Tensor Products of Fundamental Representations

1988 ◽  
Vol 40 (3) ◽  
pp. 633-648 ◽  
Author(s):  
George Kempf ◽  
Linda Ness
Keyword(s):  
Irreducible Representation ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Borel Subgroup ◽  
Reductive Group ◽  
Simple Type ◽  
Dominant Weight ◽  
Highest Weight ◽  
Image Position ◽  
Dominant Representation

Let G be a reductive group over a field of characteristic zero. Fix a Borel subgroup B of G which contains a maximal torus T. For each dominant weight X we have an irreducible representation V(X) of G with highest weight X. For two dominant representation X1 and X2 we have a decompositionThis decomposition is determined by the elementof the group ring of the group of characters of T.The objective of this paper is to compute r(X1, X2) for all pairs X1 and X2 of fundamental weights. This will be used to compute the equations for cones over homogeneous spaces. This problem immediately reduces to the case when G has simple type; An, Bn, Cn, Dn, E6, E7, E8, F4 and G2. We will give complete details for the classical types. For the case An we will work with GLn.

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

MULTIPLICITY FORMULAS FOR A CLASS OF REPRESENTATIONS OF AFFINE KAC–MOODY ALGEBRAS

1994 ◽  
Vol 06 (01) ◽  
pp. 97-114
Author(s):  
RICHARD H. CAPPS ◽  
MICHAEL A. LYONS
Keyword(s):  
Irreducible Representation ◽  
Finite Number ◽  
Null Vector ◽  
Affine Algebra ◽  
Integral Index ◽  
Dominant Weight ◽  
Highest Weight ◽  
Dominant Weights ◽  
Simple Recursion ◽  
Recursion Formulas

The dominant weight of any highest weight irreducible representation (irrep) of an indecomposable affine algebra may be written in the form Λi – qδ, where the integral index i runs from 1 to some finite number (called the width of the representation), q is any non-negative integer, and δ is a vector called the null vector. All the width-two irreps of all the affine algebras are enumerated. Techniques used in an earlier paper on the width-one irreps are generalized and used to compute simple recursion formulas for the multiplicities of all these dominant weights. These formulas are suitable for rapid calculations. Numerical tables of the multiplicities of the highest 33 dominant weights are given for all the width-two irreps of twisted algebras.

ROOT SEMIGROUPS IN REDUCTIVE MONOIDS

2011 ◽  
Vol 21 (03) ◽  
pp. 433-448 ◽  
Author(s):  
MOHAN S. PUTCHA
Keyword(s):  
Maximal Torus ◽  
Borel Subgroup ◽  
Positive Root ◽  
Algebraic Groups ◽  
Reductive Group ◽  
Jordan Decomposition ◽  
One Dimensional ◽  
Reductive Monoids ◽  
The One ◽  
Renner Monoid

It is well known that in a reductive group, the Borel subgroup is a product of the maximal torus and the one-dimensional positive root subgroups. The purpose of this paper is to find an analog of this result for reductive monoids. Via a study of reductive monoids of semisimple rank 1, we introduce the concept of root semigroups. By analyzing the associated root elements in the Renner monoid, we show that the closure of the Borel subgroup is generated by the maximal torus and positive root semigroups. Along the way we generalize the Jordan decomposition of algebraic groups to reductive monoids.

Harish-Chandra Modules over ℤ

2019 ◽  
pp. 126-167
Author(s):  
Günter Harder
Keyword(s):  
Fixed Points ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Reductive Group ◽  
Group Scheme ◽  
Finitely Generated ◽  
Cartan Involution ◽  
Split Maximal Torus ◽  
Definition Of

This chapter shows that certain classes of Harish-Chandra modules have in a natural way a structure over ℤ. The Lie group is replaced by a split reductive group scheme G/ℤ, its Lie algebra is denoted by 𝖌ℤ. On the group scheme G/ℤ there is a Cartan involution 𝚯 that acts by t ↦ t −1 on the split maximal torus. The fixed points of G/ℤ under 𝚯 is a flat group scheme 𝒦/ℤ. A Harish-Chandra module over ℤ is a ℤ-module 𝒱 that comes with an action of the Lie algebra 𝖌ℤ, an action of the group scheme 𝒦, and some compatibility conditions is required between these two actions. Finally, 𝒦-finiteness is also required, which is that 𝒱 is a union of finitely generated ℤ modules 𝒱I that are 𝒦-invariant. The definitions imitate the definition of a Harish-Chandra modules over ℝ or over ℂ.

scholarly journals A Remark on the Orthogonality Relations in the Representation Theory of Finite Groups

1959 ◽  
Vol 11 ◽  
pp. 59-60 ◽  
Author(s):  
Hirosi Nagao
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Representation Theory ◽  
Finite Groups ◽  
Characteristic Zero ◽  
Orthogonality Relations ◽  
Schur’S Lemma ◽  
Image Position ◽  
A Finite Group ◽  
Schur's Lemma

Let G be a finite group of order g, andbe an absolutely irreducible representation of degree fμ over a field of characteristic zero. As is well known, by using Schur's lemma (1), we can prove the following orthogonality relations for the coefficients :1It is easy to conclude from (1) the following orthogonality relations for characters:whereand is 1 or 0 according as t and s are conjugate in G or not, and n(t) is the order of the normalize of t.

Introduction to the Schwartz Space of T\G

1989 ◽  
Vol 41 (2) ◽  
pp. 285-320 ◽  
Author(s):  
W. Casselman
Keyword(s):  
Reductive Group ◽  
Rational Points ◽  
Schwartz Space ◽  
Parabolic Subgroups ◽  
Arithmetic Groups ◽  
Similar Question ◽  
Arithmetic Subgroup ◽  
Cohomology Of Arithmetic Groups ◽  
Image Position

Let G be the group of R-rational points on a reductive group defined over Q and T an arithmetic subgroup. The aim of this paper is to describe in some detail the Schwartz space (whose definition I recall in Section 1) and in particular to explain a decomposition of this space into constituents parametrized by the T-associate classes of rational parabolic subgroups of G. This is analogous to the more elementary of the two well known decompositions of L2 (T\G) in [20](or [17]), and a proof of something equivalent was first sketched by Langlands himself in correspondence with A. Borel in 1972. (Borel has given an account of this in [8].)Langlands’ letter was in response to a question posed by Borel concerning a decomposition of the cohomology of arithmetic groups, and the decomposition I obtain here was motivated by a similar question, which is dealt with at the end of the paper.

scholarly journals On Formality of Some Homogeneous Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1011
Author(s):  
Aleksy Tralle
Keyword(s):  
Homogeneous Space ◽  
Root System ◽  
Lie Algebra ◽  
Lie Group ◽  
Maximal Torus ◽  
Homogeneous Spaces ◽  
Direct Proof ◽  
Classical Type ◽  
Sasakian Manifolds ◽  
Full Analysis

Let G / H be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus T H of H is conjugate to a torus T β whose Lie algebra t β is the kernel of the maximal root β of the root system of the complexified Lie algebra g c . We prove that such homogeneous space is formal. As an application, we give a short direct proof of the formality property of compact homogeneous 3-Sasakian spaces of classical type. This is a complement to the work of Fernández, Muñoz, and Sanchez which contains a full analysis of the formality property of S O ( 3 ) -bundles over the Wolf spaces and the proof of the formality property of homogeneous 3-Sasakian manifolds as a corollary.

Symmetrized Kronecker Products of Group Representations

1974 ◽  
Vol 26 (02) ◽  
pp. 328-339 ◽  
Author(s):  
P. H. Butler ◽  
R. C. King
Keyword(s):  
Irreducible Representation ◽  
Simple Groups ◽  
Kronecker Products ◽  
Group Representations ◽  
Symmetry Properties ◽  
Image Position

Certain phases are associated with the Kronecker squares and cubes of representations of the finite and of the compact semi-simple groups. These phases are important in giving the symmetry properties of the 1 — jm and 3 — jm symbols of the groups [4; 9]. It is our primary purpose to evaluate these phases. The Frobenius-Schur invariant [12, p. 142] for an irreducible representation of group G (1.1)

scholarly journals MOTIVIC EULER CHARACTERISTICS AND WITT-VALUED CHARACTERISTIC CLASSES

2019 ◽  
Vol 236 ◽  
pp. 251-310 ◽  
Author(s):  
MARC LEVINE
Keyword(s):  
Euler Characteristic ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Vector Bundles ◽  
Reductive Group ◽  
Characteristic Classes ◽  
Euler Characteristics ◽  
Symmetric Powers ◽  
Maximal Tori ◽  
General Calculus

This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker–Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from $\operatorname{GL}_{n}$ or $\operatorname{SL}_{n}$ to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy’s splitting principle reduces questions about characteristic classes of vector bundles in $\operatorname{SL}$-oriented, $\unicode[STIX]{x1D702}$-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for $\operatorname{SL}_{2}$ to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.

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