecke algebra ,Functor ,Mathematics - Number Theory ,11E95, 20G25, 20C08, 22E50 ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Commutative ring ,Reductive group ,Unipotent ,01 natural sciences ,Combinatorics ,0103 physical sciences ,FOS: Mathematics ,Parabolic induction ,Number Theory (math.NT) ,010307 mathematical physics ,Locally compact space ,Representation Theory (math.RT) ,0101 mathematics ,Algebraically closed field ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $$\mathrm{F}$$ (resp. $$\mathbb F$$ ) be a nonarchimedean locally compact field with residue characteristic p (resp. a finite field with characteristic p). For $$k=\mathrm{F}$$ or $$k=\mathbb F$$ , let $$\mathbf {G}$$ be a connected reductive group over k and R be a commutative ring. We denote by $$\mathrm{Rep}( \mathbf G(k)) $$ the category of smooth R-representations of $$ \mathbf G(k) $$ . To a parabolic k-subgroup $${\mathbf P}=\mathbf {MN}$$ of $$\mathbf G$$ corresponds the parabolic induction functor $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}:\mathrm{Rep}( \mathbf M(k)) \rightarrow \mathrm{Rep}( \mathbf G(k))$$ . This functor has a left and a right adjoint. Let $${{\mathcal {U}}}$$ (resp. $${\mathbb {U}}$$ ) be a pro-p Iwahori (resp. a p-Sylow) subgroup of $$ \mathbf G(k) $$ compatible with $${\mathbf P}(k)$$ when $$k=\mathrm{F}$$ (resp. $$\mathbb F$$ ). Let $${H_{ \mathbf G(k)}}$$ denote the pro-p Iwahori (resp. unipotent) Hecke algebra of $$ \mathbf G(k) $$ over R and $$\mathrm{Mod}({H_{ \mathbf G(k)}})$$ the category of right modules over $${H_{ \mathbf G(k)}}$$ . There is a functor $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}: \mathrm{Mod}({H_{ \mathbf M(k)}}) \rightarrow \mathrm{Mod}({H_{ \mathbf G(k) }})$$ called parabolic induction for Hecke modules; it has a left and a right adjoint. We prove that the pro-p Iwahori (resp. unipotent) invariant functors commute with the parabolic induction functors, namely that $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}$$ and $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}$$ form a commutative diagram with the $${{\mathcal {U}}}$$ and $${{\mathcal {U}}}\cap \mathbf M(\mathrm{F})$$ (resp. $${\mathbb {U}}$$ and $${\mathbb {U}}\cap \mathbf M(\mathbb F) $$ ) invariant functors. We prove that the pro-p Iwahori (resp. unipotent) invariant functors also commute with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show that an irreducible admissible R-representation of $$ \mathbf G(\mathrm{F}) $$ is supercuspidal (or equivalently supersingular) if and only if the $${H_{ \mathbf G(\mathrm{F})}}$$ -module $${\mathfrak {m}}$$ of its $${{\mathcal {U}}}$$ -invariants admits a supersingular subquotient, if and only if $${\mathfrak {m}}$$ is supersingular.
- Published
- 2018
- Full Text
- View/download PDF
46. Asymptotic windings of horocycles
- Author
-
Omri Sarig and Dmitry Dolgopyat
- Subjects
Pure mathematics ,Geodesic ,General Mathematics ,Gaussian ,010102 general mathematics ,Process (computing) ,Cauchy distribution ,Unipotent ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Electromagnetic coil ,symbols ,0101 mathematics ,Algebra over a field ,Scaling ,Mathematics - Abstract
We analyze the scaling limits of the winding process for horocycles on noncompact hyperbolic surfaces with finite area. Initial conditions with precompact forward geodesics have scaling limits with gaussian and Cauchy components. Typical initial conditions have different scaling limits along different subsequences of times, and all such scaling limits can be described. Some of our results extend to other unipotent flows.
- Published
- 2018
- Full Text
- View/download PDF
47. Euler characteristic of analogues of a Deligne–Lusztig variety for GL
- Author
-
Dongkwan Kim
- Subjects
Pure mathematics ,Weyl group ,Algebra and Number Theory ,010102 general mathematics ,Unipotent ,Type (model theory) ,01 natural sciences ,Jordan decomposition ,symbols.namesake ,Conjugacy class ,Euler characteristic ,0103 physical sciences ,symbols ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Element (category theory) ,Mathematics::Representation Theory ,Mathematics - Abstract
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.
- Published
- 2018
- Full Text
- View/download PDF
48. Invariant and stationary measures for the action on Moduli space
- Author
-
Alex Eskin and Maryam Mirzakhani
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,010102 general mathematics ,Triangular matrix ,Unipotent ,Submanifold ,01 natural sciences ,Moduli space ,Number theory ,0103 physical sciences ,Ergodic theory ,010307 mathematical physics ,Affine transformation ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold. The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.
- Published
- 2018
- Full Text
- View/download PDF
49. Degeneration of Horospheres in Spherical Homogeneous Spaces
- Author
-
E. B. Vinberg and Simon Gindikin
- Subjects
Class (set theory) ,Pure mathematics ,Applied Mathematics ,Degenerate energy levels ,Unipotent ,Semisimple algebraic group ,Mathematics::Group Theory ,Simply connected space ,Variety (universal algebra) ,Mathematics::Representation Theory ,Affine variety ,Analysis ,Quotient ,Mathematics - Abstract
Horospheres for an action of a semisimple algebraic group G on an affine variety X are the generic orbits of a maximal unipotent subgroup U ⊂ G or, equivalently, the generic fibers of the categorical quotient of the variety X by the action of U, which is defined by the values of the highest weight functions. The remaining fibers of this quotient (which we call degenerate horospheres) for a certain class of spherical G-varieties containing all simply connected symmetric spaces are studied.
- Published
- 2018
- Full Text
- View/download PDF
50. Invariants of maximal tori and unipotent constituents of some quasi-projective characters for finite classical groups
- Author
-
Alexandre Zalesski
- Subjects
Classical group ,Discrete mathematics ,Weyl group ,Algebra and Number Theory ,Brauer's theorem on induced characters ,010102 general mathematics ,Group Theory (math.GR) ,Unipotent ,01 natural sciences ,Representation theory ,010101 applied mathematics ,Combinatorics ,symbols.namesake ,Algebraic group ,Irreducible representation ,FOS: Mathematics ,symbols ,Maximal torus ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Mathematics - Abstract
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.
- Published
- 2018
- Full Text
- View/download PDF
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