Description of costandard modules for Schur superalgebra S(2|2) in positive characteristic

2018 ◽  
Vol 17 (02) ◽  
pp. 1850038
Author(s):  
A. N. Grishkov ◽  
F. Marko
Keyword(s):  
Positive Characteristic ◽  
Simple Module ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Highest Weight ◽  
Decomposition Numbers ◽  
Schur Superalgebra ◽  
Composition Factors

In this paper, we shall describe costandard modules [Formula: see text] of restricted highest weight [Formula: see text] for Schur superalgebra [Formula: see text] over an algebraically closed field [Formula: see text] of positive characteristic [Formula: see text]. Additionally, for a restricted highest weight [Formula: see text], we determine all composition factors of the costandard module [Formula: see text]; in particular we compute the decomposition numbers in the process of the modular reduction of a simple module with a restricted highest weight.

scholarly journals DESCRIPTION OF SIMPLE MODULES FOR SCHUR SUPERALGEBRA S(2|2)

2013 ◽  
Vol 55 (3) ◽  
pp. 695-719 ◽  
Author(s):  
A. N. GRISHKOV ◽  
F. MARKO
Keyword(s):  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Modules ◽  
Schur Superalgebra

AbstractThe goal of this paper is to describe explicitly simple modules for Schur superalgebra S(2|2) over an algebraically closed field K of characteristic zero or positive characteristic p>2.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Fibrations with moving cuspidal singularities

1991 ◽  
Vol 122 ◽  
pp. 161-179 ◽  
Author(s):  
Yoshifumi Takeda
Keyword(s):  
Algebraic Geometry ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Closed Field ◽  
Projective Surface ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Algebraically Closed Field ◽  
Smooth Projective Surface ◽  
Almost All

Let f: V → C be a fibration from a smooth projective surface onto a smooth projective curve over an algebraically closed field k. In the case of characteristic zero, almost all fibres of f are nonsingular. In the case of positive characteristic, it is, however, known that there exist fibrations whose general fibres have singularities. Moreover, it seems that such fibrations often have pathological phenomena of algebraic geometry in positive characteristic (see M. Raynaud [7], W. Lang [4]).

THE LOEWY STRUCTURE OF -VERMA MODULES OF SINGULAR HIGHEST WEIGHTS

2015 ◽  
Vol 16 (4) ◽  
pp. 887-898
Author(s):  
Noriyuki Abe ◽  
Masaharu Kaneda
Keyword(s):  
Algebraic Group ◽  
Maximal Torus ◽  
Positive Characteristic ◽  
Closed Field ◽  
Verma Modules ◽  
Reductive Algebraic Group ◽  
Algebraically Closed Field ◽  
Loewy Length ◽  
Frobenius Kernel ◽  
Field Of Positive Characteristic

Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_{1}$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the parabolically induced $G_{1}T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan–Lusztig $P$- and $Q$-polynomials. We assume that the characteristic of the field is sufficiently large that, in particular, Lusztig’s conjecture for the irreducible $G_{1}T$-characters holds.

Centers and Azumaya loci for finite W-algebras in positive characteristic

2021 ◽  
pp. 1-32
Author(s):  
BIN SHU ◽  
YANG ZENG
Keyword(s):  
Lie Algebra ◽  
Nilpotent Element ◽  
Positive Characteristic ◽  
Irreducible Representations ◽  
Maximal Dimension ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Simple Lie Algebra ◽  
Maximal Spectrum ◽  
Smooth Locus

Abstract In this paper, we study the center Z of the finite W-algebra $${\mathcal{T}}({\mathfrak{g}},e)$$ associated with a semi-simple Lie algebra $$\mathfrak{g}$$ over an algebraically closed field $$\mathbb{k}$$ of characteristic p≫0, and an arbitrarily given nilpotent element $$e \in{\mathfrak{g}} $$ We obtain an analogue of Veldkamp’s theorem on the center. For the maximal spectrum Specm(Z), we show that its Azumaya locus coincides with its smooth locus of smooth points. The former locus reflects irreducible representations of maximal dimension for $${\mathcal{T}}({\mathfrak{g}},e)$$ .

scholarly journals The construction problem for Hodge numbers modulo an integer in positive characteristic

2020 ◽  
Vol 8 ◽  
Author(s):  
Remy van Dobben de Bruyn ◽  
Matthias Paulsen
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Construction Problem ◽  
Algebraically Closed Field ◽  
Serre Duality ◽  
Hodge Numbers ◽  
Field Of Positive Characteristic

Abstract Let k be an algebraically closed field of positive characteristic. For any integer $m\ge 2$ , we show that the Hodge numbers of a smooth projective k-variety can take on any combination of values modulo m, subject only to Serre duality. In particular, there are no non-trivial polynomial relations between the Hodge numbers.

ENRIQUES’ CLASSIFICATION IN CHARACTERISTIC 0$" height="12pt">: THE -THEOREM

2018 ◽  
Vol 235 ◽  
pp. 201-226
Author(s):  
FABRIZIO CATANESE ◽  
BINRU LI
Keyword(s):  
Positive Characteristic ◽  
Kodaira Dimension ◽  
Algebraic Surfaces ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Precise Version

The main goal of this paper is to show that Castelnuovo–Enriques’ $P_{12}$ - theorem (a precise version of the rough classification of algebraic surfaces) also holds for algebraic surfaces $S$ defined over an algebraically closed field $k$ of positive characteristic ( $\text{char}(k)=p>0$ ). The result relies on a main theorem describing the growth of the plurigenera for properly elliptic or properly quasielliptic surfaces (surfaces with Kodaira dimension equal to 1). We also discuss the limit cases, i.e., the families of surfaces which show that the result of the main theorem is sharp.

scholarly journals GENERAL HYPERPLANE SECTIONS OF THREEFOLDS IN POSITIVE CHARACTERISTIC

2018 ◽  
Vol 19 (2) ◽  
pp. 647-661 ◽  
Author(s):  
Kenta Sato ◽  
Shunsuke Takagi
Keyword(s):  
Projective Variety ◽  
Positive Characteristic ◽  
Hyperplane Section ◽  
Three Dimensional ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Canonical Singularities ◽  
Hyperplane Sections

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

scholarly journals Which rational double points occur on del Pezzo surfaces?

2021 ◽  
Vol Volume 5 ◽  
Author(s):  
Claudia Stadlmayr
Keyword(s):  
Positive Characteristic ◽  
Del Pezzo Surfaces ◽  
Closed Field ◽  
Arbitrary Degree ◽  
Algebraically Closed Field ◽  
Double Points ◽  
Arbitrary Characteristic ◽  
Classical Work ◽  
Del Pezzo

We determine all configurations of rational double points that occur on RDP del Pezzo surfaces of arbitrary degree and Picard rank over an algebraically closed field $k$ of arbitrary characteristic ${\rm char}(k)=p \geq 0$, generalizing classical work of Du Val to positive characteristic. Moreover, we give simplified equations for all RDP del Pezzo surfaces of degree $1$ containing non-taut rational double points.

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