scholarly journals Super-representations of quivers and related polynomial semi-invariants

2020 ◽  
Vol 30 (04) ◽  
pp. 883-902
Author(s):  
V. A. Bovdi ◽  
A. N. Zubkov
Keyword(s):  
Characteristic Zero ◽  
Polynomial Invariants ◽  
Representations Of Quivers ◽  
Related Polynomial

We introduce the notion of a super-representation of a quiver. For super-representations of quivers over a field of characteristic zero, we describe the corresponding (super)algebras of polynomial semi-invariants and polynomial invariants.

scholarly journals Invariants of Finite Reflection Groups

1963 ◽  
Vol 22 ◽  
pp. 57-64 ◽  
Author(s):  
Louis Solomon
Keyword(s):  
Vector Space ◽  
Characteristic Zero ◽  
Polynomial Functions ◽  
Dimensional Vector ◽  
Reflection Groups ◽  
Linear Transformations ◽  
Graded Ring ◽  
Polynomial Invariants ◽  
Dimensional Vector Space ◽  
Group Of Automorphisms

Let K be a field of characteristic zero. Let V be an n-dimensional vector space over K and let S be the graded ring of polynomial functions on V. If G is a group of linear transformations of V, then G acts naturally as a group of automorphisms of S if we defineThe elements of S invariant under all γ ∈ G constitute a homogeneous subring I(S) of S called the ring of polynomial invariants of G.

Polynomial invariants for virtual marked graphs and virtual surface-link theory I

2017 ◽  
Vol 26 (12) ◽  
pp. 1750081
Author(s):  
Sang Youl Lee
Keyword(s):  
Natural Extension ◽  
Polynomial Invariants ◽  
Kauffman Bracket ◽  
Virtual Links ◽  
Link Theory ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Ideal

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

scholarly journals On the Voevodsky motive of the moduli space of Higgs bundles on a curve

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Victoria Hoskins ◽  
Simon Pepin Lehalleur
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Characteristic Zero ◽  
Rational Point ◽  
Triangulated Category ◽  
Higgs Bundles ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
De Rham

AbstractWe study the motive of the moduli space of semistable Higgs bundles of coprime rank and degree on a smooth projective curve C over a field k under the assumption that C has a rational point. We show this motive is contained in the thick tensor subcategory of Voevodsky’s triangulated category of motives with rational coefficients generated by the motive of C. Moreover, over a field of characteristic zero, we prove a motivic non-abelian Hodge correspondence: the integral motives of the Higgs and de Rham moduli spaces are isomorphic.

scholarly journals Conjectures on Stably Newton Degenerate Singularities

2021 ◽  
Author(s):  
Jan Stevens
Keyword(s):  
Characteristic Zero ◽  
Arbitrary Number ◽  
Milnor Number ◽  
Plane Curves ◽  
Newton Diagram ◽  
Finite Characteristic ◽  
Vanishing Cycles

AbstractWe discuss a problem of Arnold, whether every function is stably equivalent to one which is non-degenerate for its Newton diagram. We argue that the answer is negative. We describe a method to make functions non-degenerate after stabilisation and give examples of singularities where this method does not work. We conjecture that they are in fact stably degenerate, that is not stably equivalent to non-degenerate functions.We review the various non-degeneracy concepts in the literature. For finite characteristic, we conjecture that there are no wild vanishing cycles for non-degenerate singularities. This implies that the simplest example of singularities with finite Milnor number, $$x^p+x^q$$ x p + x q in characteristic p, is not stably equivalent to a non-degenerate function. We argue that irreducible plane curves with an arbitrary number of Puiseux pairs (in characteristic zero) are stably non-degenerate. As the stabilisation involves many variables, it becomes very difficult to determine the Newton diagram in general, but the form of the equations indicates that the defining functions are non-degenerate.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

A Module-theoretic Characterization of Algebraic Hypersurfaces

2018 ◽  
Vol 61 (1) ◽  
pp. 166-173
Author(s):  
Cleto B. Miranda-Neto
Keyword(s):  
Algebraic Variety ◽  
Characteristic Zero ◽  
Vector Fields ◽  
Exterior Power ◽  
Algebraic Hypersurfaces ◽  
Theoretic Characterization

AbstractIn this note we prove the following surprising characterization: if X ⊂ is an (embedded, non-empty, proper) algebraic variety deûned over a field k of characteristic zero, then X is a hypersurface if and only if the module of logarithmic vector fields of X is a reflexive -module. As a consequence of this result, we derive that if is a free -module, which is shown to be equivalent to the freeness of the t-th exterior power of for some (in fact, any) t ≤ n, then necessarily X is a Saito free divisor.

scholarly journals Projective representations of quivers

2002 ◽  
Vol 31 (2) ◽  
pp. 97-101 ◽  
Author(s):  
Sangwon Park
Keyword(s):  
Projective Representation ◽  
Representations Of Quivers ◽  
Direct Sum ◽  
Projective Representations

We prove thatP1 →f P2is a projective representation of a quiverQ=•→•if and only ifP1andP2are projective leftR-modules,fis an injection, andf (P 1)⊂P 2is a summand. Then, we generalize the result so that a representationM1 →f1  M2  →f2⋯→fn−2  Mn−1→fn−1  Mnof a quiverQ=•→•→•⋯•→•→•is projective representation if and only if eachMiis a projective leftR-module and the representation is a direct sum of projective representations.

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