Irreducible representation | EPFL Graph Search

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho|_W,W), with W \subset V closed under the action of { \rho(a) : a\in A }. Every finite-dimensional unitary representation on a Hilbert space V is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible. History Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space

Eigenvalue multiplicities of group elements in irreducible representations of simple linear algebraic groups

Ana-Maria Retegan

Let k be an algebraically closed field of arbitrary characteristic, let G be a simple simply connected linear algebraic group and let V be a rational irreducible tensor-indecomposable finite-dimensional kG-module. For an element g of G we denote by

V_{g}(x)

the eigenspace corresponding to the eigenvalue x of g on V. We define N to be the minimum difference between the dimension of V and the dimension of

V_{g}(x)

, where g is a non-central element of G. In this thesis we identify pairs (G,V) with the property that

N\leq \sqrt{\dim(V)}

. This problem is an extension of the classification result obtained by Guralnick and Saxl for the condition

N\leq \max\bigg\{2,\frac{\sqrt{\dim(V)}}{2}\bigg\}

. Moreover, for all the pairs (G,V) we had to consider in our classification, we will determine the value of N.

EPFL2022

Maximal subgroups acting with two composition factors on irreducible representations of exceptional algebraic groups

Nathan Scheinmann

Let

Y

be a simply connected simple algebraic group over an algebraically closed field

k

of characteristic

p

and let

X

be a maximal closed connected simple subgroup of

Y

. Excluding some small primes in specific cases, we classify the

p

-restricted irreducible representations of

Y

on which

X

acts with exactly two composition factors. This work follows on naturally from the classification of irreducible subgroups of exceptional algebraic groups given by Testerman.

EPFL2019

On canonical representation of convex polyhedra

Stefano Picozzi

Convex polyhedra are important objects in various areas of mathematics and other disciplines. A fundamental result, known as Minkowski-Weyl theorem, states that every polyhedron admits two types of representations, either as the solution set to a finite system of linear inequalities or the Minkowski sum of a finite set of points and half-rays. These are usually referred to as its H-representations and its V-representations, respectively. Neither H-representations nor V-representations are unique. Hence, deciding whether two H-representations, or two V-representations describe the same polyhedron is a nontrivial problem. We identify particular representations which are easy to determine, are compact, and reflect the geometrical properties of the underlying polyhedron. Key ingredients to this discussion are affine transformations of polyhedra, duality and complexity of polyhedral decision problems. In this dissertation, we discuss in detail the problem of refining the definitions of Hand V- representations such as to guarantee a one to one correspondence between polyhedra and their representations. As a convenient formalism, we introduce the notion of the canonical representation rule, which is any function assigning to each polyhedron P a particular H-representation, and a particular V-representation. The orthogonal representation rule is a well-known example of such a function. We define the lexico-smallest representation rule, an improved canonical representation rule that produce representations which are nonredundant, make the underlying geometry transparent and, furthermore, are sparse. These rules also exhibit nice polarity properties that can be exploited for computation. The key computational tool is linear programing. Furthermore, we show that the lexico-smallest H-representation of a perfect bipartite matching polyhedron is easy to determine using the combinatorial properties of the underlying graph. Finally, we discuss the complexity of various polyhedral verification problems related to representation of convex polyhedra. The standard technique to identify the redundancies within an H- or a V-representation is to use linear programming. We discuss the complexity of the converse reduction. More precisely, we show that deciding the feasibility of a system of linear inequalities can be reduced to deciding redundancy in an H- or in a V- representation in linear time. Also, we will study the equivalence of these problems with those of identifying the implicit linearities within an H- or a V- representation, and the boundedness of a polyhedron.

EPFL2005