Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn Some Applications of Group Representation Theory to Algebraic Problems Related to the Congruence Principle for Equivariant Maps
112
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Given a finite group $G$ and two unitary $G$-representations $V$ and $W$, possible restrictions on Brouwer degrees of equivariant maps between representation spheres $S(V)$ and $S(W)$ are usually expressed in a form of congruences modulo the greatest common divisor of lengths of orbits in $S(V)$ (denoted $alpha(V)$). Effective applications of these congruences is limited by answers to the following questions: (i) under which conditions, is ${alpha}(V)>1$? and (ii) does there exist an equivariant map with the degree easy to calculate? In the present paper, we address both questions. We show that ${alpha}(V)>1$ for each irreducible non-trivial $C[G]$-module if and only if $G$ is solvable. For non-solvable groups, we use 2-transitive actions to construct complex representations with non-trivial ${alpha}$-characteristic. Regarding the second question, we suggest a class of Norton algebras without 2-nilpotents giving rise to equivariant quadratic maps, which admit an explicit formula for the Brouwer degree.
rate research
Read More
Let $mathcal{J}$ be the exceptional Jordan algebra and $V=mathcal{J}oplus mathcal{J}$. We construct an equivariant map from $V$ to $mathrm{Hom}_k(mathcal{J}otimes mathcal{J},mathcal{J})$ defined by homogeneous polynomials of degree $8$ such that if $xin V$ is a generic point, then the image of $x$ is the structure constant of the isotope of $mathcal{J}$ corresponding to $x$. We also give an alternative way to define the isotope corresponding to a generic point of $mathcal{J}$ by an equivariant map from $mathcal{J}$ to the space of trilinear forms.
These are notes of a talk to the International Conference on Algebra in honor of A. I. Maltsev, Novosibirsk, USSR, 1989 (to appear in Contemporary Mathematics). The concept of a divisor with complex coefficients on an algebraic curve is introduced. We consider families of complex divisors, or, equivalently, families of invertible sheaves and define Arakelov-type metrics on some invertible sheaves produced from them on the base. We apply this technique to obtain a formula for the measure on the moduli space that gives tachyon correlators in string theory.
We give applications of Foliation Theory to the Classical Invariant Theory of real orthogonal representations, including: The solution of the Inverse Invariant Theory problem for finite groups. An if-and-only-if criterion for when a separating set is a generating set. And the introduction of a class of generalized polarizations which, in the case of representations of finite groups, always generates the algebra of invariants of their diagonal representations.
This paper is a continuations of the project initiated in the book string topology for stacks. We construct string operations on the SO(2)-equivariant homology of the (free) loop space $L(X)$ of an oriented differentiable stack $X$ and show that $H^{SO(2)}_{*+dim(X) -2}(L(X))$ is a graded Lie algebra. In the particular case where $X$ is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack of a group acting on a stack and show that it is a topological stack. Then use its homotopy type to define equivariant (co)homology for stacks, transfer maps, and so on.
Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $pi$-point style rank variety for the Drinfeld double, identify $pi$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
