Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAn inductive approach to representations of general linear groups over compact discrete valuation rings
88
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In his seminal Lecture Notes in Mathematics published in 1981, Andrey Zelevinsky introduced a new family of Hopf algebras which he called {em PSH-algebras}. These algebras were designed to capture the representation theory of the symmetric groups and of classical groups over finite fields. The gist of this construction is to translate representation-theoretic operations such as induction and restriction and their parabolic variants to algebra and coalgebra operations such as multiplication and comultiplication. The Mackey formula, for example, is then reincarnated as the Hopf axiom on the algebra side. In this paper we take substantial steps to adapt these ideas for general linear groups over compact discrete valuation rings. We construct an analogous bi-algebra that contains a large PSH-algebra that extends Zelevinskys algebra for the case of general linear groups over finite fields. We prove several base change results relating algebras over extensions of discrete valuation rings.
rate research
Read More
We construct, for any finite commutative ring $R$, a family of representations of the general linear group $mathrm{GL}_n(R)$ whose intertwining properties mirror those of the principal series for $mathrm{GL}_n$ over a finite field.
We classify all triples $(G,V,H)$ such that $SL_n(q)leq Gleq GL_n(q)$, $V$ is a representation of $G$ of dimension greater than one over an algebraically closed field $FF$ of characteristic coprime to $q$, and $H$ is a proper subgroup of $G$ such that the restriction $Vdar_{H}$ is irreducible. This problem is a natural part of the Aschbacher-Scott program on maximal subgroups of finite classical groups.
The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special linear representation for next using.
We study multiplicities of unipotent characters in tensor products of unipotent characters of GL(n,q). We prove that these multiplicities are polynomials in q with non-negative integer coefficients. We study the degree of these polynomials and give a necessary and sufficient condition in terms of the representation theory of symmetric groups for these polynomials to be non-zero.
In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{Gamma}$ where $Gamma$ is a product of general linear groups over a field $K$ of characteristic zero, and $U$ is a finite dimensional rational representation of $Gamma$. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where $U=text{End}(W^{oplus k})$ and $Gamma = text{GL}(W)$ and on the case where $U=text{End}(Votimes W)$ and $Gamma = text{GL}(V)times text{GL}(W)$, though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
