Problem Set 7
... the Weyl group W . Give an explicit construction of the irreducible representations of G, compute their characters, and use the Weyl integration formula to
show that they are orthonormal.
Problem 3: For G = SU (3), explicitly define an infinite sequence of irreducible
representations on spaces of ho ...
Some Notes on Compact Lie Groups
... SO(3) subgroup of G. This defines a map ϕα : S 3 → G. The main point is that if α is a
long root, ϕα induces an isomorphism at the level of the third homotopy groups.
For classical groups, this can be shown “directly” as follows. The main tool is the
homotopy exact sequence (for a Lie group G and it ...
... • Associative: a(bc) = (ab)c
• Identity: 1 G, 1a = a1 = a,
• Inverse: a-1 G, a-1a = aa-1
= 1, a G
... ALGEBRAIC D-MODULES, part II
The theory of algebraic D-modules, also known as modules over rings of differential operators (whose creation
began in the 1970’s in the works of J. Bernstein and M. Kashiwara) is essentially a branch of algebraic geometry
but it has deep connections with analysis and ap ...
... d) Determine the degrees of the irreducible representations of G, assuming the existence of an
irreducible representation of degree 6, which we will construct later.
(This representation comes from the doubly-transitive action of G on the nonzero vectors of F32 .)
18) Let G be a finite group and K a ...
... You might draw a picture of it on a chalkboard that would look like this.
These are physical representations of the rectangle. When you think
about it you may visualize a very similar picture.
You may represent this rectangle in the real plane by giving coordinates of its corners, for example (0, 0) ...
INTRODUCTION TO C* ALGEBRAS - I Introduction : In this talk, we
... Suppose T ∈ B(H), we define the adjoint T ∗ by the formula
< T (x), y >=< x, T ∗ y >
In a course in functional analysis, one learns that T ∗ behaves like a complex-conjugate
of T in that there are many interesting properties of T that one can obtain from T ∗ . For
(1) (T ∗ )∗ = T
(2) λ is ...
Math 461/561 Week 2 Solutions 1.7 Let L be a Lie algebra. The
... It is clear from [(x, y), (a, b)] = ([x, a], [y, b]) that every member of the spanning set of (L1 ⊕ L2 )0
lies in L01 ⊕ L02 and vice versa, so they are equal.
The generalizations to finitely many summands are obvious and follow by induction.
(iii). Absolutely not! For example consider a 2-dimensiona ...
850 Oberwolfach Report 15 Equivariant Sheaves on Flag Varieties
... projective normal toric variety ([Lun95]), and for a complex semisimple adjoint
group acting on a smooth complete symmetric variety (in the sense of de Concini
and Procesi) ([Gui05]). We recently became aware of a related result for the loop
rotation equivariant derived Satake category of the aﬃne l ...
... The proof of this assertion is straightforward. Each of the brackets in the lefthand side expands to 4 terms, and then everything cancels.
In categorical terms, what we have here is a functor from the category of
associative algebras to the category of Lie algebras over a fixed field. The action
UNT UTA Algebra Symposium University of North Texas November
... Abstract: Infinite reflection groups arise naturally as simple generalizations of finite reflection
groups. This natural connection causes many of the same questions to be answered for the
infinite reflection groups that arouse from the study of their finite counterparts. Braid groups,
which were fi ...
LIE-ADMISSIBLE ALGEBRAS AND THE VIRASORO
... for x, y ∈ L, where x ◦ y = 2 (x y + yx). In particular, L with product x y
is Lie-admissible. A central problem in the study of Lie-admissible algebras
is to determine all compatible multiplications defined on Lie algebras. This
problem has been resolved for finite-dimensional third power-associati ...
Lie Groups, Lie Algebras and the Exponential Map
... On g = End(V ) there is a non-associative bilinear skew-symmetric product
given by taking commutators
(X, Y ) ∈ g × g → [X, Y ] = XY − Y X ∈ g
While matrix groups and their subgroups comprise most examples of Lie
groups that one is interested in, we will be defining Lie groups in geometrical
terms f ...
OPEN PROBLEM SESSION FROM THE CONFERENCE
... XFv is a trivial torsor over all residue fields Fv of discrete valuations v, is X trivial?
Is it true if we further suppose that G is rational? Probably one needs the local
domain to be regular.
The motivation for this problem is that one knows the answer is yes in
a similar situation, namely when F ...
The infinite fern of Galois representations of type U(3) Gaëtan
... the choice of F corresponds to a choice of a triangulation of the (ϕ, Γ)-module of
V over the Robba ring, and XV,F parameterizes the deformations such that this
triangulation lifts. When the ϕ-stable complete flag of Dcris (V ) defined by F is in
general position compared to the Hodge filtration, we ...
Math 8669 Introductory Grad Combinatorics Spring 2010, Vic Reiner
... Math 8669 Introductory Grad Combinatorics
Spring 2010, Vic Reiner
Homework 3- Friday May 7
Hand in at least 6 of the 10 problems.
1. Construct all the irreducible representations/characters for the symmetric group S4 according to the following plan (and using a labelling
convention, to be explained ...
Group representation theory
... This course is a mix of group theory and linear algebra, with probably more of the latter than
the former. You may need to revise your 2nd year vector space notes!
In mathematics the word “representation” basically means “structure-preserving function”.
Thus—in group theory and ring theory at least— ...
A Complex Analytic Study on the Theory of Fourier Series on
... Here let us review the history of the studies concerning this subject. The
concept of hyperfunctions was first introduced by Sato [Sa] in 1958. In [Sa],
among other results, he already got the characterization of Fourier series of real
analytic functions and hyperfunctions on the one dimensional sph ...
8. The Lie algebra and the exponential map for general Lie groups
... It is then easy to check that ct (x′ , 0) = 0 for t ≥ a + 1; indeed, for
r ≥ a + 1, ar (x′ , 0) = br (x′ , 0) = 0, while, for r ≤ a, (∂bt /∂xr )(x′ , 0) =
(∂at /∂xr )(x′ , 0) = 0 because bt (x′ , 0) = at (x′ , 0) = 0.
Remark. If X, Y ∈ Lie(G), the bracket [X, Y ] depends only on the values
of X and ...
A Noncommutatlve Marclnkiewlcz Theorem
... moments (6?a-point functions') are defined by <^ 0 , A\ ••• Anfa), Ai^erf, and generalized cumulants
('truncated «-point functions') are defined in analogy to probability theory. The classical
Marcinkiewicz Theorem states that if the characteristic function of a random variable f is the
... (groups which are also smooth manifold where the operation is a differentiable function between
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operation is matrix multiplication.Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. Furthermore, the vector space on which a group (for example) is represented can be infinite-dimensional, and by allowing it to be, for instance, a Hilbert space, methods of analysis can be applied to the theory of groups. Representation theory is also important in physics because, for example, it describes how the symmetry group of a physical system affects the solutions of equations describing that system.A feature of representation theory is its pervasiveness in mathematics. There are two sides to this. First, the applications of representation theory are diverse: in addition to its impact on algebra, representation theory: illuminates and generalizes Fourier analysis via harmonic analysis, is connected to geometry via invariant theory and the Erlangen program, has an impact in number theory via automorphic forms and the Langlands program.The second aspect is the diversity of approaches to representation theory. The same objects can be studied using methods from algebraic geometry, module theory, analytic number theory, differential geometry, operator theory, algebraic combinatorics and topology.The success of representation theory has led to numerous generalizations. One of the most general is in category theory. The algebraic objects to which representation theory applies can be viewed as particular kinds of categories, and the representations as functors from the object category to the category of vector spaces. This description points to two obvious generalizations: first, the algebraic objects can be replaced by more general categories; second, the target category of vector spaces can be replaced by other well-understood categories.A representation should not be confused with a presentation.