Irreducible polynomial | EPFL Graph Search

In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted for the possible factors, that is, the field to which the coefficients of the polynomial and its possible factors are supposed to belong. For example, the polynomial x2 − 2 is a polynomial with integer coefficients, but, as every integer is also a real number, it is also a polynomial with real coefficients. It is irreducible if it is considered as a polynomial with integer coefficients, but it factors as \left(x - \sqrt{2}\right)\left(x + \sqrt{2}\right) if it is considered as a polynomial with real coefficients. One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with coefficients in an integral domain, and there are two common defi

Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

Mikko Tapani Korhonen

Let

G

be a classical group with natural module

V

over an algebraically closed field of good characteristic. For every unipotent element

u

G

, we describe the Jordan block sizes of

u

on the irreducible

G

-modules which occur as composition factors of

V \otimes V^*

\wedge ^2(V)

, and

S^2(V)

. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of

u

, for which recursive formulae are known.

2019

Invariant forms on irreducible modules of simple algebraic groups

Mikko Tapani Korhonen

Let G be a simple linear algebraic group over an algebraically dosed field K of characteristic p >= 0 and let V be an irreducible rational G-module with highest weight A. When is self-dual, a basic question to ask is whether V has a non-degenerate G-invariant alternating bilinear form or a non degenerate G-invariant quadratic form. If p not equal 2, the answer is well known and easily described in terms of A. In the case where p = 2, we know that if is self-dual, it always has a non-degenerate G-invariant alternating bilinear form. However, determining when V has a non-degenerate G-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where G is of classical type and A is a fundamental highest weight omega(i), and in the case where G is of type A(i) and lambda = omega(r) + omega(s) for 1

Elsevier2017

Reductive overgroups of distinguished unipotent elements in simple algebraic groups

Mikko Tapani Korhonen

Let

G

be a simple linear algebraic group over an algebraically closed field

K

of characteristic

p \geq 0

. In this thesis, we investigate closed connected reductive subgroups

X < G

that contain a given distinguished unipotent element

u

G

. Our main result is the classification of all such

X

that are maximal among the closed connected subgroups of

G

. When

G

is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where

G

is simple of classical type, say

G = \operatorname{SL}(V)

G = \operatorname{Sp}(V)

, or

G = \operatorname{SO}(V)

. We begin by considering the maximal closed connected subgroups

X

G

which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where

X

is the stabilizer of a tensor decomposition of

V

. For

p = 2

and

X = \operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)

, we solve the problem with explicit calculations; for the other tensor product subgroups we apply a result of Barry (Comm. Algebra, 2015). After the geometric subgroups, the maximal closed connected subgroups that remain are the

X < G

such that

X

is simple and

V

is an irreducible and tensor indecomposable

X

-module. The bulk of this thesis is concerned with this case. We determine all triples

(X, u, \varphi)

where

X

is a simple algebraic group,

u \in X

is a unipotent element, and

\varphi: X \rightarrow G

is a rational irreducible representation such that

\varphi(u)

is a distinguished unipotent element of

G

. When

p = 0

, this was done in previous work by Liebeck, Seitz and Testerman (Pac. J. Math, 2015). In the final chapter of the thesis, we consider the more general problem of finding all connected reductive subgroups

X

G

that contain a distinguished unipotent element

u

G

. This leads us to consider connected reductive overgroups

X

u

which are contained in some proper parabolic subgroup of

G

. Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when

u

is a regular unipotent element of

G

, no such

X

exists. We give several examples which show that their result does not generalize to distinguished unipotent elements. As an extension of the Testerman-Zalesski result, we show that except for two known examples which occur in the case where

(G, p) = (C_2, 2)

, a connected reductive overgroup of a distinguished unipotent element of order

p

cannot be contained in a proper parabolic subgroup of

G

EPFL2018

Jordan blocks of unipotent elements in some irreducible representations of classical groups in good characteristic

Mikko Tapani Korhonen

Let

G

be a classical group with natural module

V

over an algebraically closed field of good characteristic. For every unipotent element

u

G

, we describe the Jordan block sizes of

u

on the irreducible

G

-modules which occur as composition factors of

V \otimes V^*

\wedge ^2(V)

, and

S^2(V)

. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of

u

, for which recursive formulae are known.

2019

Reductive overgroups of distinguished unipotent elements in simple algebraic groups

Mikko Tapani Korhonen

Let

G

be a simple linear algebraic group over an algebraically closed field

K

of characteristic

p \geq 0

. In this thesis, we investigate closed connected reductive subgroups

X < G

that contain a given distinguished unipotent element

u

G

. Our main result is the classification of all such

X

that are maximal among the closed connected subgroups of

G

. When

G

is simple of exceptional type, the result is easily read from the tables computed by Lawther (J. Algebra, 2009). Our focus is then on the case where

G

is simple of classical type, say

G = \operatorname{SL}(V)

G = \operatorname{Sp}(V)

, or

G = \operatorname{SO}(V)

. We begin by considering the maximal closed connected subgroups

X

G

which belong to one of the families of the so-called \emph{geometric subgroups}. Here the only difficult case is the one where

X

is the stabilizer of a tensor decomposition of

V

. For

p = 2

and

X = \operatorname{Sp}(V_1) \otimes \operatorname{Sp}(V_2)

X < G

such that

X

is simple and

V

is an irreducible and tensor indecomposable

X

-module. The bulk of this thesis is concerned with this case. We determine all triples

(X, u, \varphi)

where

X

is a simple algebraic group,

u \in X

is a unipotent element, and

\varphi: X \rightarrow G

is a rational irreducible representation such that

\varphi(u)

is a distinguished unipotent element of

G

. When

p = 0

X

G

that contain a distinguished unipotent element

u

G

. This leads us to consider connected reductive overgroups

X

u

which are contained in some proper parabolic subgroup of

G

. Testerman and Zalesski (Proc. Am. Math. Soc, 2013) have shown that when

u

is a regular unipotent element of

G

, no such

X

(G, p) = (C_2, 2)

, a connected reductive overgroup of a distinguished unipotent element of order

p

cannot be contained in a proper parabolic subgroup of

G

EPFL2018