2 edition of **Minimal parabolic subgroups in the symmetric groups** found in the catalog.

Minimal parabolic subgroups in the symmetric groups

Sandra Covello

- 355 Want to read
- 15 Currently reading

Published
**2000**
by University of Birmingham in Birmingham
.

Written in English

**Edition Notes**

Thesis (Ph.D) - University of Birmingham, School of Mathematics and Statistics, Faculty of Science.

Statement | by Sandra Covello. |

The Physical Object | |
---|---|

Pagination | vi,206p. : |

Number of Pages | 206 |

ID Numbers | |

Open Library | OL21292226M |

The Structure of Parabolic Subgroups Kenneth D. Johnson Communicated by E. B. Vinberg Abstract. Suppose G is a real connected simple noncompact Lie group with (using standard notation) Iwasawa decomposition G = KAN. If M = Z(A)∩K, the group B = MAN is . A vector subspace S ⊂V is isotropic if for any v,w ∈S, the symmetric bilinear form satisfies: B v,w 0 Definition 3. A maximal parabolic subgroup in an orthogonal group O V is the stabilizer of an isotropic subspace S ⊂V in O V. We now propose the following variation of Witt’s Extension Theorem (proved in the text for quadratic forms on.

This naturally leads to the study of a larger class of subgroups, called parabolic subgroups. These are closed subgroups which contain a Borel subgroup of G. When the group Gis connected and reductive, which means it has no non-trivial proper closed connected unipotent normal subgroups, the structure of its parabolic sub-groups is well understood. A representation induced from a parabolic k-subgroup of G generically contributes to the Plancherel decomposition of L 2 (G k /H k) if and only if the parabolic k-subgroup is #-split. So for a study of these induced representations a detailed description of the H k -conjucagy classes of these #-split parabolic k-subgroups is needed.

In On the Sylow subgroups of the symmetric and alternating groups, Amer. J. Math. 47 (), –, Louis Weisner gives a counting argument that, specialized to, claims to show that has exactly Sylow -subgroups, where, as above. The exponent of is . In this second part, we occasionally make use of some structure results for reductive groups (the open Bruhat cell, minimal parabolic subgroups), for which we refer to Springer’s book [13]. But apart from that, the prerequisites are still minimal. The books by Grosshans (see [3]) and.

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[Bo] N. Bourbaki, "Groupes et algèbres de Lie", Hermann () pp. Chapts. VII-VIII MR MR MR MR MR Zbl [Bo2] A. Borel, "Linear algebraic groups", Benjamin () MR Zbl Zbl [BoTi]. Maximal parabolic subgroups of symplectic groups These notes are intended as an outline for a long presentation to be made early The goal is to work out the structure of what are called maximal parabolic sub-groups of Sp(V), and to look at the corresponding geometry.

with group law given by addition of symmetric Size: 66KB. The geometry of the orbits of a minimal parabolic k -subgroup acting on a symmetric k -variety is essential in several areas, but its main importance is in the study of the representations.

the notion of a parabolic subgroup. Deﬁnition A subgroup P of a group G is called a parabolic sub-group if it properly contains a Borel subgroup B of G.

With these few abstract notions it is hard to tell what step to take next. Nevertheless, we may now set G = GL n(C) and describe the general form of that group’s parabolic subgroups. 1File Size: 91KB. Solvability, Structure, and Analysis for Minimal Parabolic Subgroups Joseph A.

Wolf1 Received: 6 November / Published online: 24 April are the adjoint groups of G and GC. We write P = M A N ⊂ G, P = M A N ⊂ G, and P = MAN ⊂ G for the corresponding minimal parabolic subgroups, aligned so that G File Size: 1MB. We study the structure of minimal parabolic subgroups of the classical infinite-dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras.

This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite-dimensional by: 3. We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras.

Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share.

Applications. The symmetric group on a set of size n is the Galois group of the general polynomial of degree n and plays an important role in Galois invariant theory, the symmetric group acts on the variables of a multi-variate function, and the functions left invariant are the so-called symmetric the representation theory of Lie groups, the representation theory of the.

n is the disjoint union of the parabolic double cosets W I\S n/W J = {W IwW J |w ∈S n}. I Every parabolic double coset has a unique minimal and a unique maximal length element. Thm.(Kobayashi ) Every parabolic double coset is an interval in Bruhat order.

The follow polynomials are palindromic P I,w,J(q) = X v∈W IwW J q‘(v). This is a book about arithmetic subgroups of semisimple Lie groups, which means that we will discuss the group SL(n;Z), and certain of its subgroups. By de nition, the subject matter combines algebra (groups of matrices) with number theory (properties of the integers).

However, it also has important applications in geometry. This article gives specific information, namely, subgroup structure, about a family of groups, namely: symmetric group.

View subgroup structure of group families | View other specific information about symmetric group. The symmetric group on a set is the group, under multiplication, of permutations of that set.

The symmetric group of degree is the symmetric group on a set of size. Thanks for contributing an answer to Mathematics Stack Exchange.

Please be sure to answer the question. Provide details and share your research. But avoid Asking for help, clarification, or responding to other answers. Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. This chapter discusses closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups. It presents an assumption where G is a connected Lie group, σ an involutive automorphism of G, and η a subgroup of G such that where G σ = {x ∈G│ σ x = x} and is the connected component of G σ containing the.

Parabolic subgroups P are also characterized, among algebraic subgroups, by the condition that G/P is a complete variety. Working over algebraically closed fields, the Borel subgroups turn out to be the minimal parabolic subgroups in this sense. Thus B is a Borel subgroup when the homogeneous space G/B is a complete variety which is "as large.

The principal aim of this paper is to show that every maximal parabolic subgroup P of a classical reductive algebraic group G operates with a finite number of orbits on its unipotent radical.

This is a consequence of the fact that each parabolic subgroup of a group of type A n whose unipotent radical is of nilpotent class at most two has this finiteness by: 4. In this handout we will determine the parabolic k-subgroups of symplectic groups, as well as of special orthogonal groups, in the latter case allowing arbitrary (nite-dimensional) non-degenerate quadratic spaces.

For a speci c minimal P 0, we will also describe the resulting standard parabolic Size: KB. This chapter discusses closure relations for orbits on affine symmetric spaces under the action of minimal parabolic subgroups.

It presents an assumption where G is a connected Lie group, σ an involutive automorphism of G, and η a subgroup of G such that where G σ = {x ∈G│ σ x = x} and is the connected component of G σ containing the identity. The factor space H/G is called an affine Cited by: The following property can be evaluated for a closed subgroup of an algebraic group.

Definition Symbol-free definition. A closed subgroup of an algebraic group is termed a parabolic subgroup if it satisfies the following equivalent conditions. It contains a Borel subgroup of the whole group; The quotient space, with the induced variety structure, is a complete variety.

subgroup of Sylow 2-subgroups of symmetric and alternating groups. Also minimal generic sets of Sylow 2-subgroups of A2k were founded. Elements presentation of (Syl2A2 k)′, (Syl2S2)′ was investigated.

We prove that the commutator width [1] of an arbitrary element of. nin terms of transitive groups, as GAP has a library listing all the transitive groups up to S We knew that it was unlikely for us to gure out the subgroups for symmetric groups on more than 30 points, so this library was more than enough for our purposes.

We can also think of intransitive groups as Date: VIGRE Summer Program 1.This paper determines the structure of certain classes of maximal subgroups of symmetric groups.

Maximal shall always mean proper maximal. It happens that classification of maximal subgroups is convenient in terms of concepts of transitivity.

A subset H of SÍM) is transitive if for each x and yeM, there is se H such that s(x) = y.The Atlas of Finite Groups, published inhas proved itself to be an indispensable tool to all researchers in group theory and many related areas.

The present book is the proceedings of a conference organized to mark the tenth anniversary of the publication of the Atlas, and contains twenty articles by leading experts in the field, covering many aspects of group theory and its applications.