IRREDUCIBLE DECOMPOSITIONS OF DEGENERACY LOCI OF MATRICES

2008 ◽  
Vol 18 (02) ◽  
pp. 257-270
Author(s):  
XU-AN ZHAO ◽  
HONGZHU GAO
Keyword(s):  
Degeneracy Loci ◽  
Irreducible Components ◽  
Degeneracy Locus ◽  
Rank Conditions ◽  
Rank Functions

In this paper, we study the irreducible decompositions of the degeneracy loci of matrices which are defined by rank conditions on upper left submatrices. By introducing the concepts of standard and essential rank functions, we give an explicit classification of these degeneracy loci. Based on standard rank functions, we design an algorithm to write a degeneracy locus as a union of its irreducible components. This gives an answer to a problem raised by Sturmfels.

scholarly journals The Algebraic and Geometric Classification of Four Dimensional Superalgebras

2021 ◽  
Author(s):  
◽  
Aaron Armour
Keyword(s):  
Classification Problem ◽  
Associative Algebras ◽  
New Variety ◽  
Graded Algebras ◽  
Finite Representation ◽  
Algebraic Classification ◽  
Irreducible Components ◽  
Module Algebras ◽  
Sweedler’S Hopf Algebra

<p><b>The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12].</b></p> <p>Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras.</p> <p>This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties.</p> <p>The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.</p>

scholarly journals Diagrams and Essential Sets for Signed Permutations

10.37236/8106 ◽  
2018 ◽  
Vol 25 (3) ◽  
Author(s):  
David Anderson
Keyword(s):  
Schubert Variety ◽  
Signed Permutation ◽  
Signed Permutations ◽  
Degeneracy Locus ◽  
Essential Set ◽  
Diagrammatic Method ◽  
Essential Sets ◽  
Rank Conditions ◽  
Theoretic Version

We introduce diagrams and essential sets for signed permutations, extending the analogous notions for ordinary permutations.  In particular, we show that the essential set provides a minimal list of rank conditions defining the Schubert variety or degeneracy locus corresponding to a signed permutation.  Our essential set is in bijection with the poset-theoretic version defined by Reiner, Woo, and Yong, and thus gives an explicit, diagrammatic method for computing the latter.

scholarly journals Structure and classification of Hom-associative algebras

2020 ◽  
Vol 24 (1) ◽  
pp. 79-102
Author(s):  
Abdenacer Makhlouf ◽  
Ahmed Zahari
Keyword(s):  
Deformation Theory ◽  
Matrix Algebra ◽  
Algebraic Varieties ◽  
Associative Algebras ◽  
Irreducible Components

The purpose of this paper is to study the structure and the algebraic varieties of Hom-associative algebras. We characterize multiplicative simple Hom-associative algebras and give some examples deforming the 2 × 2-matrix algebra to simple Hom-associative algebras. We provide a classification of n-dimensional Hom-associative algebras for n ≤ 3. Then we study irreducible components using deformation theory.

scholarly journals The geometric classification of nilpotent ℭ𝔇-algebras

2020 ◽  
pp. 2150198 ◽  
Author(s):  
Ivan Kaygorodov ◽  
Mykola Khrypchenko
Keyword(s):  
Parameter Family ◽  
Irreducible Components ◽  
Two Parameter

We give a geometric classification of complex 4-dimensional nilpotent [Formula: see text]-algebras. The corresponding geometric variety has dimension 18 and decomposes into 2 irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras. In particular, there are no rigid 4-dimensional complex nilpotent [Formula: see text]-algebras.

scholarly journals Moduli of toric vector bundles

2008 ◽  
Vol 144 (5) ◽  
pp. 1199-1213 ◽  
Author(s):  
Sam Payne
Keyword(s):  
Linear Group ◽  
Group Action ◽  
Chern Class ◽  
Vector Bundles ◽  
Closed Subscheme ◽  
Flag Varieties ◽  
Moduli Stack ◽  
Rank Conditions ◽  
Partial Flag Varieties

AbstractWe give a presentation of the moduli stack of toric vector bundles with fixed equivariant total Chern class as a quotient of a fine moduli scheme of framed bundles by a linear group action. This fine moduli scheme is described explicitly as a locally closed subscheme of a product of partial flag varieties cut out by combinatorially specified rank conditions. We use this description to show that the moduli of rank three toric vector bundles satisfy Murphy’s law, in the sense of Vakil. The preliminary sections of the paper give a self-contained introduction to Klyachko’s classification of toric vector bundles.

Decomposition and Classification of Generic Quantum Markov Semigroups: The Gaussian Gauge Invariant Case

2012 ◽  
Vol 19 (02) ◽  
pp. 1250010 ◽  
Author(s):  
Franco Fagnola ◽  
Skander Hachicha
Keyword(s):  
Irreducible Component ◽  
Discrete System ◽  
Jump Process ◽  
Invariant State ◽  
Markov Semigroups ◽  
Markov Jump ◽  
Gauge Invariant ◽  
Irreducible Components ◽  
Invariant States

We study the structure of generic quantum Markov semigroups, arising from the stochastic limit of a discrete system with generic Hamiltonian interacting with a Gaussian gauge invariant reservoir. We show that they can be essentially written as the sum of their irreducible components determined by closed classes of states of the associated classical Markov jump process. Each irreducible component turns out to be recurrent, transient or have an invariant state if and only if its classical (diagonal) restriction is recurrent, transient or has an invariant state, respectively. We classify invariant states and study convergence towards invariant states as time goes to infinity.

scholarly journals The Algebraic and Geometric Classification of Four Dimensional Superalgebras

2021 ◽  
Author(s):  
◽  
Aaron Armour
Keyword(s):  
Classification Problem ◽  
Associative Algebras ◽  
New Variety ◽  
Graded Algebras ◽  
Finite Representation ◽  
Algebraic Classification ◽  
Irreducible Components ◽  
Module Algebras ◽  
Sweedler’S Hopf Algebra

<p><b>The algebraic and geometric classification of k-algbras, of dimension fouror less, was started by Gabriel in “Finite representation type is open” [12].</b></p> <p>Several years later Mazzola continued in this direction with his paper “Thealgebraic and geometric classification of associative algebras of dimensionfive” [21]. The problem we attempt in this thesis, is to extend the resultsof Gabriel to the setting of super (or Z2-graded) algebras — our main effortsbeing devoted to the case of superalgebras of dimension four. Wegive an algebraic classification for superalgebras of dimension four withnon-trivial Z2-grading. By combining these results with Gabriel’s we obtaina complete algebraic classification of four dimensional superalgebras.</p> <p>This completes the classification of four dimensional Yetter-Drinfeld modulealgebras over Sweedler’s Hopf algebra H4 given by Chen and Zhangin “Four dimensional Yetter-Drinfeld module algebras over H4” [9]. Thegeometric classification problem leads us to define a new variety, Salgn —the variety of n-dimensional superalgebras—and study some of its properties.</p> <p>The geometry of Salgn is influenced by the geometry of the varietyAlgn yet it is also more complicated, an important difference being thatSalgn is disconnected. While we make significant progress on the geometricclassification of four dimensional superalgebras, it is not complete. Wediscover twenty irreducible components of Salg4 — however there couldbe up to two further irreducible components.</p>

Classification of superstable theories by rank functions

1985 ◽  
Vol 24 (1) ◽  
pp. 27-40
Author(s):  
T. G. Mustafin
Keyword(s):  
Superstable Theories ◽  
Rank Functions

Varieties of nilpotent Lie superalgebras of dimension ≤ 5

2020 ◽  
Vol 32 (3) ◽  
pp. 641-661 ◽  
Author(s):  
María Alejandra Alvarez ◽  
Isabel Hernández
Keyword(s):  
Arbitrary Dimension ◽  
Lie Superalgebras ◽  
Algebraic Classification ◽  
Irreducible Components

AbstractIn this paper, we study the varieties of nilpotent Lie superalgebras of dimension {\leq 5}. We provide the algebraic classification of these superalgebras and obtain the irreducible components in every variety. As a byproduct, we construct rigid nilpotent Lie superalgebras of arbitrary dimension.

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