scholarly journals Combinatorial study of stable categories of graded Cohen–Macaulay modules over skew quadric hypersurfaces

2021 ◽  
Author(s):  
Akihiro Higashitani ◽  
Kenta Ueyama
Keyword(s):  
Representation Theory ◽  
Polynomial Algebra ◽  
Derived Category ◽  
Stable Category ◽  
Irreducible Components ◽  
Point Scheme ◽  
Skew Polynomial ◽  
Stable Categories

AbstractIn this paper, we present a new connection between representation theory of noncommutative hypersurfaces and combinatorics. Let S be a graded ($$\pm 1$$ ± 1 )-skew polynomial algebra in n variables of degree 1 and $$f =x_1^2 + \cdots +x_n^2 \in S$$ f = x 1 2 + ⋯ + x n 2 ∈ S . We prove that the stable category $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) of graded maximal Cohen–Macaulay module over S/(f) can be completely computed using the four graphical operations. As a consequence, $$\mathsf {\underline{CM}}^{\mathbb Z}(S/(f))$$ CM ̲ Z ( S / ( f ) ) is equivalent to the derived category $$\mathsf {D}^{\mathsf {b}}({\mathsf {mod}}\,k^{2^r})$$ D b ( mod k 2 r ) , and this r is obtained as the nullity of a certain matrix over $${\mathbb F}_2$$ F 2 . Using the properties of Stanley–Reisner ideals, we also show that the number of irreducible components of the point scheme of S that are isomorphic to $${\mathbb P}^1$$ P 1 is less than or equal to $$\left( {\begin{array}{c}r+1\\ 2\end{array}}\right) $$ r + 1 2 .

scholarly journals On Knörrer Periodicity for Quadric Hypersurfaces in Skew Projective Spaces

2018 ◽  
Vol 62 (4) ◽  
pp. 896-911
Author(s):  
Kenta Ueyama
Keyword(s):  
Polynomial Algebra ◽  
Projective Spaces ◽  
Stable Category ◽  
Periodicity Theorem ◽  
Irreducible Components ◽  
Point Scheme ◽  
Skew Polynomial

AbstractWe study the structure of the stable category $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ of graded maximal Cohen–Macaulay module over $S/(f)$ where $S$ is a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree 1, and $f=x_{1}^{2}+\cdots +x_{n}^{2}$. If $S$ is commutative, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is well known by Knörrer’s periodicity theorem. In this paper, we prove that if $n\leqslant 5$, then the structure of $\text{}\underline{\mathsf{CM}}^{\mathbb{Z}}(S/(f))$ is determined by the number of irreducible components of the point scheme of $S$ which are isomorphic to $\mathbb{P}^{1}$.

scholarly journals A $\lambda$-ring Frobenius Characteristic for $G\wr S_n$

10.37236/1809 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Anthony Mendes ◽  
Jeffrey Remmel ◽  
Jennifer Wagner
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Reproducing Kernel ◽  
Irreducible Representations ◽  
Inner Product ◽  
Irreducible Components ◽  
Lambda Ring

A $\lambda$-ring version of a Frobenius characteristic for groups of the form $G \wr S_n$ is given. Our methods provide natural analogs of classic results in the representation theory of the symmetric group. Included is a method decompose the Kronecker product of two irreducible representations of $G\wr S_n$ into its irreducible components along with generalizations of the Murnaghan-Nakayama rule, the Hall inner product, and the reproducing kernel for $G\wr S_n$.

scholarly journals Algebras stably equivalent to selfinjective algebras whose Auslander-Reiten quivers consist only of generalized standard components

1998 ◽  
Vol 40 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Zygmunt Pogorzały
Keyword(s):  
Unit Element ◽  
Global Dimension ◽  
Closed Field ◽  
Modular Representation Theory ◽  
Stable Category ◽  
Finite Global Dimension ◽  
Finite Dimensional ◽  
Stable Equivalences ◽  
Stable Categories

Throughout the paper K denotes a fixed algebraically closed field. All algebras considered are finite-dimensional associative K-algebras with a unit element. Moreover, they are assumed to be basic and connected. For an algebra A we denote by mod(A) the category of all finitely generated right A-modules, and mod(A) denotes the stable category of mod(A), i.e. mod(A)/℘ where ℘ is the two-sided ideal in mod(A) of all morphisms that factorize through projective A-modules. Two algebras A and B are said to be stably equivalent if the stable categories mod(A) and mod(B) are equivalent. The study of stable equivalences of algebras has its sources in modular representation theory of finite groups. It is of importance in this theory whether two stably equivalent algebras have the same number of pairwise non-isomorphic nonprojective simple modules. Another motivation for studying stable equivalences appears in the following context. If E is a K-algebra of finite global dimension then its derived category Db(E) is equivalent to the stable category mod(Ê) of the repetitive category Ê of E [15]. Thus the problem of a classification of derived equivalent algebras leads in many cases to a classification of stably equivalent selfinjective algebras.

Parabolic Subgroups with Abelian Unipotent Radical as a Testing Site for Invariant Theory

1999 ◽  
Vol 51 (3) ◽  
pp. 616-635 ◽  
Author(s):  
Dmitri I. Panyushev
Keyword(s):  
Polynomial Algebra ◽  
Free Module ◽  
Unipotent Radical ◽  
Parabolic Subgroups ◽  
Simple Algebraic Group ◽  
Orbit Closures ◽  
Determinantal Varieties ◽  
Commuting Variety ◽  
Irreducible Components ◽  
Testing Site

AbstractLet L be a simple algebraic group and P a parabolic subgroup with Abelian unipotent radical Pu. Many familiar varieties (determinantal varieties, their symmetric and skew-symmetric analogues) arise as closures of P-orbits in Pu. We give a unified invariant-theoretic treatment of various properties of these orbit closures. We also describe the closures of the conormal bundles of these orbits as the irreducible components of some commuting variety and show that the polynomial algebra k[Pu] is a free module over the algebra of covariants.

The existence of n-Auslander–Reiten sequences via determined morphisms

2021 ◽  
Author(s):  
Zongyang Xie ◽  
Zhongkui Liu ◽  
Xiaoyan Yang
Keyword(s):  
Artinian Ring ◽  
Finitely Generated ◽  
Stable Category ◽  
Equivalent Characterization ◽  
Stable Categories

Let [Formula: see text] be a commutative artinian ring and [Formula: see text] a small Ext-finite Krull–Schmidt [Formula: see text]-abelian [Formula: see text]-category with enough projectives and injectives. We introduce two full subcategories [Formula: see text] and [Formula: see text] of [Formula: see text] in terms of the representable functors from the stable category of [Formula: see text] to category of finitely generated [Formula: see text]-modules. Moreover, we define two additive functors [Formula: see text] and [Formula: see text], which are mutually quasi-inverse equivalences between the stable categories of this two full subcategories. We give an equivalent characterization on the existence of [Formula: see text]-Auslander–Reiten sequences using determined morphisms.

Triangulated Equivalences Involving Gorenstein Projective Modules

2017 ◽  
Vol 60 (4) ◽  
pp. 879-890 ◽  
Author(s):  
Yuefei Zheng ◽  
Zhaoyong Huang
Keyword(s):  
Noetherian Ring ◽  
Derived Category ◽  
Projective Modules ◽  
Injective Dimension ◽  
Stable Category ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Gorenstein Projective Modules ◽  
Left And Right

AbstractFor any ring R, we show that, in the bounded derived category Db(Mod R) of left R-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category (resp. ) of Gorenstein projective (resp. injective) modules. As a consequence, we get that if R is a left and right noetherian ring admitting a dualizing complex, then and are equivalent.

scholarly journals On representation theory of total (co)integrals

2016 ◽  
Vol 15 (10) ◽  
pp. 1650195
Author(s):  
Mohammad Hassanzadeh
Keyword(s):  
Representation Theory ◽  
Hopf Algebra ◽  
Polynomial Algebra ◽  
Universal Enveloping Algebra ◽  
Drinfeld Modules ◽  
Enveloping Algebra

In this paper, we show that total integrals and cointegrals are new sources of stable anti-Yetter–Drinfeld modules. We explicitly show that how special types of total (co)integrals can be used to provide both (stable) anti Yetter–Drinfeld and Yetter–Drinfeld modules. We use these modules to classify total (co)integrals and (cleft) Hopf Galois (co)extensions of the Connes–Moscovici Hopf algebra, and some examples of universal enveloping algebra and polynomial algebra.

scholarly journals DEGREE CONES AND MONOMIAL BASES OF LIE ALGEBRAS AND QUANTUM GROUPS

2017 ◽  
Vol 59 (3) ◽  
pp. 595-621 ◽  
Author(s):  
TEODOR BACKHAUS ◽  
XIN FANG ◽  
GHISLAIN FOURIER
Keyword(s):  
Lie Algebras ◽  
Classical Limit ◽  
Polynomial Algebra ◽  
Monomial Ideals ◽  
Lattice Points ◽  
Graded Algebra ◽  
Negative Part ◽  
Graded Modules ◽  
Finite Dimensional ◽  
Skew Polynomial

AbstractWe provide ℕ-filtrations on the negative part Uq($\mathfrak{n}$−) of the quantum group associated to a finite-dimensional simple Lie algebra $\mathfrak{g}$, such that the associated graded algebra is a skew-polynomial algebra on $\mathfrak{n}$−. The filtration is obtained by assigning degrees to Lusztig's quantum PBW root vectors. The possible degrees can be described as lattice points in certain polyhedral cones. In the classical limit, such a degree induces an ℕ-filtration on any finite-dimensional simple $\mathfrak{g}$-module. We prove for type An, Cn, B3, D4 and G2 that a degree can be chosen such that the associated graded modules are defined by monomial ideals, and conjecture that this is true for any $\mathfrak{g}$.

scholarly journals STABILITY PATTERNS IN REPRESENTATION THEORY

2015 ◽  
Vol 3 ◽  
Author(s):  
STEVEN V SAM ◽  
ANDREW SNOWDEN
Keyword(s):  
Representation Theory ◽  
Linear Theory ◽  
Symmetric Groups ◽  
General Linear ◽  
Stable Category ◽  
Comprehensive Theory ◽  
Stable Representation

We develop a comprehensive theory of the stable representation categories of several sequences of groups, including the classical and symmetric groups, and their relation to the unstable categories. An important component of this theory is an array of equivalences between the stable representation category and various other categories, each of which has its own flavor (representation theoretic, combinatorial, commutative algebraic, or categorical) and offers a distinct perspective on the stable category. We use this theory to produce a host of specific results: for example, the construction of injective resolutions of simple objects, duality between the orthogonal and symplectic theories, and a canonical derived auto-equivalence of the general linear theory.

m-Noncrossing trees

2018 ◽  
Vol 17 (10) ◽  
pp. 1850187 ◽  
Author(s):  
Kiyoshi Igusa
Keyword(s):  
Representation Theory ◽  
Derived Category ◽  
Point Of View ◽  
Module Category ◽  
Cluster Category ◽  
Cluster Theory ◽  
Type Formula ◽  
Exceptional Sequences ◽  
The Relationship

We examine [Formula: see text]-cluster theory from an elementary point of view using a generalization of [Formula: see text]-ary trees which we call [Formula: see text]-noncrossing trees. We show that these trees are in bijection with [Formula: see text]-clusters in the [Formula: see text]-cluster category of a quiver of type [Formula: see text]. Similar trees are in bijection with complete exceptional sequences. Most of this paper is expository, explaining definitions and known results about these topics in representation theory. One application of these trees is that the mutation formula for [Formula: see text]-clusters is derived from the more elementary mutation of trees. The main new result is that the natural map of an [Formula: see text]-noncrossing tree into the plane is an embedding. We also explain the relationship between [Formula: see text]-noncrossing trees and finite Harder–Narasimhan systems in the derived category of the module category of type [Formula: see text].

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